[{"Name":"Numeric data and number bases","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction to number systems","Duration":"6m 1s","ChapterTopicVideoID":28553,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/28553.jpeg","UploadDate":"2022-02-28T16:16:55.1470000","DurationForVideoObject":"PT6M1S","Description":null,"MetaTitle":"Introduction to number systems: Video + Workbook | Proprep","MetaDescription":"Machine Level Representation of Data - Numeric data and number bases. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/principles-of-programing/machine-level-representation-of-data/numeric-data-and-number-bases/vid30071","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"Welcome to this video on number systems."},{"Start":"00:02.490 ","End":"00:03.600","Text":"By the end of this section,"},{"Start":"00:03.600 ","End":"00:05.070","Text":"you\u0027ll be able to explain the need for"},{"Start":"00:05.070 ","End":"00:08.340","Text":"alternative number systems, convert between decimal,"},{"Start":"00:08.340 ","End":"00:10.635","Text":"binary, hexadecimal, and octal,"},{"Start":"00:10.635 ","End":"00:15.165","Text":"and select between appropriate number bases according to need."},{"Start":"00:15.165 ","End":"00:18.270","Text":"I\u0027ve been working with computers for nearly 40 years and"},{"Start":"00:18.270 ","End":"00:20.985","Text":"there\u0027s still something I find amazing after all this time,"},{"Start":"00:20.985 ","End":"00:23.880","Text":"which is the simple fact that any type of information can be"},{"Start":"00:23.880 ","End":"00:27.690","Text":"reduced to just 2 symbols, 1 or 0."},{"Start":"00:27.690 ","End":"00:32.640","Text":"That means any number can be represented solely by 1s and 0s."},{"Start":"00:32.640 ","End":"00:36.990","Text":"Any texts can be represented with 1s and 0s."},{"Start":"00:36.990 ","End":"00:41.345","Text":"An image can be represented with 1s and 0s."},{"Start":"00:41.345 ","End":"00:44.000","Text":"The sounds you\u0027re hearing now and the video that you\u0027re"},{"Start":"00:44.000 ","End":"00:47.135","Text":"watching is made up entirely of just 1s and 0s."},{"Start":"00:47.135 ","End":"00:49.580","Text":"Even the software that\u0027s running on your computer that"},{"Start":"00:49.580 ","End":"00:51.800","Text":"enables you to see and hear the video and interact"},{"Start":"00:51.800 ","End":"00:56.600","Text":"with it is also ultimately made up solely of 1s and 0s."},{"Start":"00:56.600 ","End":"00:59.015","Text":"Once you get your head around this revelation,"},{"Start":"00:59.015 ","End":"01:01.650","Text":"it would be a perfectly reasonable question to ask,"},{"Start":"01:01.650 ","End":"01:05.990","Text":"why would you want to represent information in terms of 1 and 0?"},{"Start":"01:05.990 ","End":"01:08.000","Text":"The answer is that computers are made up of"},{"Start":"01:08.000 ","End":"01:10.865","Text":"highly complex arrangements of electronic circuits,"},{"Start":"01:10.865 ","End":"01:13.835","Text":"specifically a type called a digital circuit."},{"Start":"01:13.835 ","End":"01:19.340","Text":"Each element within a digital circuit can be in 1 of 2 states, on or off."},{"Start":"01:19.340 ","End":"01:24.289","Text":"In this case, 1 element of the circuit might be a light-emitting diode or LED,"},{"Start":"01:24.289 ","End":"01:27.140","Text":"which can be made to turn on and in this case emit"},{"Start":"01:27.140 ","End":"01:31.475","Text":"green light or turn off and emit no light."},{"Start":"01:31.475 ","End":"01:35.060","Text":"It\u0027s convenient to represent the state of on being equivalent to"},{"Start":"01:35.060 ","End":"01:39.950","Text":"the digit 1 and the state of off being equivalent to the digit 0."},{"Start":"01:39.950 ","End":"01:44.075","Text":"When you hear the word digital being mentioned in relation to technology,"},{"Start":"01:44.075 ","End":"01:48.965","Text":"what we\u0027re saying is that information is being represented by digits."},{"Start":"01:48.965 ","End":"01:55.520","Text":"Digital systems are systems in which data is represented using discrete numerical values."},{"Start":"01:55.520 ","End":"01:58.730","Text":"More specifically, these discrete numerical values are"},{"Start":"01:58.730 ","End":"02:02.015","Text":"represented as digits that can only be 0 or 1."},{"Start":"02:02.015 ","End":"02:05.240","Text":"There is no value in between 0 and 1,"},{"Start":"02:05.240 ","End":"02:08.960","Text":"and that\u0027s what we mean by discrete values."},{"Start":"02:08.960 ","End":"02:11.780","Text":"You may have noticed we\u0027ve used the term data here,"},{"Start":"02:11.780 ","End":"02:15.365","Text":"which is sometimes used interchangeably with information."},{"Start":"02:15.365 ","End":"02:19.410","Text":"However, the 2 terms are technically not the same thing."},{"Start":"02:19.580 ","End":"02:26.495","Text":"Data are facts which have been coded in a form suitable for processing."},{"Start":"02:26.495 ","End":"02:30.005","Text":"Information is data that has been processed,"},{"Start":"02:30.005 ","End":"02:34.030","Text":"stored, analyzed, or interpreted."},{"Start":"02:34.030 ","End":"02:37.320","Text":"To give 1 example to illustrate the difference,"},{"Start":"02:37.320 ","End":"02:40.070","Text":"if we asked individuals how they were going to vote in"},{"Start":"02:40.070 ","End":"02:43.850","Text":"an upcoming election, we\u0027re collecting data."},{"Start":"02:43.850 ","End":"02:46.640","Text":"We would code this in the form of a list of"},{"Start":"02:46.640 ","End":"02:51.350","Text":"all available candidates and an extra possible response of undecided."},{"Start":"02:51.350 ","End":"02:53.990","Text":"A valid response would be to tick 1 box,"},{"Start":"02:53.990 ","End":"02:55.985","Text":"and 1 box only."},{"Start":"02:55.985 ","End":"02:59.720","Text":"If we then process all the data we collected,"},{"Start":"02:59.720 ","End":"03:05.160","Text":"we can produce information showing what the voting intentions are."},{"Start":"03:06.380 ","End":"03:11.450","Text":"Now, we\u0027ve established why there\u0027s a need to represent data in terms of 1s and 0s."},{"Start":"03:11.450 ","End":"03:14.210","Text":"Let\u0027s have a look at the idea of number systems."},{"Start":"03:14.210 ","End":"03:16.130","Text":"Perhaps the easiest way to understand"},{"Start":"03:16.130 ","End":"03:19.745","Text":"a number system is to consider the one that we\u0027re already familiar with"},{"Start":"03:19.745 ","End":"03:27.050","Text":"called decimal or denary This is a number system based on 10 symbols between 0 and 9."},{"Start":"03:27.050 ","End":"03:31.225","Text":"Each place value in a decimal number is a power of 10."},{"Start":"03:31.225 ","End":"03:33.660","Text":"Here the first 4 powers of 10,"},{"Start":"03:33.660 ","End":"03:38.630","Text":"each one gives the weighting of a digit in a 4-digit decimal number."},{"Start":"03:38.630 ","End":"03:41.360","Text":"Each digit of a decimal is"},{"Start":"03:41.360 ","End":"03:46.860","Text":"10 times the previous one as we move from the right to the left."},{"Start":"03:47.290 ","End":"03:54.705","Text":"In mathematical terms, the number 2049 is actually 2 times 10^3,"},{"Start":"03:54.705 ","End":"03:56.865","Text":"plus 0 times 10^2,"},{"Start":"03:56.865 ","End":"03:58.785","Text":"plus 4 times 10^1,"},{"Start":"03:58.785 ","End":"04:01.785","Text":"plus 9 times 10^0."},{"Start":"04:01.785 ","End":"04:07.640","Text":"A binary number system is a system based on the 2 symbols, 0 and 1."},{"Start":"04:07.640 ","End":"04:13.865","Text":"Each digit of a binary number is 2 times the previous one as we move from right to left."},{"Start":"04:13.865 ","End":"04:15.590","Text":"Given this particular number,"},{"Start":"04:15.590 ","End":"04:18.725","Text":"the temptation is to say that it\u0027s 1011,"},{"Start":"04:18.725 ","End":"04:22.415","Text":"but that\u0027s actually interpreting it as a decimal number."},{"Start":"04:22.415 ","End":"04:27.260","Text":"What we\u0027re really saying is that this is a binary number, 1011."},{"Start":"04:27.260 ","End":"04:35.405","Text":"In mathematical terms, that would be 1 times 2^3 plus 0 times 2^2 plus 1 times 2^1,"},{"Start":"04:35.405 ","End":"04:38.215","Text":"plus 1 times 2^0."},{"Start":"04:38.215 ","End":"04:42.608","Text":"Because we are only ever multiplying the column value by 1 or 0,"},{"Start":"04:42.608 ","End":"04:46.145","Text":"it\u0027s simple to convert this binary number into a decimal number."},{"Start":"04:46.145 ","End":"04:49.190","Text":"We just take the column values given in decimal"},{"Start":"04:49.190 ","End":"04:53.795","Text":"here and add the relevant values together."},{"Start":"04:53.795 ","End":"04:55.370","Text":"Anyway, where there\u0027s a 0,"},{"Start":"04:55.370 ","End":"04:57.680","Text":"we ignore that column and anywhere where there\u0027s a 1,"},{"Start":"04:57.680 ","End":"05:01.310","Text":"we include that column in our final results."},{"Start":"05:01.310 ","End":"05:06.470","Text":"Here we ignore the column width 4 and add the remaining column values together,"},{"Start":"05:06.470 ","End":"05:08.630","Text":"giving us 8 plus 2 plus 1,"},{"Start":"05:08.630 ","End":"05:10.585","Text":"which is 11 obviously."},{"Start":"05:10.585 ","End":"05:17.030","Text":"The binary number 1011 is the same as the decimal number 11."},{"Start":"05:17.030 ","End":"05:19.610","Text":"You can see that when written down this way,"},{"Start":"05:19.610 ","End":"05:21.320","Text":"it can be quite confusing."},{"Start":"05:21.320 ","End":"05:22.790","Text":"When we write down a number,"},{"Start":"05:22.790 ","End":"05:26.630","Text":"we should really annotate it to make clear which number base we\u0027re using."},{"Start":"05:26.630 ","End":"05:31.685","Text":"We do this by putting a subscript after the number indicating the base."},{"Start":"05:31.685 ","End":"05:34.205","Text":"Binary numbers have a subscript of 2,"},{"Start":"05:34.205 ","End":"05:36.965","Text":"decimal numbers, a subscript of 10."},{"Start":"05:36.965 ","End":"05:38.540","Text":"This is much clearer."},{"Start":"05:38.540 ","End":"05:43.790","Text":"1011 is a binary number which is the same as 11 in decimal."},{"Start":"05:43.790 ","End":"05:46.745","Text":"Sometimes you\u0027ll see this written with brackets as well,"},{"Start":"05:46.745 ","End":"05:50.390","Text":"with the digits inside the brackets and the number base outside."},{"Start":"05:50.390 ","End":"05:52.670","Text":"Let\u0027s pause there and absorb what we\u0027ve learned so"},{"Start":"05:52.670 ","End":"05:55.250","Text":"far about encoding data and number systems."},{"Start":"05:55.250 ","End":"05:58.070","Text":"In the next video, we\u0027ll learn how to move between"},{"Start":"05:58.070 ","End":"06:02.370","Text":"the decimal and binary number systems. See you then."}],"ID":30071},{"Watched":false,"Name":"Binary","Duration":"8m 9s","ChapterTopicVideoID":28554,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.025","Text":"Hello? Welcome back."},{"Start":"00:02.025 ","End":"00:07.695","Text":"The range of numbers available to us in binary is dependent on how many digits we have,"},{"Start":"00:07.695 ","End":"00:09.750","Text":"if we have 8 binary digits,"},{"Start":"00:09.750 ","End":"00:13.935","Text":"we can express a number from 0-255."},{"Start":"00:13.935 ","End":"00:17.762","Text":"Because of 1 in every column is the binary number 1,"},{"Start":"00:17.762 ","End":"00:19.118","Text":"1, 1, 1,"},{"Start":"00:19.118 ","End":"00:20.474","Text":"1, 1, 1,"},{"Start":"00:20.474 ","End":"00:25.320","Text":"1 that in decimal would be 255,"},{"Start":"00:25.320 ","End":"00:28.590","Text":"a 0 in every column would give us 0,"},{"Start":"00:28.590 ","End":"00:34.935","Text":"so 8 bits allows us to store any number between 0 and 255."},{"Start":"00:34.935 ","End":"00:37.080","Text":"Rather than saying binary digit,"},{"Start":"00:37.080 ","End":"00:38.760","Text":"every time we use those words,"},{"Start":"00:38.760 ","End":"00:43.205","Text":"we can use the abbreviation bit for binary digit."},{"Start":"00:43.205 ","End":"00:48.425","Text":"When we collect 8 bits together as we did just now to make the number 255,"},{"Start":"00:48.425 ","End":"00:52.150","Text":"we call that a byte or octet,"},{"Start":"00:52.150 ","End":"00:55.820","Text":"that\u0027s a grouping of 8 binary digits."},{"Start":"00:55.820 ","End":"00:59.315","Text":"Historically, a byte can mean a larger grouping than 8,"},{"Start":"00:59.315 ","End":"01:03.590","Text":"so you might hear the word octet used instead, which is unambiguous."},{"Start":"01:03.590 ","End":"01:08.480","Text":"However, more generally you will hear the term byte used and in a modern context,"},{"Start":"01:08.480 ","End":"01:11.475","Text":"it almost always means 8 bits."},{"Start":"01:11.475 ","End":"01:15.035","Text":"A half a byte is known as a nibble,"},{"Start":"01:15.035 ","End":"01:18.095","Text":"which is a grouping of 4 binary digits."},{"Start":"01:18.095 ","End":"01:19.910","Text":"When we write down a binary number,"},{"Start":"01:19.910 ","End":"01:24.335","Text":"we often separate it into 2 nibbles just to make it easier to read,"},{"Start":"01:24.335 ","End":"01:27.065","Text":"for example, this number here,"},{"Start":"01:27.065 ","End":"01:33.415","Text":"1101 1010, is much easier to read partitioned into 2 halves."},{"Start":"01:33.415 ","End":"01:36.620","Text":"In a binary number of any size the bit on"},{"Start":"01:36.620 ","End":"01:41.060","Text":"the extreme right is known as the least significant bit,"},{"Start":"01:41.060 ","End":"01:45.700","Text":"because it has the lowest significance in terms of the size of the overall number,"},{"Start":"01:45.700 ","End":"01:49.340","Text":"and the bits on the extreme left is known as"},{"Start":"01:49.340 ","End":"01:51.890","Text":"the most significant bit because it has"},{"Start":"01:51.890 ","End":"01:55.570","Text":"the most significance in terms of the size of the overall number."},{"Start":"01:55.570 ","End":"01:58.550","Text":"How would you know how many bits you need to encode"},{"Start":"01:58.550 ","End":"02:02.405","Text":"any particular decimal number in binary?"},{"Start":"02:02.405 ","End":"02:05.570","Text":"You first have to make a decision on what would be"},{"Start":"02:05.570 ","End":"02:08.210","Text":"the highest number you want to express,"},{"Start":"02:08.210 ","End":"02:14.370","Text":"so you knew that the maximum number you\u0027d ever store would be 10,000 in decimal."},{"Start":"02:14.370 ","End":"02:18.620","Text":"In binary, because we\u0027re dealing with powers of 2,"},{"Start":"02:18.620 ","End":"02:23.845","Text":"we need to find the nearest power of 2-10,000."},{"Start":"02:23.845 ","End":"02:26.555","Text":"Each time you add a new bit,"},{"Start":"02:26.555 ","End":"02:29.585","Text":"the value of the MSB has doubled,"},{"Start":"02:29.585 ","End":"02:36.575","Text":"so we can just keep adding bits until we get to a number that if we doubled it again,"},{"Start":"02:36.575 ","End":"02:40.850","Text":"will be bigger than the maximum number we\u0027re looking to encode."},{"Start":"02:40.850 ","End":"02:42.890","Text":"Here, we just keep adding bits,"},{"Start":"02:42.890 ","End":"02:49.710","Text":"doubling the MSB value each time until we reach 8192,"},{"Start":"02:49.710 ","End":"02:51.185","Text":"if we go any further,"},{"Start":"02:51.185 ","End":"02:55.415","Text":"the MSB will be bigger than 10,000 and that\u0027s more than we need,"},{"Start":"02:55.415 ","End":"02:57.370","Text":"so we can stop here."},{"Start":"02:57.370 ","End":"02:59.625","Text":"We have 14 columns now,"},{"Start":"02:59.625 ","End":"03:05.945","Text":"so the number of binary digits we need to encode the decimal value 10,000 is 14."},{"Start":"03:05.945 ","End":"03:09.920","Text":"We can also calculate the highest number that can be stored with a given number of"},{"Start":"03:09.920 ","End":"03:14.570","Text":"bits using the formula 2^n minus 1,"},{"Start":"03:14.570 ","End":"03:17.470","Text":"where n is the number of bits,"},{"Start":"03:17.470 ","End":"03:23.544","Text":"so 2^14 minus 1"},{"Start":"03:23.544 ","End":"03:26.050","Text":"would give us 16,383 which is more than enough range to store"},{"Start":"03:26.050 ","End":"03:29.645","Text":"the maximum number 10,000 that we set out to store."},{"Start":"03:29.645 ","End":"03:32.820","Text":"As we\u0027ve already seen for 8 bits,"},{"Start":"03:32.820 ","End":"03:37.630","Text":"2^8 minus 1 would give us 255."},{"Start":"03:37.630 ","End":"03:40.900","Text":"Let\u0027s stick with the example number of 10,000 and go through"},{"Start":"03:40.900 ","End":"03:44.515","Text":"the process of converting that number to binary."},{"Start":"03:44.515 ","End":"03:49.555","Text":"Earlier, we put a comma in to make this decimal number clearer when written out,"},{"Start":"03:49.555 ","End":"03:52.375","Text":"as is conventional in general usage in the UK,"},{"Start":"03:52.375 ","End":"03:54.100","Text":"but this can vary in other countries,"},{"Start":"03:54.100 ","End":"03:56.630","Text":"so we\u0027ll dispense with a comma from now on."},{"Start":"03:56.630 ","End":"04:01.640","Text":"We start converting from decimal to binary by looking at the MSB,"},{"Start":"04:01.640 ","End":"04:04.820","Text":"deciding whether we need to put a 1 in that column,"},{"Start":"04:04.820 ","End":"04:07.550","Text":"and at each point as we move from left to right,"},{"Start":"04:07.550 ","End":"04:09.260","Text":"we ask ourselves the question,"},{"Start":"04:09.260 ","End":"04:11.615","Text":"do we need this bit or not?"},{"Start":"04:11.615 ","End":"04:15.545","Text":"If we do we insert a 1 if we don\u0027t we insert a 0."},{"Start":"04:15.545 ","End":"04:19.280","Text":"The number you\u0027re looking to convert is bigger than the column that you\u0027re looking at,"},{"Start":"04:19.280 ","End":"04:21.395","Text":"then you do need that column,"},{"Start":"04:21.395 ","End":"04:25.100","Text":"so as 10,000 is bigger than 8,192,"},{"Start":"04:25.100 ","End":"04:26.795","Text":"we do need that value,"},{"Start":"04:26.795 ","End":"04:29.725","Text":"so we write a 1 in that position."},{"Start":"04:29.725 ","End":"04:34.410","Text":"We now have 8,192 out of the 10,000 we need to make up,"},{"Start":"04:34.410 ","End":"04:42.330","Text":"so we calculate what\u0027s left to make up from the remaining binary digits, which is 1,808."},{"Start":"04:42.330 ","End":"04:48.935","Text":"We don\u0027t need the 4,096 column because that\u0027s larger than 1,808,"},{"Start":"04:48.935 ","End":"04:51.890","Text":"so we put a 0 in that position."},{"Start":"04:51.890 ","End":"04:54.830","Text":"We don\u0027t need 2,048 either,"},{"Start":"04:54.830 ","End":"04:59.030","Text":"as it\u0027s also larger than 1,808,"},{"Start":"04:59.030 ","End":"05:01.175","Text":"so we put another 0 in there."},{"Start":"05:01.175 ","End":"05:04.900","Text":"However, we do need 1,024,"},{"Start":"05:04.900 ","End":"05:07.380","Text":"so 1 goes in that position,"},{"Start":"05:07.380 ","End":"05:15.465","Text":"we now subtract 1,024 from 1,808 to know what\u0027s left to make up that 784."},{"Start":"05:15.465 ","End":"05:18.405","Text":"784 is bigger than 512,"},{"Start":"05:18.405 ","End":"05:20.430","Text":"so we need 512."},{"Start":"05:20.430 ","End":"05:25.175","Text":"Now let us see what\u0027s left to make up once we\u0027ve deducted 512 is 272,"},{"Start":"05:25.175 ","End":"05:27.925","Text":"which is also bigger than 256,"},{"Start":"05:27.925 ","End":"05:29.285","Text":"so we\u0027ll need that column,"},{"Start":"05:29.285 ","End":"05:31.200","Text":"we put a 1 in and then we carry on,"},{"Start":"05:31.200 ","End":"05:34.470","Text":"16 is smaller than the next few columns,"},{"Start":"05:34.470 ","End":"05:37.420","Text":"so we put 0s in each of those."},{"Start":"05:37.420 ","End":"05:42.860","Text":"The 16 column is exactly equal to the amount we\u0027ve left to make up,"},{"Start":"05:42.860 ","End":"05:47.345","Text":"so we put a 1 in that column and there\u0027s nothing left to make up,"},{"Start":"05:47.345 ","End":"05:50.840","Text":"so we put 0s in every other column."},{"Start":"05:50.840 ","End":"05:56.675","Text":"We\u0027ve now converted the decimal number 10,000 into the binary equivalent."},{"Start":"05:56.675 ","End":"06:00.320","Text":"To convert in the other direction from binary into decimal,"},{"Start":"06:00.320 ","End":"06:02.435","Text":"we can work in 2 different ways."},{"Start":"06:02.435 ","End":"06:03.935","Text":"Looking at the number,"},{"Start":"06:03.935 ","End":"06:06.035","Text":"if we work from the LSB,"},{"Start":"06:06.035 ","End":"06:09.760","Text":"we know that it\u0027s worth 2^0 in decimal,"},{"Start":"06:09.760 ","End":"06:13.035","Text":"the next bit is 2^1 and so on."},{"Start":"06:13.035 ","End":"06:15.470","Text":"Anywhere in the binary number where there was 1,"},{"Start":"06:15.470 ","End":"06:16.820","Text":"we include that value,"},{"Start":"06:16.820 ","End":"06:19.760","Text":"otherwise, we ignore it."},{"Start":"06:19.760 ","End":"06:26.640","Text":"Here we have 2^3 plus 2^8 plus 2^9 and 2^12,"},{"Start":"06:26.640 ","End":"06:30.005","Text":"we can now use a calculator to work out each of those values"},{"Start":"06:30.005 ","End":"06:33.815","Text":"and add them up and come to a final value,"},{"Start":"06:33.815 ","End":"06:36.925","Text":"which in this case is 4,872."},{"Start":"06:36.925 ","End":"06:43.400","Text":"We\u0027ve converted the binary number to its decimal equivalent 4,872."},{"Start":"06:43.400 ","End":"06:47.060","Text":"If you don\u0027t have access to a calculator or not allowed 1,"},{"Start":"06:47.060 ","End":"06:48.605","Text":"for example in an exam,"},{"Start":"06:48.605 ","End":"06:50.875","Text":"you can convert as follows."},{"Start":"06:50.875 ","End":"06:53.625","Text":"Write out the binary number,"},{"Start":"06:53.625 ","End":"06:56.085","Text":"and then working from the LSB,"},{"Start":"06:56.085 ","End":"06:58.795","Text":"write 1 above the LSB,"},{"Start":"06:58.795 ","End":"07:00.800","Text":"and then work to the left,"},{"Start":"07:00.800 ","End":"07:03.590","Text":"doubling the value for each column,"},{"Start":"07:03.590 ","End":"07:08.730","Text":"and carefully writing the value directly above each binary digit,"},{"Start":"07:08.990 ","End":"07:12.350","Text":"until you reach the end of the number."},{"Start":"07:12.350 ","End":"07:15.770","Text":"We can stop here as leading 0s can be"},{"Start":"07:15.770 ","End":"07:20.375","Text":"ignored as they will not contribute anything to the calculation."},{"Start":"07:20.375 ","End":"07:23.870","Text":"The decimal number can now be found simply by writing out"},{"Start":"07:23.870 ","End":"07:29.765","Text":"a column value for each position in the binary number, where there\u0027s a ball."},{"Start":"07:29.765 ","End":"07:39.480","Text":"Here, 2,048 plus 64 plus 16 plus 4 gives us 2,132,"},{"Start":"07:39.740 ","End":"07:43.716","Text":"so we\u0027ve now converted the binary number 0,1,"},{"Start":"07:43.716 ","End":"07:45.135","Text":"0, 0, 0,"},{"Start":"07:45.135 ","End":"07:47.499","Text":"0, 1, 0,1, 0,"},{"Start":"07:47.499 ","End":"07:54.685","Text":"1, 0, 0 into its decimal equivalent 2,132."},{"Start":"07:54.685 ","End":"07:57.230","Text":"Let\u0027s pause there again and absorb what"},{"Start":"07:57.230 ","End":"07:59.975","Text":"we\u0027ve learned about converting to and from binary."},{"Start":"07:59.975 ","End":"08:01.415","Text":"In the next video,"},{"Start":"08:01.415 ","End":"08:05.000","Text":"we\u0027ll look at another useful number system that has some advantages"},{"Start":"08:05.000 ","End":"08:09.610","Text":"over binary in certain situations. See you then."}],"ID":30072},{"Watched":false,"Name":"Hexadecimal","Duration":"8m 32s","ChapterTopicVideoID":28555,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.755","Text":"Hello, welcome back."},{"Start":"00:01.755 ","End":"00:03.060","Text":"In the previous video,"},{"Start":"00:03.060 ","End":"00:06.870","Text":"we established that digital circuits make it necessary for us to adopt"},{"Start":"00:06.870 ","End":"00:11.970","Text":"a binary number system based on 2 possible states, 0 or 1."},{"Start":"00:11.970 ","End":"00:14.550","Text":"We can encode a decimal number in binary simply"},{"Start":"00:14.550 ","End":"00:16.980","Text":"by using the appropriate number of binary digits,"},{"Start":"00:16.980 ","End":"00:19.350","Text":"known for short as bits with values of"},{"Start":"00:19.350 ","End":"00:22.890","Text":"0 or 1 in the relevant positions within the binary number."},{"Start":"00:22.890 ","End":"00:26.520","Text":"A problem quickly arises when dealing with numbers encoded in binary,"},{"Start":"00:26.520 ","End":"00:28.890","Text":"which is that they become quite lengthy to read,"},{"Start":"00:28.890 ","End":"00:30.705","Text":"write out, and even to say."},{"Start":"00:30.705 ","End":"00:33.615","Text":"This binary number is made up of 16 bits,"},{"Start":"00:33.615 ","End":"00:37.140","Text":"that\u0027s 16 symbols to write out for this single number."},{"Start":"00:37.140 ","End":"00:39.150","Text":"To make it easier to read or write,"},{"Start":"00:39.150 ","End":"00:42.690","Text":"the bits can be grouped in fours like so."},{"Start":"00:42.690 ","End":"00:46.070","Text":"We say we split the number up into 4 nibbles."},{"Start":"00:46.070 ","End":"00:48.770","Text":"If there isn\u0027t an exact multiple of 4 digits,"},{"Start":"00:48.770 ","End":"00:54.085","Text":"we can simply add zeros to the left hand side to make it up to a multiple of 4."},{"Start":"00:54.085 ","End":"01:01.899","Text":"For example, the 5-bit number 10111 can be turned into 2 nibbles,"},{"Start":"01:02.690 ","End":"01:08.165","Text":"0001 0111, simply by adding 3 zeros at the left-hand side."},{"Start":"01:08.165 ","End":"01:11.600","Text":"Adding these zeros to the left doesn\u0027t change the value of the number,"},{"Start":"01:11.600 ","End":"01:16.355","Text":"but we can more clearly read the numbers as equal-sized patterns."},{"Start":"01:16.355 ","End":"01:22.115","Text":"It be convenient to be able to take each nibble and encode it as a single symbol."},{"Start":"01:22.115 ","End":"01:25.790","Text":"And this is exactly what you can do with hexadecimal,"},{"Start":"01:25.790 ","End":"01:30.515","Text":"which is a number system based on 16 symbols between 0 and"},{"Start":"01:30.515 ","End":"01:36.235","Text":"F. Each place value in a hexadecimal number is a power of 16."},{"Start":"01:36.235 ","End":"01:39.165","Text":"Here are the first 4 powers of 16."},{"Start":"01:39.165 ","End":"01:44.555","Text":"Each one gives a waiting of a digit in a 4 digit hexadecimal number."},{"Start":"01:44.555 ","End":"01:47.660","Text":"Each digit in hexadecimal number is"},{"Start":"01:47.660 ","End":"01:52.740","Text":"16 times the previous one as we move from right to left."},{"Start":"01:52.870 ","End":"01:57.890","Text":"Hexadecimal numbers can seem quite strange at first because there"},{"Start":"01:57.890 ","End":"02:02.650","Text":"are letters as well as the digits we\u0027re familiar with from the decimal system."},{"Start":"02:02.650 ","End":"02:06.350","Text":"The digits from 0-9 are identical to decimal,"},{"Start":"02:06.350 ","End":"02:11.175","Text":"but there are also symbols that represent a decimal values 10-15."},{"Start":"02:11.175 ","End":"02:16.415","Text":"The letters A-F are used to represent the decimal values 10-15,"},{"Start":"02:16.415 ","End":"02:18.377","Text":"while the letters A-F, well,"},{"Start":"02:18.377 ","End":"02:21.380","Text":"they follow a well-known sequence i.e."},{"Start":"02:21.380 ","End":"02:23.100","Text":"the alphabet just like numbers,"},{"Start":"02:23.100 ","End":"02:25.580","Text":"and they can be typed on a keyboard."},{"Start":"02:25.580 ","End":"02:27.650","Text":"We only need to go up to F because"},{"Start":"02:27.650 ","End":"02:29.960","Text":"remember the range of numbers that can be expressed with"},{"Start":"02:29.960 ","End":"02:34.840","Text":"a given number of bits is given by the formula 2^n minus 1,"},{"Start":"02:34.840 ","End":"02:37.410","Text":"where n is the number of bits."},{"Start":"02:37.410 ","End":"02:40.170","Text":"For a nibble that would be 4,"},{"Start":"02:40.170 ","End":"02:44.715","Text":"and so 2^4 minus 1 would give us 15."},{"Start":"02:44.715 ","End":"02:48.410","Text":"15 is the highest number we can express so hence,"},{"Start":"02:48.410 ","End":"02:51.350","Text":"our need for 16 symbols including 0."},{"Start":"02:51.350 ","End":"02:53.960","Text":"To convert a hexadecimal number to decimal,"},{"Start":"02:53.960 ","End":"02:57.380","Text":"we multiply each hexadecimal digit by its column"},{"Start":"02:57.380 ","End":"03:02.420","Text":"waiting and sum all the parts together to get a decimal equivalent."},{"Start":"03:02.420 ","End":"03:05.120","Text":"This is the same as in binary or decimal,"},{"Start":"03:05.120 ","End":"03:10.955","Text":"but we have the extra complication here of the hexadecimal values for 10-15,"},{"Start":"03:10.955 ","End":"03:14.960","Text":"which are represented by the letters A-F. We take"},{"Start":"03:14.960 ","End":"03:20.975","Text":"the additional step of replacing those hex digits with their equivalents in decimal,"},{"Start":"03:20.975 ","End":"03:27.470","Text":"in this case giving us the final value, 41,723."},{"Start":"03:27.470 ","End":"03:32.030","Text":"We\u0027ve converted a number in hexadecimal to its decimal equivalent."},{"Start":"03:32.030 ","End":"03:34.310","Text":"Again, if we don\u0027t have access to a calculator,"},{"Start":"03:34.310 ","End":"03:37.100","Text":"we can work out the decimal value by starting at"},{"Start":"03:37.100 ","End":"03:40.325","Text":"the least significant digit on the extreme right,"},{"Start":"03:40.325 ","End":"03:42.085","Text":"which we know is 1."},{"Start":"03:42.085 ","End":"03:45.440","Text":"Once we\u0027ve written out that 1 above the relevant digit,"},{"Start":"03:45.440 ","End":"03:47.324","Text":"we know that in hexadecimal,"},{"Start":"03:47.324 ","End":"03:50.015","Text":"each column is 16 times more the 1 to the right."},{"Start":"03:50.015 ","End":"03:54.650","Text":"So 16 will be the next column value, and we write that in."},{"Start":"03:54.650 ","End":"03:56.645","Text":"Then keep going."},{"Start":"03:56.645 ","End":"03:59.825","Text":"16 by 16 is 256, and we write that in."},{"Start":"03:59.825 ","End":"04:05.015","Text":"Then final column for a 4 digit number will be 16 times 256,"},{"Start":"04:05.015 ","End":"04:08.415","Text":"which you can work out to be 4,096."},{"Start":"04:08.415 ","End":"04:12.145","Text":"Now we can multiply each digit by its column value."},{"Start":"04:12.145 ","End":"04:19.660","Text":"So 1 times 4,096, 0 lots of 256, A lots of 16, and C lots of 1."},{"Start":"04:19.660 ","End":"04:23.380","Text":"Again, we have the complication of the digits which are greater than 9,"},{"Start":"04:23.380 ","End":"04:26.125","Text":"which we need to convert to their decimal equivalents."},{"Start":"04:26.125 ","End":"04:34.535","Text":"So 10 for A and 12 for C. Now we multiply out and add up to get the final value."},{"Start":"04:34.535 ","End":"04:38.095","Text":"And we\u0027ve converted the hex value into decimal."},{"Start":"04:38.095 ","End":"04:42.160","Text":"To go in the other direction from decimal to hexadecimal,"},{"Start":"04:42.160 ","End":"04:46.465","Text":"we need to first determine how many hexadecimal digits we need."},{"Start":"04:46.465 ","End":"04:50.905","Text":"Remember the formula we saw for binary was the number based to the power of n,"},{"Start":"04:50.905 ","End":"04:53.590","Text":"where n is the number of digits we\u0027re using."},{"Start":"04:53.590 ","End":"04:56.839","Text":"For 4 hexadecimal digits,"},{"Start":"04:57.330 ","End":"05:06.350","Text":"16^n minus 1, and the range will give us 0 to 65,535."},{"Start":"05:06.350 ","End":"05:08.180","Text":"If 4 hexadecimal digits is"},{"Start":"05:08.180 ","End":"05:11.000","Text":"enough range to encourage that decimal number we\u0027re looking at,"},{"Start":"05:11.000 ","End":"05:12.545","Text":"then we can go ahead."},{"Start":"05:12.545 ","End":"05:15.890","Text":"Otherwise we add more digits until we do have enough range."},{"Start":"05:15.890 ","End":"05:19.895","Text":"Let\u0027s encode the decimal number 10,000 again as we did previously."},{"Start":"05:19.895 ","End":"05:23.465","Text":"That\u0027ll fit comfortably into 4 hex digits."},{"Start":"05:23.465 ","End":"05:27.975","Text":"We start in the leftmost column and ask ourselves if we need this value."},{"Start":"05:27.975 ","End":"05:31.645","Text":"10,000 is greater than 4,096,"},{"Start":"05:31.645 ","End":"05:33.175","Text":"so we need it."},{"Start":"05:33.175 ","End":"05:36.410","Text":"We find out how many times it goes in."},{"Start":"05:36.410 ","End":"05:41.670","Text":"By dividing using integer division where we ignore the remainder,"},{"Start":"05:42.820 ","End":"05:45.590","Text":"10,000/4,096, which gives us 2."},{"Start":"05:45.590 ","End":"05:48.395","Text":"We write that into the relevant column."},{"Start":"05:48.395 ","End":"05:52.610","Text":"We can now get the remainder by doing what\u0027s called a modulo operation,"},{"Start":"05:52.610 ","End":"05:55.235","Text":"which should be available on most calculators."},{"Start":"05:55.235 ","End":"06:01.460","Text":"If not, we just take 2 times 4,096 from 10,000 to find the remainder."},{"Start":"06:01.460 ","End":"06:03.395","Text":"So that\u0027s 1,808,"},{"Start":"06:03.395 ","End":"06:07.219","Text":"and that\u0027s what we have to make up from the remaining columns."},{"Start":"06:07.219 ","End":"06:09.530","Text":"We repeat the process asking"},{"Start":"06:09.530 ","End":"06:12.365","Text":"ourselves is the next column less than what\u0027s left to make up?"},{"Start":"06:12.365 ","End":"06:14.090","Text":"Which in this case it is."},{"Start":"06:14.090 ","End":"06:16.385","Text":"So we need the 256 column,"},{"Start":"06:16.385 ","End":"06:21.990","Text":"but this time we do an integer division of 256, giving us 7."},{"Start":"06:21.990 ","End":"06:25.835","Text":"We write that into a relevant place and work out the remainder,"},{"Start":"06:25.835 ","End":"06:28.265","Text":"which gives us 16."},{"Start":"06:28.265 ","End":"06:31.085","Text":"We now only have 16 to make up."},{"Start":"06:31.085 ","End":"06:33.800","Text":"We repeat the process again for the next column,"},{"Start":"06:33.800 ","End":"06:36.800","Text":"which is 16, giving us 1,"},{"Start":"06:36.800 ","End":"06:38.660","Text":"which we write in."},{"Start":"06:38.660 ","End":"06:42.230","Text":"Now there\u0027s no remainder left in this case,"},{"Start":"06:42.230 ","End":"06:45.950","Text":"so the last digit is going to be 0."},{"Start":"06:45.950 ","End":"06:53.360","Text":"And we\u0027ve converted the decimal value 10,000 into its hexadecimal equivalent 2,710."},{"Start":"06:53.360 ","End":"06:56.120","Text":"Binary to hexadecimal is very straightforward."},{"Start":"06:56.120 ","End":"06:58.400","Text":"You simply divide up the binary into"},{"Start":"06:58.400 ","End":"07:02.825","Text":"nibbles and give each nibble a hexadecimal representation."},{"Start":"07:02.825 ","End":"07:05.510","Text":"If there isn\u0027t an exact multiple of 4, it\u0027s not problem."},{"Start":"07:05.510 ","End":"07:09.635","Text":"You can just assume that the most significant bits in that nibble are 0."},{"Start":"07:09.635 ","End":"07:15.080","Text":"Each nibble has bit values which are powers of 2 up to 8."},{"Start":"07:15.080 ","End":"07:18.575","Text":"These are the same for each individual nibble,"},{"Start":"07:18.575 ","End":"07:23.135","Text":"except in our case for the last nibble which doesn\u0027t have 4-bits,"},{"Start":"07:23.135 ","End":"07:27.560","Text":"we assume that most significant bit is 0 and we only go up to 4 not 8."},{"Start":"07:27.560 ","End":"07:30.770","Text":"Now we can convert each nibble into a hex digit."},{"Start":"07:30.770 ","End":"07:33.365","Text":"The most significant digit here is 4 plus 1,"},{"Start":"07:33.365 ","End":"07:36.755","Text":"which is 5 in decimal or hex. We write that in."},{"Start":"07:36.755 ","End":"07:38.315","Text":"Then 2 plus 1,"},{"Start":"07:38.315 ","End":"07:41.525","Text":"which is 3 in decimal and hex."},{"Start":"07:41.525 ","End":"07:44.000","Text":"Now this nibble is 8 plus 4,"},{"Start":"07:44.000 ","End":"07:45.320","Text":"which is 12 in decimal,"},{"Start":"07:45.320 ","End":"07:46.730","Text":"but we can\u0027t write that in."},{"Start":"07:46.730 ","End":"07:48.845","Text":"We need a single hexadecimal digit,"},{"Start":"07:48.845 ","End":"07:51.170","Text":"which will be C. Remember, A is 10,"},{"Start":"07:51.170 ","End":"07:53.945","Text":"B is 11, C is 12, and so on up to 15."},{"Start":"07:53.945 ","End":"07:58.025","Text":"This final nibble is 8 plus 4 plus 2, which is 14."},{"Start":"07:58.025 ","End":"08:01.370","Text":"In hex, we write that as E. Now we\u0027ve"},{"Start":"08:01.370 ","End":"08:04.940","Text":"converted the binary number into its hexadecimal equivalent."},{"Start":"08:04.940 ","End":"08:08.210","Text":"Notice the difference in the number of symbols used."},{"Start":"08:08.210 ","End":"08:15.625","Text":"The binary representation needed 15 symbols and the hexadecimal equivalent only 4."},{"Start":"08:15.625 ","End":"08:17.790","Text":"Let\u0027s pause 1 final time,"},{"Start":"08:17.790 ","End":"08:22.850","Text":"and in the next video we will look at our general number bases and"},{"Start":"08:22.850 ","End":"08:26.180","Text":"1 specific one and how we would choose"},{"Start":"08:26.180 ","End":"08:29.795","Text":"between the different number bases given any particular situation."},{"Start":"08:29.795 ","End":"08:33.360","Text":"So see you soon for that one. Goodbye."}],"ID":30073},{"Watched":false,"Name":"Selecting between number systems","Duration":"6m 23s","ChapterTopicVideoID":28556,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.665","Text":"Hello, welcome back."},{"Start":"00:01.665 ","End":"00:03.690","Text":"By now we should have got used to the idea of"},{"Start":"00:03.690 ","End":"00:07.500","Text":"number basis and can spot the general pattern that they all follow."},{"Start":"00:07.500 ","End":"00:09.570","Text":"For any given number base B,"},{"Start":"00:09.570 ","End":"00:13.905","Text":"we can express a range of numbers determined by the number of digits"},{"Start":"00:13.905 ","End":"00:19.100","Text":"n. The highest value that can be expressed is B^n minus 1."},{"Start":"00:19.100 ","End":"00:22.265","Text":"Because 0 is a valid combination too."},{"Start":"00:22.265 ","End":"00:26.170","Text":"Let\u0027s try this with a number base of 8 and 4 digits,"},{"Start":"00:26.170 ","End":"00:30.875","Text":"8^4 minus 1 would give us 4,095."},{"Start":"00:30.875 ","End":"00:33.560","Text":"We could express the decimal values from"},{"Start":"00:33.560 ","End":"00:39.845","Text":"0-4,095 using 4 digits if we had a number system based on 8."},{"Start":"00:39.845 ","End":"00:41.690","Text":"We\u0027d need 8 different symbols,"},{"Start":"00:41.690 ","End":"00:47.554","Text":"so we may as well use the existing decimal digits for 0-7 to make up these 8 symbols."},{"Start":"00:47.554 ","End":"00:52.030","Text":"This number system based on 8 and using the symbols 0-7"},{"Start":"00:52.030 ","End":"00:56.705","Text":"is actually the next most common system after the ones that we\u0027ve already seen."},{"Start":"00:56.705 ","End":"00:58.700","Text":"Although it is rare these days,"},{"Start":"00:58.700 ","End":"01:00.800","Text":"and it\u0027s called octal,"},{"Start":"01:00.800 ","End":"01:05.945","Text":"each place value in an octal number is a power of 8."},{"Start":"01:05.945 ","End":"01:08.585","Text":"Given this particular octal number,"},{"Start":"01:08.585 ","End":"01:12.530","Text":"we can convert it to a decimal in the usual way,"},{"Start":"01:12.530 ","End":"01:15.950","Text":"multiplying out the digit by its place value."},{"Start":"01:15.950 ","End":"01:24.560","Text":"The octal number 4,107 is equivalent to 2,119, in decimal."},{"Start":"01:24.560 ","End":"01:31.385","Text":"Octal has the convenient property that each octal digit can represent 3 binary digits."},{"Start":"01:31.385 ","End":"01:35.330","Text":"We call a grouping of 3 binary digits a triad."},{"Start":"01:35.330 ","End":"01:39.425","Text":"Converting between octal and binary is a simple case of converting"},{"Start":"01:39.425 ","End":"01:45.080","Text":"each octal digit into binary and writing it above the relevant digit."},{"Start":"01:45.080 ","End":"01:55.740","Text":"Here, 4 would be 100,1 in the next triad would be 001,0 would just be 3 zeros,"},{"Start":"01:55.740 ","End":"01:58.875","Text":"and 7 would be 3 1\u0027s."},{"Start":"01:58.875 ","End":"02:02.655","Text":"The binary number, 1 4 zeros,"},{"Start":"02:02.655 ","End":"02:04.620","Text":"1, 3 zeros 1, 1,"},{"Start":"02:04.620 ","End":"02:10.395","Text":"1 is equivalent to 4107 in octal."},{"Start":"02:10.395 ","End":"02:12.800","Text":"Which number bases most appropriate for"},{"Start":"02:12.800 ","End":"02:17.510","Text":"a given situation depends on what the data is and who is working with it."},{"Start":"02:17.510 ","End":"02:22.790","Text":"An end-user of most digital systems is almost always going to want to deal in decimal,"},{"Start":"02:22.790 ","End":"02:26.645","Text":"since that\u0027s the only number system most people are familiar with."},{"Start":"02:26.645 ","End":"02:29.495","Text":"For input and output intended for humans,"},{"Start":"02:29.495 ","End":"02:32.270","Text":"decimal is the way to go."},{"Start":"02:32.270 ","End":"02:37.490","Text":"Where computer hardware is being interacted with at the bit level, for example,"},{"Start":"02:37.490 ","End":"02:42.920","Text":"the internal memory of a CPU or a chip that interfaces with a CPU,"},{"Start":"02:42.920 ","End":"02:47.105","Text":"binary is often the most appropriate number base to work in."},{"Start":"02:47.105 ","End":"02:49.880","Text":"This CPU contains, for example,"},{"Start":"02:49.880 ","End":"02:52.790","Text":"a flag register, an area of memory within it."},{"Start":"02:52.790 ","End":"02:55.490","Text":"With individual bits indicating the status"},{"Start":"02:55.490 ","End":"02:58.955","Text":"of various operations taking place within the CPU."},{"Start":"02:58.955 ","End":"03:01.640","Text":"If, say, the result of a calculation couldn\u0027t"},{"Start":"03:01.640 ","End":"03:04.205","Text":"be contained in a particular memory location,"},{"Start":"03:04.205 ","End":"03:09.125","Text":"bit 2, the overflow bit would be set to 1."},{"Start":"03:09.125 ","End":"03:13.610","Text":"If an addition of 2 numbers resulted in a carryover to a new digit,"},{"Start":"03:13.610 ","End":"03:17.915","Text":"bit 0, the carry bit would be set to 1."},{"Start":"03:17.915 ","End":"03:21.530","Text":"Read these individual bits in isolation from all the others."},{"Start":"03:21.530 ","End":"03:26.045","Text":"It\u0027s more convenient to deal in binary than decimal."},{"Start":"03:26.045 ","End":"03:30.470","Text":"Networking is another area where sometimes binary can be useful."},{"Start":"03:30.470 ","End":"03:32.180","Text":"Using a network mask,"},{"Start":"03:32.180 ","End":"03:36.515","Text":"we can partition something called an IP address into 2 halves,"},{"Start":"03:36.515 ","End":"03:39.845","Text":"the network ID and the host ID."},{"Start":"03:39.845 ","End":"03:42.080","Text":"When looking at it as a decimal number,"},{"Start":"03:42.080 ","End":"03:44.705","Text":"it can be hard to see which part is which."},{"Start":"03:44.705 ","End":"03:50.855","Text":"Because decimal 240 includes some bits that are 1 and some bits that are 0."},{"Start":"03:50.855 ","End":"03:55.040","Text":"Seeing it in binary makes it much easier to spot which is"},{"Start":"03:55.040 ","End":"03:59.285","Text":"part of the host address and which is part of the network address."},{"Start":"03:59.285 ","End":"04:03.680","Text":"The one shows the region that\u0027s part of the network ID and"},{"Start":"04:03.680 ","End":"04:09.005","Text":"where there are zeros that shows the host ID range."},{"Start":"04:09.005 ","End":"04:12.620","Text":"Hexadecimal is most often used by programmers or"},{"Start":"04:12.620 ","End":"04:15.990","Text":"perhaps graphic designers or digital artists."},{"Start":"04:15.990 ","End":"04:19.755","Text":"Color is often expressed as a 24-bit value,"},{"Start":"04:19.755 ","End":"04:21.725","Text":"split into 3 bytes,"},{"Start":"04:21.725 ","End":"04:23.945","Text":"red, green, and blue."},{"Start":"04:23.945 ","End":"04:29.360","Text":"The specific value is given to each of these determine the exact color being expressed."},{"Start":"04:29.360 ","End":"04:32.810","Text":"This blue color here being used in the font,"},{"Start":"04:32.810 ","End":"04:35.810","Text":"happens to be this precise color."},{"Start":"04:35.810 ","End":"04:42.080","Text":"Here we can see a color dialogue showing this blue color as a single hex value here."},{"Start":"04:42.080 ","End":"04:47.135","Text":"Also individual decimal values for red, green, and blue."},{"Start":"04:47.135 ","End":"04:52.580","Text":"Because hexadecimal uses a maximum of 2 symbols for an 8-bit value."},{"Start":"04:52.580 ","End":"04:58.080","Text":"A hexadecimal RGB value can always be expressed with no more than 6 symbols,"},{"Start":"04:58.080 ","End":"05:04.345","Text":"written out as decimal it could be a maximum of 9 symbols."},{"Start":"05:04.345 ","End":"05:06.335","Text":"For the same reason,"},{"Start":"05:06.335 ","End":"05:10.220","Text":"anywhere where binary values are grouped in 4\u0027s or multiples of 4,"},{"Start":"05:10.220 ","End":"05:13.205","Text":"or when display spaces are premium,"},{"Start":"05:13.205 ","End":"05:15.575","Text":"hexadecimal can be useful."},{"Start":"05:15.575 ","End":"05:20.990","Text":"Here we see a display of the contents of memory in hex called a hex dump."},{"Start":"05:20.990 ","End":"05:24.635","Text":"Each memory location takes up a single byte."},{"Start":"05:24.635 ","End":"05:29.164","Text":"It can be represented by 2 hexadecimal symbols."},{"Start":"05:29.164 ","End":"05:32.855","Text":"For example, this one here, or any of these."},{"Start":"05:32.855 ","End":"05:37.580","Text":"The 32-bit hex memory location is shown"},{"Start":"05:37.580 ","End":"05:44.380","Text":"here and each byte of memory is then shown after the address."},{"Start":"05:44.380 ","End":"05:50.525","Text":"Each full line shows 16 bytes of memory in this type of view."},{"Start":"05:50.525 ","End":"05:56.240","Text":"It\u0027s a nice compact way of displaying memory contents."},{"Start":"05:56.240 ","End":"05:59.405","Text":"That\u0027s it now on number systems."},{"Start":"05:59.405 ","End":"06:02.780","Text":"In this section, we learned how to explain the need for"},{"Start":"06:02.780 ","End":"06:06.080","Text":"alternative number systems to convert between decimal, binary,"},{"Start":"06:06.080 ","End":"06:07.820","Text":"hexadecimal, and octal,"},{"Start":"06:07.820 ","End":"06:12.950","Text":"and to select between appropriate number bases according to need."},{"Start":"06:12.950 ","End":"06:16.355","Text":"Be sure to complete the exercises in the workbook"},{"Start":"06:16.355 ","End":"06:20.390","Text":"to consolidate your understanding of everything we\u0027ve heard about."},{"Start":"06:20.390 ","End":"06:23.490","Text":"Thank you very much and see you soon."}],"ID":30074},{"Watched":false,"Name":"Exercise 1 parts A-B","Duration":"3m 39s","ChapterTopicVideoID":25440,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"Hello, everyone. In this video,"},{"Start":"00:02.970 ","End":"00:09.015","Text":"we\u0027re going to see how you convert a binary number into a decimal representation."},{"Start":"00:09.015 ","End":"00:12.570","Text":"There are 2 simple exercises to start with,"},{"Start":"00:12.570 ","End":"00:14.580","Text":"and then the next exercise,"},{"Start":"00:14.580 ","End":"00:17.265","Text":"we\u0027ll move on to slightly longer numbers."},{"Start":"00:17.265 ","End":"00:20.190","Text":"I\u0027m first going to explain how we\u0027d go about"},{"Start":"00:20.190 ","End":"00:23.910","Text":"the process and then we\u0027re going to do the exercises in turn."},{"Start":"00:23.910 ","End":"00:26.700","Text":"Starting with the first part,"},{"Start":"00:26.700 ","End":"00:30.390","Text":"part A, we can see a short binary number."},{"Start":"00:30.390 ","End":"00:31.590","Text":"There are 1, 2,"},{"Start":"00:31.590 ","End":"00:32.760","Text":"3, 4, 5,"},{"Start":"00:32.760 ","End":"00:35.730","Text":"6, 7 columns in this number."},{"Start":"00:35.730 ","End":"00:39.945","Text":"Each column, you\u0027ll recall in binary has a waiting."},{"Start":"00:39.945 ","End":"00:42.600","Text":"We start from this side,"},{"Start":"00:42.600 ","End":"00:49.295","Text":"the left-hand side and we write above each column the waiting for that column."},{"Start":"00:49.295 ","End":"00:54.200","Text":"Let\u0027s go ahead and do that. The next part of the process then is to simply"},{"Start":"00:54.200 ","End":"00:59.295","Text":"look at where there is a 1 under a particular value,"},{"Start":"00:59.295 ","End":"01:01.910","Text":"and we include anywhere where we have a 1 in"},{"Start":"01:01.910 ","End":"01:07.100","Text":"our final calculation and ignore anywhere where there\u0027s a 0 in the calculation."},{"Start":"01:07.100 ","End":"01:09.980","Text":"Working from left to right,"},{"Start":"01:09.980 ","End":"01:16.400","Text":"we have got a value for 64 because there is a 1 in this column."},{"Start":"01:16.400 ","End":"01:19.820","Text":"We skip the 32 because there\u0027s a 0."},{"Start":"01:19.820 ","End":"01:24.355","Text":"We do include the 16 because there is a 1."},{"Start":"01:24.355 ","End":"01:29.250","Text":"We also include the 8 because there\u0027s a 1 there."},{"Start":"01:29.250 ","End":"01:31.410","Text":"Ignore the 4,"},{"Start":"01:31.410 ","End":"01:33.150","Text":"ignore the 2,"},{"Start":"01:33.150 ","End":"01:35.940","Text":"but include the 1, and we,"},{"Start":"01:35.940 ","End":"01:39.450","Text":"therefore, are left with 64 plus 16,"},{"Start":"01:39.450 ","End":"01:41.370","Text":"which is 80, plus 8,"},{"Start":"01:41.370 ","End":"01:45.095","Text":"88, plus 1 is 89."},{"Start":"01:45.095 ","End":"01:49.055","Text":"We have now completed the conversion from"},{"Start":"01:49.055 ","End":"01:53.570","Text":"the binary representation here to the decimal representation here."},{"Start":"01:53.570 ","End":"01:57.140","Text":"Part B is exactly the same process."},{"Start":"01:57.140 ","End":"01:58.880","Text":"However, this time we have more columns."},{"Start":"01:58.880 ","End":"02:00.440","Text":"If we look at how many columns, we got 1,"},{"Start":"02:00.440 ","End":"02:02.130","Text":"2, 3, 4,"},{"Start":"02:02.130 ","End":"02:03.825","Text":"5, 6, 7,"},{"Start":"02:03.825 ","End":"02:06.090","Text":"8, 9 columns this time."},{"Start":"02:06.090 ","End":"02:11.510","Text":"We\u0027d expect the values on the left-hand side to go higher."},{"Start":"02:11.510 ","End":"02:15.660","Text":"Let\u0027s go ahead and see where we get to with that."},{"Start":"02:15.660 ","End":"02:16.745","Text":"As we can see now,"},{"Start":"02:16.745 ","End":"02:22.160","Text":"the most significant value here on the extreme left-hand side is 256 now,"},{"Start":"02:22.160 ","End":"02:24.770","Text":"whereas previously it was only 64."},{"Start":"02:24.770 ","End":"02:28.490","Text":"We proceed in exactly the same fashion as we did before."},{"Start":"02:28.490 ","End":"02:34.370","Text":"Include any columns where we have a 1 and ignore any columns where we have a 0."},{"Start":"02:34.370 ","End":"02:39.530","Text":"This will give us therefore 256 because there\u0027s a 1 in that column,"},{"Start":"02:39.530 ","End":"02:44.030","Text":"128 because there\u0027s a 1 in that column, skip the 64,"},{"Start":"02:44.030 ","End":"02:48.410","Text":"but include the 32 and 16,"},{"Start":"02:48.410 ","End":"02:49.820","Text":"skip the 8,"},{"Start":"02:49.820 ","End":"02:51.460","Text":"include the 4,"},{"Start":"02:51.460 ","End":"02:54.105","Text":"skip the 2, but include the 1."},{"Start":"02:54.105 ","End":"02:58.885","Text":"Probably is more straightforward to add up in pairs here now,"},{"Start":"02:58.885 ","End":"03:06.420","Text":"so 256 plus 128 is 384, plus,"},{"Start":"03:06.420 ","End":"03:10.920","Text":"32 and 16 is 48 plus 5,"},{"Start":"03:10.920 ","End":"03:13.620","Text":"and that will give us the final answer,"},{"Start":"03:13.620 ","End":"03:21.190","Text":"384 plus 48 plus 5 is 437."},{"Start":"03:21.190 ","End":"03:25.503","Text":"We have completed the first 2 parts of the exercise now."},{"Start":"03:25.503 ","End":"03:30.890","Text":"This binary value has been converted to 437 in decimal."},{"Start":"03:30.890 ","End":"03:32.570","Text":"In the next video, we\u0027ll look at"},{"Start":"03:32.570 ","End":"03:36.050","Text":"slightly longer bodies written out"},{"Start":"03:36.050 ","End":"03:40.530","Text":"in binary and how we might best approach those. See you in the next video."}],"ID":30075},{"Watched":false,"Name":"Exercise 2 part A","Duration":"4m ","ChapterTopicVideoID":25441,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.860","Text":"Welcome back. For this next exercise we\u0027ve been asked to do the reverse conversion."},{"Start":"00:07.860 ","End":"00:12.450","Text":"Compared to the previous exercise where we were asked to convert from binary to decimal,"},{"Start":"00:12.450 ","End":"00:16.050","Text":"we\u0027re now being asked to convert from decimal to binary."},{"Start":"00:16.050 ","End":"00:17.925","Text":"There\u0027s 2 methods of doing this."},{"Start":"00:17.925 ","End":"00:19.080","Text":"In this first video,"},{"Start":"00:19.080 ","End":"00:20.925","Text":"I\u0027m going to show you 1 method."},{"Start":"00:20.925 ","End":"00:23.715","Text":"In the next video I\u0027ll show you an alternative method."},{"Start":"00:23.715 ","End":"00:26.730","Text":"You can choose which one suits you."},{"Start":"00:26.730 ","End":"00:30.090","Text":"We begin by looking at the number"},{"Start":"00:30.090 ","End":"00:33.495","Text":"itself and deciding how many columns we\u0027re going to need."},{"Start":"00:33.495 ","End":"00:35.820","Text":"It\u0027s a fairly simple decision to make."},{"Start":"00:35.820 ","End":"00:41.700","Text":"We start on the right hand side writing out the values for each column."},{"Start":"00:41.700 ","End":"00:46.760","Text":"We keep going until we find we don\u0027t need the columns anymore."},{"Start":"00:46.760 ","End":"00:49.805","Text":"To give you an example, 237."},{"Start":"00:49.805 ","End":"00:50.750","Text":"We start by"},{"Start":"00:50.750 ","End":"01:00.443","Text":"writing 1,"},{"Start":"01:00.443 ","End":"01:01.470","Text":"2, 4, 8, 16, 32, 64, 1, 2, 8."},{"Start":"01:04.960 ","End":"01:08.425","Text":"If I was to move on to the next column,"},{"Start":"01:08.425 ","End":"01:12.130","Text":"256 is actually a bigger number than 237,"},{"Start":"01:12.130 ","End":"01:13.615","Text":"so I don\u0027t need it."},{"Start":"01:13.615 ","End":"01:16.600","Text":"So this is the point at which I can stop."},{"Start":"01:16.600 ","End":"01:20.680","Text":"Now it\u0027s a simple case of doing"},{"Start":"01:20.680 ","End":"01:24.395","Text":"a simple test to determine whether I need the column or not."},{"Start":"01:24.395 ","End":"01:26.150","Text":"The test is basically,"},{"Start":"01:26.150 ","End":"01:29.545","Text":"is the number I\u0027m looking at greater than this column,"},{"Start":"01:29.545 ","End":"01:32.477","Text":"or is it not greater than this column?"},{"Start":"01:32.477 ","End":"01:37.055","Text":"For the first one, we can see that 237 is indeed greater than 128."},{"Start":"01:37.055 ","End":"01:38.330","Text":"So we need this column,"},{"Start":"01:38.330 ","End":"01:40.535","Text":"so I\u0027m going to write a 1 there."},{"Start":"01:40.535 ","End":"01:44.075","Text":"Before we move on, we are actually going to have to subtract the"},{"Start":"01:44.075 ","End":"01:48.750","Text":"128 from 237 to see how much we\u0027ve got left."},{"Start":"01:48.750 ","End":"01:50.430","Text":"I\u0027m going to write it at the side here,"},{"Start":"01:50.430 ","End":"01:53.315","Text":"so 237 is what we started with,"},{"Start":"01:53.315 ","End":"01:59.080","Text":"take off 128 and we are given 109."},{"Start":"01:59.080 ","End":"02:02.505","Text":"109 is what remains to be made up."},{"Start":"02:02.505 ","End":"02:04.490","Text":"I then carry on as before,"},{"Start":"02:04.490 ","End":"02:06.560","Text":"I look at the number I\u0027ve got to make up,"},{"Start":"02:06.560 ","End":"02:09.860","Text":"which is 109. Do I need the 64?"},{"Start":"02:09.860 ","End":"02:12.290","Text":"Yes, 109 is greater than 64,"},{"Start":"02:12.290 ","End":"02:13.925","Text":"so I need that column."},{"Start":"02:13.925 ","End":"02:16.160","Text":"Then I do the same step again,"},{"Start":"02:16.160 ","End":"02:21.740","Text":"remove the 64 from the 109 and that would give me 45."},{"Start":"02:21.740 ","End":"02:22.970","Text":"Ask the question again."},{"Start":"02:22.970 ","End":"02:24.050","Text":"Do I need this column?"},{"Start":"02:24.050 ","End":"02:26.269","Text":"Is 45 greater than 32?"},{"Start":"02:26.269 ","End":"02:29.225","Text":"Yes, it is. So I need it as well."},{"Start":"02:29.225 ","End":"02:32.390","Text":"Subtract the 32 from 45,"},{"Start":"02:32.390 ","End":"02:34.705","Text":"which gives me 13."},{"Start":"02:34.705 ","End":"02:37.053","Text":"I\u0027m moving on to the next column."},{"Start":"02:37.053 ","End":"02:38.850","Text":"Do I need the 16 column?"},{"Start":"02:38.850 ","End":"02:44.235","Text":"No, in this case I don\u0027t because 13 is less than 16, not greater."},{"Start":"02:44.235 ","End":"02:47.870","Text":"I put a 0 here. Moving on."},{"Start":"02:47.870 ","End":"02:50.285","Text":"Do I need the next column, 8?"},{"Start":"02:50.285 ","End":"02:52.310","Text":"Yes, because 13 is greater than 8,"},{"Start":"02:52.310 ","End":"02:53.780","Text":"so I need that."},{"Start":"02:53.780 ","End":"02:58.840","Text":"Subtract the 8 from 13, given me 5."},{"Start":"02:58.840 ","End":"03:00.530","Text":"Move on to the next column."},{"Start":"03:00.530 ","End":"03:02.480","Text":"Do I need the 4?"},{"Start":"03:02.480 ","End":"03:04.915","Text":"Yes, indeed, I need the 4."},{"Start":"03:04.915 ","End":"03:07.545","Text":"Take away 4 from 5,"},{"Start":"03:07.545 ","End":"03:09.915","Text":"I have now only got 1 left to make up."},{"Start":"03:09.915 ","End":"03:14.660","Text":"So it\u0027s going to be 0 in this column because 2 is greater than 1."},{"Start":"03:14.660 ","End":"03:18.390","Text":"Although 1 is not greater than 1,"},{"Start":"03:18.390 ","End":"03:20.130","Text":"it is equal to 1,"},{"Start":"03:20.130 ","End":"03:21.765","Text":"so I do need it."},{"Start":"03:21.765 ","End":"03:24.635","Text":"The final value there is going to be 1."},{"Start":"03:24.635 ","End":"03:26.810","Text":"If I go backwards through this and just do"},{"Start":"03:26.810 ","End":"03:30.635","Text":"a little quick check to see whether I\u0027ve done the right operations."},{"Start":"03:30.635 ","End":"03:39.665","Text":"128 plus 64 will give me 192 plus 32 will give me 224,"},{"Start":"03:39.665 ","End":"03:43.880","Text":"plus 8 would give me 232,"},{"Start":"03:43.880 ","End":"03:47.090","Text":"236, 237,"},{"Start":"03:47.090 ","End":"03:50.195","Text":"which is what I was originally asked to convert."},{"Start":"03:50.195 ","End":"03:53.290","Text":"That looks like I\u0027ve done the job."},{"Start":"03:53.290 ","End":"04:00.450","Text":"Here is the binary conversion of 237 decimal."}],"ID":30076},{"Watched":false,"Name":"Exercise 2 part B","Duration":"3m 9s","ChapterTopicVideoID":25442,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"Let\u0027s now look at part B."},{"Start":"00:02.910 ","End":"00:05.550","Text":"We\u0027re going to use exactly the same method as before."},{"Start":"00:05.550 ","End":"00:08.865","Text":"The first thing to do is determine how many columns we need."},{"Start":"00:08.865 ","End":"00:11.280","Text":"Starting on the right-hand side,"},{"Start":"00:11.280 ","End":"00:13.530","Text":"we write the column values."},{"Start":"00:13.530 ","End":"00:17.340","Text":"Now, we realize that if we went any further,"},{"Start":"00:17.340 ","End":"00:20.040","Text":"we\u0027d have 512 and that\u0027s bigger than 359,"},{"Start":"00:20.040 ","End":"00:22.905","Text":"so we don\u0027t actually need to proceed any further."},{"Start":"00:22.905 ","End":"00:25.170","Text":"We\u0027ve got as many columns as we need."},{"Start":"00:25.170 ","End":"00:30.915","Text":"There\u0027s now 9 bits rather than 8-bits tree had in the previous part A."},{"Start":"00:30.915 ","End":"00:36.600","Text":"Now we\u0027re going to proceed just as we did before to ask ourselves the question,"},{"Start":"00:36.600 ","End":"00:39.105","Text":"do we need the column or do we not need the column?"},{"Start":"00:39.105 ","End":"00:41.025","Text":"The first column here,"},{"Start":"00:41.025 ","End":"00:44.330","Text":"359 is indeed bigger than 256,"},{"Start":"00:44.330 ","End":"00:45.980","Text":"so I do need that column,"},{"Start":"00:45.980 ","End":"00:51.120","Text":"but then I need to take away from 359,"},{"Start":"00:51.120 ","End":"00:55.785","Text":"256 and that will give me 103."},{"Start":"00:55.785 ","End":"01:01.225","Text":"I have 103 remaining to make up from the other columns."},{"Start":"01:01.225 ","End":"01:05.275","Text":"So 103 is not greater than 128,"},{"Start":"01:05.275 ","End":"01:07.370","Text":"so I do not need that column."},{"Start":"01:07.370 ","End":"01:10.130","Text":"However, it is bigger than 64."},{"Start":"01:10.130 ","End":"01:11.855","Text":"I do need that column."},{"Start":"01:11.855 ","End":"01:14.935","Text":"Now, I need to remove 64 from 103,"},{"Start":"01:14.935 ","End":"01:17.500","Text":"which gives me 39."},{"Start":"01:17.500 ","End":"01:21.245","Text":"Carrying on, 39 is bigger than 32,"},{"Start":"01:21.245 ","End":"01:23.240","Text":"so I will need that column."},{"Start":"01:23.240 ","End":"01:25.970","Text":"If I remove the 32 from 39,"},{"Start":"01:25.970 ","End":"01:28.780","Text":"I\u0027ve only got 7 left to make up."},{"Start":"01:28.780 ","End":"01:34.035","Text":"I won\u0027t need the 6 column because 7 is not greater than 6,"},{"Start":"01:34.035 ","End":"01:36.370","Text":"not 6, 16."},{"Start":"01:36.370 ","End":"01:39.905","Text":"I won\u0027t need the 8 column either."},{"Start":"01:39.905 ","End":"01:42.370","Text":"However, I will need the 4."},{"Start":"01:42.370 ","End":"01:44.715","Text":"I\u0027ve got 3 left to make up."},{"Start":"01:44.715 ","End":"01:46.845","Text":"I will need the 2."},{"Start":"01:46.845 ","End":"01:49.070","Text":"Only 1 left to make up now,"},{"Start":"01:49.070 ","End":"01:50.720","Text":"and I will need the 1."},{"Start":"01:50.720 ","End":"01:55.670","Text":"I\u0027m now done. I\u0027ve got nothing else to include."},{"Start":"01:55.670 ","End":"01:58.790","Text":"Let\u0027s just do a quick check to see whether that\u0027s right."},{"Start":"01:58.790 ","End":"02:03.330","Text":"You\u0027ve got 256 plus 264,"},{"Start":"02:03.330 ","End":"02:07.650","Text":"it gives us 320 plus 32,"},{"Start":"02:07.650 ","End":"02:17.670","Text":"will gives us 352"},{"Start":"02:17.670 ","End":"02:24.660","Text":"plus 4 would give us 356, 358, 359."},{"Start":"02:24.660 ","End":"02:30.685","Text":"We\u0027ve arrived at the binary conversion"},{"Start":"02:30.685 ","End":"02:36.070","Text":"from decimal and using a method which I call the greater than method,"},{"Start":"02:36.070 ","End":"02:37.300","Text":"so you\u0027re, each time,"},{"Start":"02:37.300 ","End":"02:45.055","Text":"inspecting whether the value you\u0027re looking for is greater than the column or not."},{"Start":"02:45.055 ","End":"02:46.675","Text":"If it is greater than the column,"},{"Start":"02:46.675 ","End":"02:48.284","Text":"you need to include a 1 there,"},{"Start":"02:48.284 ","End":"02:52.220","Text":"otherwise, you include a zero and you keep doing that,"},{"Start":"02:52.220 ","End":"02:56.075","Text":"reducing by any of the columns you\u0027ve already used"},{"Start":"02:56.075 ","End":"03:00.455","Text":"and you will eventually end up producing the binary number that you\u0027re looking for."},{"Start":"03:00.455 ","End":"03:02.465","Text":"Now there is another method of doing this,"},{"Start":"03:02.465 ","End":"03:05.420","Text":"and we\u0027ll look at that in the next video,"},{"Start":"03:05.420 ","End":"03:10.290","Text":"which looks at parts C and D of the exercise."}],"ID":30077},{"Watched":false,"Name":"Exercise 2 part C","Duration":"3m 34s","ChapterTopicVideoID":25443,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"Welcome back. In this next video,"},{"Start":"00:03.060 ","End":"00:06.420","Text":"we\u0027re going to continue with the conversions from decimal to binary,"},{"Start":"00:06.420 ","End":"00:09.045","Text":"but we\u0027re going to use a slightly different method."},{"Start":"00:09.045 ","End":"00:12.255","Text":"Actually, we\u0027ve got 2 larger numbers here as well."},{"Start":"00:12.255 ","End":"00:14.805","Text":"Let\u0027s see how this method works."},{"Start":"00:14.805 ","End":"00:18.795","Text":"Essentially, this method involves dividing by 2."},{"Start":"00:18.795 ","End":"00:22.110","Text":"You repeatedly do that recording the remainder when"},{"Start":"00:22.110 ","End":"00:25.140","Text":"you divide by 2 with an integer division."},{"Start":"00:25.140 ","End":"00:26.640","Text":"With an integer division,"},{"Start":"00:26.640 ","End":"00:29.880","Text":"you cannot have fractional parts."},{"Start":"00:29.880 ","End":"00:34.485","Text":"For example, if we divide 1856 by 2,"},{"Start":"00:34.485 ","End":"00:37.815","Text":"we would actually get 928.5,"},{"Start":"00:37.815 ","End":"00:42.090","Text":"but we discard the half and record it as a remainder of 1."},{"Start":"00:42.090 ","End":"00:45.575","Text":"Let\u0027s go ahead and see how this process works."},{"Start":"00:45.575 ","End":"00:47.090","Text":"I\u0027ll start, as I said,"},{"Start":"00:47.090 ","End":"00:52.100","Text":"by dividing 1857 by 2,"},{"Start":"00:52.100 ","End":"00:56.585","Text":"and that will give us 928 with a remainder of 1."},{"Start":"00:56.585 ","End":"00:59.485","Text":"1928 divided by 2,"},{"Start":"00:59.485 ","End":"01:06.018","Text":"will give us 464 with a remainder of 0."},{"Start":"01:06.018 ","End":"01:12.558","Text":"464 divided by 2 would give us 232 with a remainder of 0."},{"Start":"01:12.558 ","End":"01:19.255","Text":"232 divided by 2 would give us 116 with a remainder of 0."},{"Start":"01:19.255 ","End":"01:23.140","Text":"116 divided by 2 is 58,"},{"Start":"01:23.140 ","End":"01:25.175","Text":"with a remainder of 0."},{"Start":"01:25.175 ","End":"01:30.410","Text":"58 divided by 2 is 29, with a remainder of 0."},{"Start":"01:30.410 ","End":"01:32.300","Text":"Now we\u0027ve got an odd number,"},{"Start":"01:32.300 ","End":"01:34.295","Text":"so we\u0027re going to have a remainder,"},{"Start":"01:34.295 ","End":"01:36.980","Text":"29 divided by 2 would be 14,"},{"Start":"01:36.980 ","End":"01:38.695","Text":"with a remainder of 1."},{"Start":"01:38.695 ","End":"01:43.490","Text":"14 divided by 2 would give us 7 and remainder of 0."},{"Start":"01:43.490 ","End":"01:45.695","Text":"7, again, is an odd number."},{"Start":"01:45.695 ","End":"01:47.885","Text":"We divide that by 2,"},{"Start":"01:47.885 ","End":"01:51.740","Text":"and we would get 3 and a remainder of 1."},{"Start":"01:51.740 ","End":"01:57.510","Text":"We divide 3 by 2 we get 1 with a remainder of 1,"},{"Start":"01:57.510 ","End":"01:59.700","Text":"and if we divide 1 by 2,"},{"Start":"01:59.700 ","End":"02:03.015","Text":"we get 0 with a remainder of 1."},{"Start":"02:03.015 ","End":"02:05.910","Text":"We\u0027ve now completed the conversion."},{"Start":"02:05.910 ","End":"02:10.430","Text":"The simple, almost magical trick you\u0027re"},{"Start":"02:10.430 ","End":"02:14.765","Text":"using here is to read-out the remainders in reverse order."},{"Start":"02:14.765 ","End":"02:16.340","Text":"That actually gives you"},{"Start":"02:16.340 ","End":"02:23.060","Text":"the binary conversion of the original number we started with. Let\u0027s do that."},{"Start":"02:23.060 ","End":"02:24.440","Text":"We have"},{"Start":"02:24.440 ","End":"02:32.415","Text":"111 0100"},{"Start":"02:32.415 ","End":"02:37.380","Text":"0001."},{"Start":"02:37.380 ","End":"02:42.819","Text":"If we were to convert this back into decimal,"},{"Start":"02:42.819 ","End":"02:46.345","Text":"just to check whether we\u0027ve done this right."},{"Start":"02:46.345 ","End":"02:49.045","Text":"We just write the column values on top."},{"Start":"02:49.045 ","End":"02:53.170","Text":"What we have is 1024 plus"},{"Start":"02:53.170 ","End":"03:00.885","Text":"512 plus 256 plus 64 plus 1,"},{"Start":"03:00.885 ","End":"03:07.695","Text":"which gives us, do that in pairs 1536 plus"},{"Start":"03:07.695 ","End":"03:17.080","Text":"320 would be 1856 plus 1 would indeed be 1857."},{"Start":"03:17.080 ","End":"03:25.730","Text":"We have managed to convert the original decimal value into its binary representation."},{"Start":"03:25.730 ","End":"03:32.460","Text":"The final answer here is 111 0100 0001."}],"ID":30078},{"Watched":false,"Name":"Exercise 2 part D","Duration":"3m 43s","ChapterTopicVideoID":25444,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.840","Text":"Well, now do part d in exactly the same manner as we did part c. We start with"},{"Start":"00:06.840 ","End":"00:09.810","Text":"a number that we\u0027re looking to convert and we have"},{"Start":"00:09.810 ","End":"00:14.040","Text":"it each time recording whether there\u0027s a remainder or not."},{"Start":"00:14.040 ","End":"00:16.755","Text":"Let\u0027s start with the first division,"},{"Start":"00:16.755 ","End":"00:22.995","Text":"give us 22050 with a remainder of 0."},{"Start":"00:22.995 ","End":"00:31.725","Text":"Dividing that in half would give us 11025 with a remainder of 0."},{"Start":"00:31.725 ","End":"00:37.010","Text":"Having that would give us 5512,"},{"Start":"00:37.010 ","End":"00:38.770","Text":"but we\u0027d have a remainder now,"},{"Start":"00:38.770 ","End":"00:42.755","Text":"of 1 given that 11025 is an odd number."},{"Start":"00:42.755 ","End":"00:47.615","Text":"Take 5512 and halve it would give us"},{"Start":"00:47.615 ","End":"00:54.080","Text":"2756 with a remainder of 0."},{"Start":"00:54.080 ","End":"00:59.585","Text":"2756 halved, would give us 1378,"},{"Start":"00:59.585 ","End":"01:02.320","Text":"with a remainder of 0."},{"Start":"01:02.320 ","End":"01:06.560","Text":"1378 halved, would give us 689,"},{"Start":"01:06.560 ","End":"01:09.345","Text":"with a remainder of 0."},{"Start":"01:09.345 ","End":"01:10.940","Text":"We now have an odd number here,"},{"Start":"01:10.940 ","End":"01:13.415","Text":"so we\u0027re going to expect a remainder."},{"Start":"01:13.415 ","End":"01:17.615","Text":"689 halved, the nearest number would be 344,"},{"Start":"01:17.615 ","End":"01:20.306","Text":"with a remainder of 1."},{"Start":"01:20.306 ","End":"01:25.988","Text":"344 halved, would give us 172 with a remainder of 0."},{"Start":"01:25.988 ","End":"01:30.485","Text":"172 halved would give us 86,"},{"Start":"01:30.485 ","End":"01:32.120","Text":"with a remainder of 0."},{"Start":"01:32.120 ","End":"01:38.585","Text":"86 halved, would give us 43 and the remainder of 0."},{"Start":"01:38.585 ","End":"01:40.100","Text":"43, clearly,"},{"Start":"01:40.100 ","End":"01:47.810","Text":"is an odd number so halving that would give us 21 with a remainder of 1."},{"Start":"01:47.810 ","End":"01:49.550","Text":"21 is, again,"},{"Start":"01:49.550 ","End":"01:58.560","Text":"an odd number and that would give us 10 with a remainder of 1."},{"Start":"01:58.560 ","End":"02:02.640","Text":"10 halved would give us 5 with a remainder of 0."},{"Start":"02:02.640 ","End":"02:06.890","Text":"5 halved would give us 2 with a remainder of 1."},{"Start":"02:06.890 ","End":"02:11.790","Text":"2 divided by 2 would give us 1 with a remainder of 0."},{"Start":"02:11.790 ","End":"02:13.290","Text":"We\u0027ve got 1 left,"},{"Start":"02:13.290 ","End":"02:17.275","Text":"1 divided by 2 gives us 0 with a remainder of 1."},{"Start":"02:17.275 ","End":"02:23.460","Text":"We\u0027ve now got all of our binary digits reading them from the bottom as before."},{"Start":"02:23.460 ","End":"02:28.435","Text":"It might be helpful here to group them into 4 bits at a time,"},{"Start":"02:28.435 ","End":"02:32.015","Text":"just so that we don\u0027t make any mistakes and shift the column."},{"Start":"02:32.015 ","End":"02:36.660","Text":"Let\u0027s just mark off every 4 bits here."},{"Start":"02:36.660 ","End":"02:40.690","Text":"Our final answer then is"},{"Start":"02:49.040 ","End":"02:54.790","Text":"1,0,1,0,1,1,0,0,0,1,0,0,0,1,0,0, again. If we wanted to check this, again,"},{"Start":"02:54.790 ","End":"02:57.790","Text":"we could use the method that we already know,"},{"Start":"02:57.790 ","End":"03:03.250","Text":"which is to include the column values above each binary digit."},{"Start":"03:03.250 ","End":"03:13.440","Text":"Our final answer is in this column 8192 from these 2 columns."},{"Start":"03:13.440 ","End":"03:15.580","Text":"If we were to add all those up,"},{"Start":"03:15.580 ","End":"03:20.800","Text":"we would indeed obtain 44100."},{"Start":"03:20.800 ","End":"03:26.510","Text":"Once again, we have managed to do the decimal"},{"Start":"03:26.510 ","End":"03:32.095","Text":"to binary conversion and our final answer is here."},{"Start":"03:32.095 ","End":"03:35.540","Text":"That\u0027s decimal to binary conversions using"},{"Start":"03:35.540 ","End":"03:39.020","Text":"2 different methods and we will move on to"},{"Start":"03:39.020 ","End":"03:43.830","Text":"another number base in the next video. See you then."}],"ID":30079},{"Watched":false,"Name":"Exercise 3 parts A-B","Duration":"9m 52s","ChapterTopicVideoID":25445,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.250","Text":"Hello everybody. In this question,"},{"Start":"00:02.250 ","End":"00:04.680","Text":"we\u0027ve been asked to convert numbers from"},{"Start":"00:04.680 ","End":"00:08.624","Text":"a hexadecimal representation this time into binary."},{"Start":"00:08.624 ","End":"00:10.755","Text":"Let\u0027s start with part a."},{"Start":"00:10.755 ","End":"00:15.570","Text":"We\u0027ve been asked to convert the number 42 in hexadecimal to binary."},{"Start":"00:15.570 ","End":"00:18.240","Text":"It\u0027s actually a very straightforward process."},{"Start":"00:18.240 ","End":"00:23.715","Text":"The whole purpose of hexadecimal is to take 4 bits at a time."},{"Start":"00:23.715 ","End":"00:28.935","Text":"Those 4 bits can be represented by a single character in hexadecimal."},{"Start":"00:28.935 ","End":"00:34.920","Text":"What we will do is go ahead and write the number first of all, which is 42."},{"Start":"00:34.920 ","End":"00:37.055","Text":"Then what we\u0027re going to do is we\u0027re going to take"},{"Start":"00:37.055 ","End":"00:41.000","Text":"each individual hexadecimal digit and break it out into binary."},{"Start":"00:41.000 ","End":"00:43.310","Text":"What I\u0027m going to do is I\u0027m going to write out"},{"Start":"00:43.310 ","End":"00:47.532","Text":"the column values for each 1 of those digits, 8, 4, 2,"},{"Start":"00:47.532 ","End":"00:51.080","Text":"and 1 and start again 8,4,2,1,"},{"Start":"00:51.080 ","End":"00:55.770","Text":"because each of these digits is to be represented by a 4-bit pattern."},{"Start":"00:55.770 ","End":"00:58.190","Text":"The overall number therefore,"},{"Start":"00:58.190 ","End":"01:00.350","Text":"is the combination of these 2,"},{"Start":"01:00.350 ","End":"01:04.740","Text":"4-bit values. This is really straightforward."},{"Start":"01:04.740 ","End":"01:08.120","Text":"This question, 4 represented in binary,"},{"Start":"01:08.120 ","End":"01:10.415","Text":"if we look at this individual nibble here,"},{"Start":"01:10.415 ","End":"01:14.060","Text":"is going to be 0,1,0,0,"},{"Start":"01:14.060 ","End":"01:16.925","Text":"because we need the 4 and nothing else."},{"Start":"01:16.925 ","End":"01:20.555","Text":"Similarly to represent the 2 in binary,"},{"Start":"01:20.555 ","End":"01:22.285","Text":"we just need the 2,"},{"Start":"01:22.285 ","End":"01:25.050","Text":"everything else is going to be a 0,"},{"Start":"01:25.050 ","End":"01:26.585","Text":"so we\u0027ve done it."},{"Start":"01:26.585 ","End":"01:29.360","Text":"That\u0027s as simple as that."},{"Start":"01:29.360 ","End":"01:34.715","Text":"We\u0027ve got 42 hexadecimal into a binary bit pattern."},{"Start":"01:34.715 ","End":"01:39.505","Text":"Now, you might want to check that you\u0027ve got the correct answer."},{"Start":"01:39.505 ","End":"01:43.385","Text":"The way we would check this is to convert into"},{"Start":"01:43.385 ","End":"01:46.640","Text":"decimal both the hexadecimal"},{"Start":"01:46.640 ","End":"01:49.730","Text":"number and the binary number and see if you get the same number."},{"Start":"01:49.730 ","End":"01:53.840","Text":"What we need to do to convert it into decimal"},{"Start":"01:53.840 ","End":"01:59.360","Text":"is the way hexadecimal works is this first digit is the units column."},{"Start":"01:59.360 ","End":"02:03.340","Text":"The next column over is worth 16 times the previous column,"},{"Start":"02:03.340 ","End":"02:04.670","Text":"and we continue on that."},{"Start":"02:04.670 ","End":"02:08.015","Text":"Then in that manner for however many columns we have."},{"Start":"02:08.015 ","End":"02:09.650","Text":"Obviously, we\u0027ve only got 2 columns here."},{"Start":"02:09.650 ","End":"02:11.660","Text":"But if we did have another digit here,"},{"Start":"02:11.660 ","End":"02:14.135","Text":"that would be worth 16 times 16,"},{"Start":"02:14.135 ","End":"02:15.920","Text":"which is to 256."},{"Start":"02:15.920 ","End":"02:18.470","Text":"But as I said, we\u0027ve only got 2 digits here,"},{"Start":"02:18.470 ","End":"02:21.110","Text":"so what we need to do now is to multiply out"},{"Start":"02:21.110 ","End":"02:24.170","Text":"these columns and we\u0027ll have the value in decimal."},{"Start":"02:24.170 ","End":"02:29.150","Text":"It turns out to be 16 times 4 plus 1 times 2."},{"Start":"02:29.150 ","End":"02:32.300","Text":"I\u0027ll just write that as 2."},{"Start":"02:32.300 ","End":"02:40.505","Text":"What that will give us 16 times 4 is 64 plus 2 is 66."},{"Start":"02:40.505 ","End":"02:47.525","Text":"This number that we started with as a decimal number is actually 66."},{"Start":"02:47.525 ","End":"02:49.940","Text":"Now, if we go back to our binary bit pattern,"},{"Start":"02:49.940 ","End":"02:54.515","Text":"these bits here are actually not really 1, 2, 4 and 8."},{"Start":"02:54.515 ","End":"02:58.550","Text":"If we consider the whole byte together, the 2 nibbles."},{"Start":"02:58.550 ","End":"03:02.630","Text":"Really what we\u0027re looking at here is these digits are"},{"Start":"03:02.630 ","End":"03:07.220","Text":"16 times what they say in blue here."},{"Start":"03:07.220 ","End":"03:09.635","Text":"It\u0027s not really 1 here."},{"Start":"03:09.635 ","End":"03:11.600","Text":"We\u0027re talking about 16."},{"Start":"03:11.600 ","End":"03:12.830","Text":"I want to write it above in"},{"Start":"03:12.830 ","End":"03:16.640","Text":"a different color so that we can see that this is actually 16,"},{"Start":"03:16.640 ","End":"03:19.145","Text":"1 over is 32,"},{"Start":"03:19.145 ","End":"03:20.732","Text":"the next 1,"},{"Start":"03:20.732 ","End":"03:23.690","Text":"64, and then next one\u0027s 128."},{"Start":"03:23.690 ","End":"03:25.550","Text":"When we read this number here,"},{"Start":"03:25.550 ","End":"03:29.650","Text":"0,1,0,0, It\u0027s not really 8,4,2,1,"},{"Start":"03:29.650 ","End":"03:31.470","Text":"It\u0027s 128, 64,"},{"Start":"03:31.470 ","End":"03:32.730","Text":"32 and 16,"},{"Start":"03:32.730 ","End":"03:36.605","Text":"and the only 1 that\u0027s got a 1 here is actually 64."},{"Start":"03:36.605 ","End":"03:41.945","Text":"We\u0027ll include that in our crosscheck as part of the answer 64."},{"Start":"03:41.945 ","End":"03:44.685","Text":"Then we\u0027ve got nothing here,"},{"Start":"03:44.685 ","End":"03:46.485","Text":"nothing here, but we do have a 2."},{"Start":"03:46.485 ","End":"03:52.183","Text":"We need to add 2 and 64 plus 2 is 66,"},{"Start":"03:52.183 ","End":"03:54.785","Text":"as we saw across here."},{"Start":"03:54.785 ","End":"03:58.010","Text":"We\u0027re confident that we\u0027ve done the current version correctly."},{"Start":"03:58.010 ","End":"04:04.685","Text":"This binary bit pattern is representing the number 4 to in hexadecimal,"},{"Start":"04:04.685 ","End":"04:07.700","Text":"which turns out to be 66 in decimal."},{"Start":"04:07.700 ","End":"04:11.930","Text":"We\u0027ve converted both the binary to decimal and got 66."},{"Start":"04:11.930 ","End":"04:17.300","Text":"We converted the hexadecimal directly to decimal by multiplying"},{"Start":"04:17.300 ","End":"04:21.170","Text":"the first hexadecimal digit by 16 and adding"},{"Start":"04:21.170 ","End":"04:25.790","Text":"on the second hexadecimal digit, we again got 66."},{"Start":"04:25.790 ","End":"04:29.050","Text":"We\u0027re confident that we\u0027ve got the right answer for this part."},{"Start":"04:29.050 ","End":"04:30.390","Text":"For the second part,"},{"Start":"04:30.390 ","End":"04:32.970","Text":"the question, we have a different number, D9."},{"Start":"04:32.970 ","End":"04:34.655","Text":"We\u0027re going to do exactly as we did before."},{"Start":"04:34.655 ","End":"04:38.365","Text":"Let\u0027s write it down, D9."},{"Start":"04:38.365 ","End":"04:43.145","Text":"I\u0027m giving myself some space there so I can write all a bit values out for binary."},{"Start":"04:43.145 ","End":"04:48.650","Text":"Once again, if I wanted to do the conversion afterwards into decimal,"},{"Start":"04:48.650 ","End":"04:54.635","Text":"I\u0027m just going to label for myself the value of each of these hexadecimal digits."},{"Start":"04:54.635 ","End":"04:59.360","Text":"Now there\u0027s a little technique I would recommend that you use for"},{"Start":"04:59.360 ","End":"05:03.920","Text":"hexadecimal numbers just to make sure you don\u0027t make any silly errors."},{"Start":"05:03.920 ","End":"05:06.710","Text":"That is to write out at the side of your paper"},{"Start":"05:06.710 ","End":"05:09.770","Text":"just a little aide memoir so that"},{"Start":"05:09.770 ","End":"05:15.170","Text":"you see what the decimal equivalent is of a hexadecimal digit."},{"Start":"05:15.170 ","End":"05:23.120","Text":"You should know hexadecimal digits are exactly the same as decimal digits from 1-9,"},{"Start":"05:23.120 ","End":"05:26.780","Text":"so there\u0027s no conversion to do,"},{"Start":"05:26.780 ","End":"05:30.830","Text":"there\u0027s enough confusion really when we\u0027re dealing with numbers from 1-9."},{"Start":"05:30.830 ","End":"05:33.440","Text":"Where the issues might come in is,"},{"Start":"05:33.440 ","End":"05:37.670","Text":"the numbers from 10 upwards up to 15 are"},{"Start":"05:37.670 ","End":"05:42.260","Text":"not what we would expect to see a number as because they\u0027re letters,"},{"Start":"05:42.260 ","End":"05:48.860","Text":"the numbers 10-15 are the letters A to F. It"},{"Start":"05:48.860 ","End":"05:51.860","Text":"does help to have them written down at the side of your paper"},{"Start":"05:51.860 ","End":"05:55.880","Text":"so you can very quickly glance across and see what the equivalent values are."},{"Start":"05:55.880 ","End":"05:58.175","Text":"If you see a C,"},{"Start":"05:58.175 ","End":"05:59.570","Text":"you know that it\u0027s worth 12."},{"Start":"05:59.570 ","End":"06:00.605","Text":"If you see an F,"},{"Start":"06:00.605 ","End":"06:02.090","Text":"you know its 15,"},{"Start":"06:02.090 ","End":"06:03.770","Text":"with lots and lots of experience,"},{"Start":"06:03.770 ","End":"06:05.060","Text":"you\u0027ll just remember these numbers,"},{"Start":"06:05.060 ","End":"06:10.700","Text":"but even then actually it\u0027s quite easy to mistake the decimal value for C and D,"},{"Start":"06:10.700 ","End":"06:15.260","Text":"say at the extremes where we\u0027ve got F and A they\u0027re very easy to remember,"},{"Start":"06:15.260 ","End":"06:20.530","Text":"and really the only 1 you do need to remember is this one here, 10 is A,"},{"Start":"06:20.530 ","End":"06:23.390","Text":"because all the others fall in sequence afterwards,"},{"Start":"06:23.390 ","End":"06:24.905","Text":"we know the numbers after 10,"},{"Start":"06:24.905 ","End":"06:27.410","Text":"are 11,12,13,14 and 15."},{"Start":"06:27.410 ","End":"06:31.230","Text":"We also know that any letters after A,"},{"Start":"06:31.230 ","End":"06:33.030","Text":"B, are C, D, E,"},{"Start":"06:33.030 ","End":"06:37.520","Text":"and F. That\u0027s why the symbols were chosen for hexadecimal,"},{"Start":"06:37.520 ","End":"06:41.120","Text":"they\u0027re are pattern that follows in sequence that everyone remembers."},{"Start":"06:41.120 ","End":"06:43.655","Text":"That was just a little aide memoir there."},{"Start":"06:43.655 ","End":"06:47.785","Text":"What we\u0027re going to do now is we\u0027re going to actually do the conversion into binary."},{"Start":"06:47.785 ","End":"06:51.080","Text":"16, 16s column,"},{"Start":"06:51.080 ","End":"06:52.115","Text":"remember it\u0027s a nibble,"},{"Start":"06:52.115 ","End":"06:58.070","Text":"again that we start labeling as 8,4,2,1."},{"Start":"06:58.070 ","End":"07:01.235","Text":"We do the same for that lower nibble,"},{"Start":"07:01.235 ","End":"07:04.220","Text":"so D is 13."},{"Start":"07:04.220 ","End":"07:06.730","Text":"That\u0027s where our reminder here helps us."},{"Start":"07:06.730 ","End":"07:10.385","Text":"We want to make 13 out of these columns."},{"Start":"07:10.385 ","End":"07:13.235","Text":"That\u0027s going to be 12 plus 1."},{"Start":"07:13.235 ","End":"07:18.110","Text":"I need 8 and 4 to make 12 and I need the 1."},{"Start":"07:18.110 ","End":"07:20.120","Text":"There\u0027s my upper nibble done."},{"Start":"07:20.120 ","End":"07:24.620","Text":"That is D in binary 1,1,0,1,"},{"Start":"07:24.620 ","End":"07:26.675","Text":"because that\u0027s 13,"},{"Start":"07:26.675 ","End":"07:29.030","Text":"and to make 9 in a similar way,"},{"Start":"07:29.030 ","End":"07:30.980","Text":"I need 8 and 1."},{"Start":"07:30.980 ","End":"07:34.655","Text":"I put 1 in that column and a 1 in that column."},{"Start":"07:34.655 ","End":"07:36.945","Text":"The others are zeros,"},{"Start":"07:36.945 ","End":"07:39.250","Text":"and I am done."},{"Start":"07:39.250 ","End":"07:43.895","Text":"That is the conversion to binary of D9,"},{"Start":"07:43.895 ","End":"07:45.440","Text":"this binary pattern here."},{"Start":"07:45.440 ","End":"07:49.085","Text":"Once again, if I want to check this, I can do."},{"Start":"07:49.085 ","End":"07:50.930","Text":"The way I would do that, first of all,"},{"Start":"07:50.930 ","End":"07:54.770","Text":"if we convert the binary into decimal,"},{"Start":"07:54.770 ","End":"07:59.195","Text":"and remember I need to make sure that these columns are not 1,2,4 and 8,"},{"Start":"07:59.195 ","End":"08:03.650","Text":"they were to represent this one single hexadecimal digit."},{"Start":"08:03.650 ","End":"08:08.675","Text":"But if I\u0027m considering this binary pattern here as 1 number,"},{"Start":"08:08.675 ","End":"08:12.035","Text":"then this one is not actually worth 1, it\u0027s worth 16."},{"Start":"08:12.035 ","End":"08:13.865","Text":"The 2 is 32,"},{"Start":"08:13.865 ","End":"08:15.855","Text":"the 4 is 64,"},{"Start":"08:15.855 ","End":"08:19.925","Text":"the 8 is 128 and it\u0027s just carrying on this sequence here, 1, 2, 4,"},{"Start":"08:19.925 ","End":"08:21.770","Text":"8,16, 32, 64,"},{"Start":"08:21.770 ","End":"08:25.490","Text":"128 doubling each time as we\u0027ve done previously."},{"Start":"08:25.490 ","End":"08:29.865","Text":"If I was to add up all these numbers here,"},{"Start":"08:29.865 ","End":"08:35.845","Text":"what I should find is I get 128 and 64,"},{"Start":"08:35.845 ","End":"08:38.710","Text":"skip the 32 because there\u0027s a 0 there,"},{"Start":"08:38.710 ","End":"08:43.730","Text":"16 and then I\u0027ve got 81 on the end there."},{"Start":"08:43.730 ","End":"08:46.115","Text":"If you add all that up,"},{"Start":"08:46.115 ","End":"08:50.680","Text":"you will get 217 in decimal."},{"Start":"08:50.680 ","End":"08:56.885","Text":"If we do the conversion from hexadecimal directly into decimal,"},{"Start":"08:56.885 ","End":"09:01.850","Text":"we know that what we need to do is we need to multiply by 16 D,"},{"Start":"09:01.850 ","End":"09:04.700","Text":"which again is 13 looking across here,"},{"Start":"09:04.700 ","End":"09:07.955","Text":"and then we need to add to that 9,"},{"Start":"09:07.955 ","End":"09:18.225","Text":"so16 times 13 is 160 plus 3 lots of 16, which is 208."},{"Start":"09:18.225 ","End":"09:24.550","Text":"Then we add to that 9 and we will get 217 again."},{"Start":"09:24.550 ","End":"09:30.020","Text":"Because these match up, we\u0027ve converted from hexadecimal directly to decimal."},{"Start":"09:30.020 ","End":"09:33.770","Text":"We\u0027ve converted from binary to decimal."},{"Start":"09:33.770 ","End":"09:37.760","Text":"We have now proved that this bit pattern is correct,"},{"Start":"09:37.760 ","End":"09:39.425","Text":"because we\u0027ve got the same number."},{"Start":"09:39.425 ","End":"09:45.545","Text":"That\u0027s this simple example done of converting numbers from hexadecimal to binary."},{"Start":"09:45.545 ","End":"09:47.419","Text":"In the next exercise,"},{"Start":"09:47.419 ","End":"09:53.140","Text":"we\u0027ll see some slightly longer hexadecimal numbers. I\u0027ll see you then."}],"ID":30080},{"Watched":false,"Name":"Exercise 3 parts C-D","Duration":"9m 44s","ChapterTopicVideoID":25446,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"Welcome back everyone. In this second part of the question,"},{"Start":"00:04.680 ","End":"00:09.015","Text":"we\u0027re going to convert slightly longer hexadecimal numbers to binary."},{"Start":"00:09.015 ","End":"00:12.885","Text":"The method is going to be exactly the same and the cross-check,"},{"Start":"00:12.885 ","End":"00:15.390","Text":"it\u0027s just going to be slightly more long-winded."},{"Start":"00:15.390 ","End":"00:19.320","Text":"Let\u0027s have a go at doing this and let\u0027s give ourselves plenty of space this time to"},{"Start":"00:19.320 ","End":"00:24.000","Text":"write out the individual hexadecimal digits."},{"Start":"00:24.000 ","End":"00:28.695","Text":"We\u0027ve got 1, F, and A, and as before,"},{"Start":"00:28.695 ","End":"00:34.180","Text":"we\u0027re going to treat each nibbles separately."},{"Start":"00:34.220 ","End":"00:38.010","Text":"Nibbles bits are worth 8, 4, 2,"},{"Start":"00:38.010 ","End":"00:40.695","Text":"and 1 for each individual nibble,"},{"Start":"00:40.695 ","End":"00:47.705","Text":"and we just need to work out which positions to include A1 and then put zeros elsewhere."},{"Start":"00:47.705 ","End":"00:52.295","Text":"For this first hexadecimal digit of 1, we don\u0027t need the 8,"},{"Start":"00:52.295 ","End":"00:54.745","Text":"the 4, the 2, we simply need the 1,"},{"Start":"00:54.745 ","End":"00:57.530","Text":"and therefore, we put zeros everywhere else."},{"Start":"00:57.530 ","End":"01:01.820","Text":"For F, we need to work out what F is first of all in decimal."},{"Start":"01:01.820 ","End":"01:04.760","Text":"Remember, last time what we did was we wrote down"},{"Start":"01:04.760 ","End":"01:10.460","Text":"a little cheat on the side to just remind us of what the digits were."},{"Start":"01:10.460 ","End":"01:15.050","Text":"If you write 10-15 down the side of the page,"},{"Start":"01:15.050 ","End":"01:17.615","Text":"and then next to each one of those,"},{"Start":"01:17.615 ","End":"01:21.140","Text":"just write down the hexadecimal symbol."},{"Start":"01:21.140 ","End":"01:24.395","Text":"That\u0027s the equivalent for that decimal number,"},{"Start":"01:24.395 ","End":"01:27.575","Text":"the letters A-F. As I said before,"},{"Start":"01:27.575 ","End":"01:32.925","Text":"all you really got to remember is 10 is A and the rest is following sequence."},{"Start":"01:32.925 ","End":"01:35.390","Text":"F, we see therefore it\u0027s 15."},{"Start":"01:35.390 ","End":"01:38.540","Text":"We\u0027ve got to make 15 up from these bit positions here,"},{"Start":"01:38.540 ","End":"01:41.690","Text":"and that\u0027s straightforward because it\u0027s all of the digits,"},{"Start":"01:41.690 ","End":"01:45.050","Text":"8 plus 4 plus 2 is 14, plus 1 is 15."},{"Start":"01:45.050 ","End":"01:47.615","Text":"We need a 1 everywhere."},{"Start":"01:47.615 ","End":"01:52.400","Text":"Then finally, for the final hexadecimal digits of A,"},{"Start":"01:52.400 ","End":"01:53.915","Text":"we need to make up 10,"},{"Start":"01:53.915 ","End":"01:56.180","Text":"10 is made from 8 and 2."},{"Start":"01:56.180 ","End":"02:00.195","Text":"We put ones there and zeros everywhere else,"},{"Start":"02:00.195 ","End":"02:04.550","Text":"and we have completed the conversion of"},{"Start":"02:04.550 ","End":"02:09.660","Text":"this hexadecimal number 1FA into its binary equivalent,"},{"Start":"02:09.660 ","End":"02:11.560","Text":"and that\u0027s the result."},{"Start":"02:11.560 ","End":"02:14.110","Text":"As before, we could do a cross-check here,"},{"Start":"02:14.110 ","End":"02:19.405","Text":"and this depends on whether you are allowed to have a calculator or not."},{"Start":"02:19.405 ","End":"02:23.500","Text":"Then what do you need to do is to consider that"},{"Start":"02:23.500 ","End":"02:28.840","Text":"these bits here for this lower nibble are going to remain exactly as they are."},{"Start":"02:28.840 ","End":"02:34.045","Text":"But actually, this next nibble on 1 over is not going to be 1 here."},{"Start":"02:34.045 ","End":"02:37.480","Text":"It\u0027s going to be the next in the sequence of powers of 2."},{"Start":"02:37.480 ","End":"02:39.350","Text":"It\u0027s going to be 16 there,"},{"Start":"02:39.350 ","End":"02:46.065","Text":"32 there, 64 there, and 128 there."},{"Start":"02:46.065 ","End":"02:50.510","Text":"There\u0027s the 64, there\u0027s the 16, here\u0027s the 32,"},{"Start":"02:50.510 ","End":"02:52.850","Text":"and there\u0027s the 128, and similarly,"},{"Start":"02:52.850 ","End":"02:56.360","Text":"we\u0027ll carry on the next is going to be 256."},{"Start":"02:56.360 ","End":"02:58.430","Text":"We don\u0027t need to go any further next."},{"Start":"02:58.430 ","End":"02:59.450","Text":"We\u0027ve got all zeros here,"},{"Start":"02:59.450 ","End":"03:00.770","Text":"but this would be 512,"},{"Start":"03:00.770 ","End":"03:03.920","Text":"1024, and 2048."},{"Start":"03:03.920 ","End":"03:05.810","Text":"To do our little cross-check,"},{"Start":"03:05.810 ","End":"03:10.010","Text":"then what we would need to do is to add anywhere where there is a 1."},{"Start":"03:10.010 ","End":"03:17.265","Text":"We\u0027ve got 256 and 128 and 64,"},{"Start":"03:17.265 ","End":"03:23.615","Text":"32, and 16 because the whole of that nibble there has got ones,"},{"Start":"03:23.615 ","End":"03:25.730","Text":"and then this low nibble here,"},{"Start":"03:25.730 ","End":"03:27.470","Text":"it\u0027s just the 8 and the 2."},{"Start":"03:27.470 ","End":"03:29.870","Text":"If we add all those together,"},{"Start":"03:29.870 ","End":"03:34.434","Text":"what we find is, we get 506."},{"Start":"03:34.434 ","End":"03:39.515","Text":"The decimal representation of this binary bit pattern,"},{"Start":"03:39.515 ","End":"03:41.825","Text":"and if we\u0027ve done it correctly,"},{"Start":"03:41.825 ","End":"03:46.865","Text":"this hexadecimal sequence is 506 in decimal."},{"Start":"03:46.865 ","End":"03:50.480","Text":"Once again, we can work out that directly from"},{"Start":"03:50.480 ","End":"03:54.890","Text":"the hexadecimal by considering that columns worth units."},{"Start":"03:54.890 ","End":"03:56.450","Text":"This is 16\u0027s,"},{"Start":"03:56.450 ","End":"03:58.895","Text":"and the next is 16 times 16,"},{"Start":"03:58.895 ","End":"04:02.660","Text":"which you didn\u0027t see in the previous examples. It\u0027s now 256."},{"Start":"04:02.660 ","End":"04:07.360","Text":"We should get if we did that conversion, 256,"},{"Start":"04:07.360 ","End":"04:09.480","Text":"because it\u0027s 1 lot of 256,"},{"Start":"04:09.480 ","End":"04:15.050","Text":"and 16 times 15 because F is 15."},{"Start":"04:15.050 ","End":"04:18.210","Text":"Let\u0027s put brackets around that just to make it clear."},{"Start":"04:18.210 ","End":"04:25.170","Text":"Then finally, we\u0027ve got 1 lot of 10,"},{"Start":"04:25.170 ","End":"04:27.160","Text":"because A is 10."},{"Start":"04:27.160 ","End":"04:32.075","Text":"That would give us 256"},{"Start":"04:32.075 ","End":"04:38.390","Text":"plus 16 times 15 is 240, and 10."},{"Start":"04:38.390 ","End":"04:40.400","Text":"If you add those out,"},{"Start":"04:40.400 ","End":"04:47.510","Text":"you get the same results we saw before, 506 in decimal."},{"Start":"04:47.510 ","End":"04:50.450","Text":"We know we\u0027ve done a conversion correctly,"},{"Start":"04:50.450 ","End":"04:52.220","Text":"and this bit pattern here,"},{"Start":"04:52.220 ","End":"04:54.980","Text":"0011, 1111,"},{"Start":"04:54.980 ","End":"05:00.455","Text":"1010, is the conversion of 1FA into binary."},{"Start":"05:00.455 ","End":"05:04.085","Text":"Continuing on with the next part, part D,"},{"Start":"05:04.085 ","End":"05:06.650","Text":"we\u0027ve got an even longer number,"},{"Start":"05:06.650 ","End":"05:10.625","Text":"so let\u0027s give ourselves even more spaces time to write these out."},{"Start":"05:10.625 ","End":"05:12.850","Text":"We\u0027ve got C, D,"},{"Start":"05:12.850 ","End":"05:16.685","Text":"D, and B."},{"Start":"05:16.685 ","End":"05:19.880","Text":"What we\u0027re looking at here is,"},{"Start":"05:19.880 ","End":"05:26.640","Text":"if I put the column values above each hexadecimal columns,"},{"Start":"05:26.640 ","End":"05:28.790","Text":"we\u0027ve got 256 here as before,"},{"Start":"05:28.790 ","End":"05:30.305","Text":"and this time over here,"},{"Start":"05:30.305 ","End":"05:35.405","Text":"we\u0027ve got 256 times 16, which is 4,096."},{"Start":"05:35.405 ","End":"05:38.570","Text":"But we\u0027ve actually just been asked to convert to binary."},{"Start":"05:38.570 ","End":"05:40.250","Text":"That\u0027s irrelevant in some ways."},{"Start":"05:40.250 ","End":"05:42.200","Text":"We\u0027re only going to use it for our cross-check."},{"Start":"05:42.200 ","End":"05:45.770","Text":"What we really are interested in is treating each one of"},{"Start":"05:45.770 ","End":"05:49.790","Text":"these as an independent nibble of 4-bits."},{"Start":"05:49.790 ","End":"05:55.145","Text":"We\u0027re going to write out the binary for each of those nibbles as we did before."},{"Start":"05:55.145 ","End":"05:56.525","Text":"Let\u0027s do that."},{"Start":"05:56.525 ","End":"05:58.620","Text":"To get C,"},{"Start":"05:58.620 ","End":"06:05.685","Text":"which is 12, we\u0027re going to need to add 8 and 4."},{"Start":"06:05.685 ","End":"06:07.875","Text":"Put ones there,"},{"Start":"06:07.875 ","End":"06:11.330","Text":"and the remaining parts of that nibble, we\u0027re going to be zeros."},{"Start":"06:11.330 ","End":"06:14.090","Text":"To get D, which is 13,"},{"Start":"06:14.090 ","End":"06:15.890","Text":"we\u0027ll need 12 plus 1."},{"Start":"06:15.890 ","End":"06:20.040","Text":"That\u0027s 12 and there\u0027s 1 there,"},{"Start":"06:20.040 ","End":"06:22.125","Text":"so it\u0027s just a 0 for the 2,"},{"Start":"06:22.125 ","End":"06:24.590","Text":"and obviously, the next symbol is a D as well."},{"Start":"06:24.590 ","End":"06:27.395","Text":"It\u0027s just a copy of the previous 1."},{"Start":"06:27.395 ","End":"06:30.380","Text":"Then finally B, which is 11,"},{"Start":"06:30.380 ","End":"06:33.975","Text":"and that\u0027s 8 and 2 and 1."},{"Start":"06:33.975 ","End":"06:36.245","Text":"Only the 4 has a 0,"},{"Start":"06:36.245 ","End":"06:39.870","Text":"everywhere else is a 1, and that\u0027s it."},{"Start":"06:39.870 ","End":"06:43.415","Text":"No more complex really then in all the other examples."},{"Start":"06:43.415 ","End":"06:46.565","Text":"The cross-check is going to be a lot more"},{"Start":"06:46.565 ","End":"06:51.260","Text":"long-winded and we can do exactly the same method as we had before,"},{"Start":"06:51.260 ","End":"06:55.280","Text":"but I can assure you that it all comes out correct in the end."},{"Start":"06:55.280 ","End":"07:00.605","Text":"But we would have to work that out if we didn\u0027t have a calculator by hand,"},{"Start":"07:00.605 ","End":"07:04.610","Text":"and it would be exactly the same here,"},{"Start":"07:04.610 ","End":"07:07.805","Text":"but we\u0027d obviously go much further in terms of the bit values."},{"Start":"07:07.805 ","End":"07:10.355","Text":"If I write those out for us,"},{"Start":"07:10.355 ","End":"07:13.340","Text":"here, we\u0027d still have 16."},{"Start":"07:13.340 ","End":"07:15.470","Text":"That would be 32,"},{"Start":"07:15.470 ","End":"07:17.740","Text":"64 over here,"},{"Start":"07:17.740 ","End":"07:24.200","Text":"128 here, 256 here, 512 there,"},{"Start":"07:24.200 ","End":"07:32.390","Text":"1,024 there, 2048 there, 4,096 there,"},{"Start":"07:32.390 ","End":"07:42.275","Text":"8,192 there, 16,384 there, and 32,768 there."},{"Start":"07:42.275 ","End":"07:47.600","Text":"What that will give us 32,768"},{"Start":"07:47.600 ","End":"07:54.290","Text":"plus 16,384 plus 2048,"},{"Start":"07:54.290 ","End":"07:58.005","Text":"1,024, 256,"},{"Start":"07:58.005 ","End":"08:03.330","Text":"128 and 64, and then 16."},{"Start":"08:03.330 ","End":"08:05.460","Text":"The remaining bits here,"},{"Start":"08:05.460 ","End":"08:06.990","Text":"which is 8 and 2 and 1,"},{"Start":"08:06.990 ","End":"08:09.045","Text":"which together will be 11."},{"Start":"08:09.045 ","End":"08:11.825","Text":"If we were to add all those together,"},{"Start":"08:11.825 ","End":"08:17.030","Text":"what we would get is 52,699."},{"Start":"08:17.030 ","End":"08:20.390","Text":"You\u0027re just going to have to trust me on that rather than us working it all out."},{"Start":"08:20.390 ","End":"08:25.685","Text":"That\u0027s the decimal conversion of this bit pattern here."},{"Start":"08:25.685 ","End":"08:30.770","Text":"If we were to multiply out these digits here,"},{"Start":"08:30.770 ","End":"08:37.875","Text":"you would have 4,096 times C, which is 12."},{"Start":"08:37.875 ","End":"08:45.600","Text":"Add onto that 256 times 13 because that\u0027s what D is,"},{"Start":"08:45.600 ","End":"08:50.960","Text":"and 16 times 13."},{"Start":"08:50.960 ","End":"08:52.685","Text":"Again, that\u0027s what D is."},{"Start":"08:52.685 ","End":"08:55.990","Text":"Then finally, we know the end is 11,"},{"Start":"08:55.990 ","End":"08:57.740","Text":"and again, trust me,"},{"Start":"08:57.740 ","End":"09:00.965","Text":"if we were to work all this out,"},{"Start":"09:00.965 ","End":"09:05.104","Text":"52,699 is what this would produce."},{"Start":"09:05.104 ","End":"09:07.850","Text":"That\u0027s the same as the figure over here."},{"Start":"09:07.850 ","End":"09:11.600","Text":"We have proven that our conversion went out correct."},{"Start":"09:11.600 ","End":"09:14.900","Text":"If you have access to a calculator on the exam,"},{"Start":"09:14.900 ","End":"09:16.655","Text":"this is all very straightforward."},{"Start":"09:16.655 ","End":"09:18.530","Text":"But if you don\u0027t,"},{"Start":"09:18.530 ","End":"09:22.610","Text":"then you can check your answer by using this method here,"},{"Start":"09:22.610 ","End":"09:25.070","Text":"which would be quite long-winded for a number this large,"},{"Start":"09:25.070 ","End":"09:26.650","Text":"but it can be done."},{"Start":"09:26.650 ","End":"09:31.490","Text":"We have once again converted from hexadecimal to binary,"},{"Start":"09:31.490 ","End":"09:33.110","Text":"much longer numbers this time,"},{"Start":"09:33.110 ","End":"09:35.315","Text":"but the process was exactly the same."},{"Start":"09:35.315 ","End":"09:38.060","Text":"We\u0027ve found a way to prove whether we\u0027ve got"},{"Start":"09:38.060 ","End":"09:41.060","Text":"the correct answer as well before we leave the exam."},{"Start":"09:41.060 ","End":"09:44.850","Text":"That\u0027s great. I\u0027ll see you in the next one."}],"ID":30081},{"Watched":false,"Name":"Exercise 4 parts A-B","Duration":"6m 37s","ChapterTopicVideoID":25437,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.980","Text":"Hello, welcome back everybody."},{"Start":"00:01.980 ","End":"00:08.055","Text":"In this exercise we\u0027ve been asked to convert a number from decimal to hexadecimal."},{"Start":"00:08.055 ","End":"00:12.720","Text":"We\u0027ve got 2 quite short numbers here, 240 and 251."},{"Start":"00:12.720 ","End":"00:15.600","Text":"Let\u0027s think about how we would do that."},{"Start":"00:15.600 ","End":"00:20.810","Text":"The first thing to do for small number I would suggest is to convert it to binary,"},{"Start":"00:20.810 ","End":"00:24.270","Text":"and it\u0027s good practice at this stage for us to just continue"},{"Start":"00:24.270 ","End":"00:29.010","Text":"on consolidating the things that we\u0027ve already covered in these exercises."},{"Start":"00:29.010 ","End":"00:31.250","Text":"This is quite helpful thing to do."},{"Start":"00:31.250 ","End":"00:34.790","Text":"Let\u0027s try and convert this then into binary first of all."},{"Start":"00:34.790 ","End":"00:40.659","Text":"In order to do that, we will write out the column values as we have done previously."},{"Start":"00:40.659 ","End":"00:46.040","Text":"We keep going until we don\u0027t need anymore digits."},{"Start":"00:46.040 ","End":"00:48.770","Text":"If I went to 1 more, that\u0027ll be 256."},{"Start":"00:48.770 ","End":"00:53.810","Text":"256 is bigger than 240 so I don\u0027t actually need to go that far."},{"Start":"00:53.810 ","End":"00:57.350","Text":"I can stop here, and now I just need to put ones"},{"Start":"00:57.350 ","End":"01:00.560","Text":"where I need the column and zeros where I don\u0027t need it."},{"Start":"01:00.560 ","End":"01:02.270","Text":"We\u0027ve done this many times before,"},{"Start":"01:02.270 ","End":"01:04.055","Text":"but I\u0027ll do it again."},{"Start":"01:04.055 ","End":"01:06.320","Text":"Look at the number 240."},{"Start":"01:06.320 ","End":"01:09.275","Text":"We do need the column for 128,"},{"Start":"01:09.275 ","End":"01:11.405","Text":"because 240 is bigger than 128."},{"Start":"01:11.405 ","End":"01:15.365","Text":"However, now we need to subtract the 128 off 240."},{"Start":"01:15.365 ","End":"01:17.990","Text":"Let\u0027s keep track of that up here."},{"Start":"01:17.990 ","End":"01:24.845","Text":"240 minus 128 would give us 112."},{"Start":"01:24.845 ","End":"01:29.000","Text":"I now need to decide if I need the 64 column,"},{"Start":"01:29.000 ","End":"01:32.330","Text":"which I do because 112 is bigger than 64."},{"Start":"01:32.330 ","End":"01:37.085","Text":"But again, subtract 64 off the 112,"},{"Start":"01:37.085 ","End":"01:43.130","Text":"and that would give me 48 and continue on."},{"Start":"01:43.130 ","End":"01:46.235","Text":"Do I need 32 column? Yes, I do."},{"Start":"01:46.235 ","End":"01:52.090","Text":"However, I\u0027ve got to subtract the 32 away from 48 now and I\u0027m left with 16."},{"Start":"01:52.090 ","End":"01:54.145","Text":"Do I need the 16 column?"},{"Start":"01:54.145 ","End":"01:56.560","Text":"Yes, I do. I include that."},{"Start":"01:56.560 ","End":"02:00.250","Text":"Now, however, I\u0027ve got nothing left to make up"},{"Start":"02:00.250 ","End":"02:04.645","Text":"because I\u0027ve accounted for the whole number with just these 4 bits."},{"Start":"02:04.645 ","End":"02:09.069","Text":"These remaining bits here are going to be 0."},{"Start":"02:09.069 ","End":"02:12.030","Text":"There we go. Now,"},{"Start":"02:12.030 ","End":"02:13.300","Text":"I\u0027ve got the number in binary."},{"Start":"02:13.300 ","End":"02:15.790","Text":"However, I was asked to convert it to hexadecimal,"},{"Start":"02:15.790 ","End":"02:19.120","Text":"but the reason I converted into binary first is it\u0027s"},{"Start":"02:19.120 ","End":"02:22.300","Text":"very straightforward converting from binary to"},{"Start":"02:22.300 ","End":"02:26.520","Text":"hexadecimal because we simply deal with 4 bits at a time and"},{"Start":"02:26.520 ","End":"02:31.800","Text":"nibble and we assign that 4-bits hexadecimal character,"},{"Start":"02:31.800 ","End":"02:33.875","Text":"and you\u0027ll remember before as well,"},{"Start":"02:33.875 ","End":"02:37.440","Text":"I said that it was worth writing at the side of your paper."},{"Start":"02:37.440 ","End":"02:41.940","Text":"The various digits from 10 upwards."},{"Start":"02:41.940 ","End":"02:44.535","Text":"If we do that, 10, 11,"},{"Start":"02:44.535 ","End":"02:49.425","Text":"12, 13, 14, 15."},{"Start":"02:49.425 ","End":"02:53.965","Text":"Next to them, just writing the hexadecimal characters,"},{"Start":"02:53.965 ","End":"02:56.525","Text":"A to F in sequence,"},{"Start":"02:56.525 ","End":"02:58.610","Text":"just so that we don\u0027t make a silly mistake."},{"Start":"02:58.610 ","End":"03:03.140","Text":"In this case, we can see that we need to make up these 4 ones."},{"Start":"03:03.140 ","End":"03:05.450","Text":"However, we don\u0027t look at them as 128,"},{"Start":"03:05.450 ","End":"03:07.460","Text":"64, 32, and 16."},{"Start":"03:07.460 ","End":"03:13.330","Text":"We renumber these because these are independent hexadecimal digits,"},{"Start":"03:13.330 ","End":"03:15.340","Text":"each of which is numbered 8,"},{"Start":"03:15.340 ","End":"03:17.290","Text":"4, 2, and 1."},{"Start":"03:17.290 ","End":"03:20.770","Text":"In order to represent this as a hexadecimal digits,"},{"Start":"03:20.770 ","End":"03:22.840","Text":"it\u0027s the 8 and the 4 and the 2 and the 1,"},{"Start":"03:22.840 ","End":"03:24.400","Text":"which would be 15,"},{"Start":"03:24.400 ","End":"03:28.345","Text":"and so 15, we know is F. Therefore,"},{"Start":"03:28.345 ","End":"03:32.439","Text":"that hexadecimal character should be F because that\u0027s 15."},{"Start":"03:32.439 ","End":"03:36.010","Text":"The lower 4 bits are going to be 0,"},{"Start":"03:36.010 ","End":"03:39.765","Text":"because 0 is 0 in hexadecimal, as well as decimal."},{"Start":"03:39.765 ","End":"03:43.050","Text":"We\u0027ve actually got our answer now,"},{"Start":"03:43.050 ","End":"03:48.735","Text":"so 240 in decimal is F0 in hexadecimal."},{"Start":"03:48.735 ","End":"03:50.685","Text":"Let\u0027s move on to the second part now."},{"Start":"03:50.685 ","End":"03:54.665","Text":"251, we\u0027re going to use exactly the same method gives us in practice."},{"Start":"03:54.665 ","End":"04:03.529","Text":"Once again, I\u0027m going to write out the column values 128 because if I go any further,"},{"Start":"04:03.529 ","End":"04:06.650","Text":"I\u0027ll have 256 and this number is less than 256,"},{"Start":"04:06.650 ","End":"04:07.880","Text":"so I need to go that far."},{"Start":"04:07.880 ","End":"04:11.074","Text":"Once again, I\u0027ll use the same method as before,"},{"Start":"04:11.074 ","End":"04:14.630","Text":"converting to a binary number first and then looking at each"},{"Start":"04:14.630 ","End":"04:19.160","Text":"nibble and converting that nibble to a hexadecimal character."},{"Start":"04:19.160 ","End":"04:21.530","Text":"First things first, 251,"},{"Start":"04:21.530 ","End":"04:23.600","Text":"is it greater than 128? Yes, it is."},{"Start":"04:23.600 ","End":"04:25.115","Text":"I need the 128-column,"},{"Start":"04:25.115 ","End":"04:30.065","Text":"but I need to subtract from 251, 128."},{"Start":"04:30.065 ","End":"04:32.300","Text":"I know what\u0027s left to make up,"},{"Start":"04:32.300 ","End":"04:34.715","Text":"and that will be 123."},{"Start":"04:34.715 ","End":"04:37.550","Text":"Do I need the 64? Of course, I do."},{"Start":"04:37.550 ","End":"04:43.300","Text":"Now, I take away 64 and that will give me 59."},{"Start":"04:43.300 ","End":"04:45.450","Text":"Is 59 greater than 32?"},{"Start":"04:45.450 ","End":"04:48.330","Text":"Of course, it is. I need the 32 as well."},{"Start":"04:48.330 ","End":"04:51.055","Text":"Let\u0027s subtract 32 then."},{"Start":"04:51.055 ","End":"04:54.230","Text":"That would give me 27,"},{"Start":"04:54.230 ","End":"04:59.330","Text":"so 27 is obviously greater than 16. Also, I need the 16."},{"Start":"04:59.330 ","End":"05:03.400","Text":"Subtract the 16, then I\u0027ve got 11 left to make up."},{"Start":"05:03.400 ","End":"05:06.440","Text":"I need to make 11 from these remaining bits,"},{"Start":"05:06.440 ","End":"05:09.410","Text":"I can carry on through that procedure was obviously very easy for me to spot"},{"Start":"05:09.410 ","End":"05:13.295","Text":"now that 11 is made from 8, 2, and 1."},{"Start":"05:13.295 ","End":"05:15.605","Text":"I\u0027ll just put ones there."},{"Start":"05:15.605 ","End":"05:17.480","Text":"8, 2,"},{"Start":"05:17.480 ","End":"05:20.315","Text":"and 1, and a 0 there."},{"Start":"05:20.315 ","End":"05:26.230","Text":"I\u0027ve now got the binary number that is the equivalent of this decimal number,"},{"Start":"05:26.230 ","End":"05:28.530","Text":"but I need to convert it into hexadecimal."},{"Start":"05:28.530 ","End":"05:30.140","Text":"Is the question asked for that."},{"Start":"05:30.140 ","End":"05:32.690","Text":"Again, I know already what 1,"},{"Start":"05:32.690 ","End":"05:33.950","Text":"1, 1, 1 is."},{"Start":"05:33.950 ","End":"05:36.860","Text":"I don\u0027t use these headings here."},{"Start":"05:36.860 ","End":"05:39.170","Text":"I use the headings of 8,"},{"Start":"05:39.170 ","End":"05:40.365","Text":"4, 2,"},{"Start":"05:40.365 ","End":"05:42.200","Text":"and 1 because I treat this nibble,"},{"Start":"05:42.200 ","End":"05:44.790","Text":"is completely separate from this nibble."},{"Start":"05:44.920 ","End":"05:48.800","Text":"8 plus 4 plus 2 plus 1 is 15 as we previously found,"},{"Start":"05:48.800 ","End":"05:53.465","Text":"and 15 is F. This character should be F,"},{"Start":"05:53.465 ","End":"05:55.640","Text":"as in the previous answer."},{"Start":"05:55.640 ","End":"06:00.470","Text":"This time now we\u0027ve got 8 plus 2 plus 1, which is 11."},{"Start":"06:00.470 ","End":"06:03.370","Text":"Eleven is B."},{"Start":"06:03.370 ","End":"06:06.295","Text":"There is our final answer."},{"Start":"06:06.295 ","End":"06:08.690","Text":"That was a very straightforward process."},{"Start":"06:08.690 ","End":"06:11.300","Text":"We convert it to binary, first of all,"},{"Start":"06:11.300 ","End":"06:14.765","Text":"from the decimal number using method that we\u0027ve already used previously."},{"Start":"06:14.765 ","End":"06:16.730","Text":"Then once we had the binary value,"},{"Start":"06:16.730 ","End":"06:20.300","Text":"we could treat it as 2 nibbles, completely separate nibbles,"},{"Start":"06:20.300 ","End":"06:28.135","Text":"and give each nibble a hexadecimal character from a column values for that nibble."},{"Start":"06:28.135 ","End":"06:31.670","Text":"That gave us the final answer in hexadecimal."},{"Start":"06:31.670 ","End":"06:33.200","Text":"In the next exercise, we\u0027ll do"},{"Start":"06:33.200 ","End":"06:38.190","Text":"slightly longer examples and I\u0027ll show you a slightly different method. See you then."}],"ID":30082},{"Watched":false,"Name":"Exercise 4 part C","Duration":"11m 6s","ChapterTopicVideoID":25447,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.665","Text":"Hey everyone, welcome back."},{"Start":"00:01.665 ","End":"00:05.310","Text":"In these questions we\u0027ve been trying to convert decimal to hexadecimal,"},{"Start":"00:05.310 ","End":"00:09.960","Text":"and we\u0027ve had 2 very short numbers for the first 2 parts of this question."},{"Start":"00:09.960 ","End":"00:11.940","Text":"The third one is a longer number,"},{"Start":"00:11.940 ","End":"00:14.790","Text":"so we\u0027re going to use a slightly different method than we used before,"},{"Start":"00:14.790 ","End":"00:19.680","Text":"where we convert it to binary first and then each group of 4-bits,"},{"Start":"00:19.680 ","End":"00:21.780","Text":"we convert it to a hexadecimal digit."},{"Start":"00:21.780 ","End":"00:23.130","Text":"We\u0027re still going to do the same thing,"},{"Start":"00:23.130 ","End":"00:24.960","Text":"we\u0027re not going to use that method."},{"Start":"00:24.960 ","End":"00:28.410","Text":"What I\u0027d like to do is to assume,"},{"Start":"00:28.410 ","End":"00:31.845","Text":"first of all, you have a calculator and if you do, it\u0027s very straightforward."},{"Start":"00:31.845 ","End":"00:35.220","Text":"I\u0027ll give you an example of a hexadecimal number,"},{"Start":"00:35.220 ","End":"00:38.790","Text":"let\u0027s just pick a random number, say 201."},{"Start":"00:38.790 ","End":"00:41.720","Text":"The way we are expressing this number is"},{"Start":"00:41.720 ","End":"00:44.690","Text":"that there are different column values for each digit."},{"Start":"00:44.690 ","End":"00:46.760","Text":"The first one is the units,"},{"Start":"00:46.760 ","End":"00:50.435","Text":"the next column over is 16 times the units."},{"Start":"00:50.435 ","End":"00:53.210","Text":"The next one is 16 times 16,"},{"Start":"00:53.210 ","End":"00:55.070","Text":"which would be 256."},{"Start":"00:55.070 ","End":"00:56.930","Text":"If we did one another column,"},{"Start":"00:56.930 ","End":"01:01.880","Text":"that would actually be 16 times to 256, which is 4096."},{"Start":"01:01.880 ","End":"01:05.510","Text":"For this example, I\u0027ve just made up, you don\u0027t need that."},{"Start":"01:05.510 ","End":"01:13.200","Text":"What we\u0027re actually saying here is this number is equivalent to 2 lots of 256,"},{"Start":"01:13.200 ","End":"01:16.440","Text":"no lots of 16 because we\u0027ve got 0 here,"},{"Start":"01:16.440 ","End":"01:18.615","Text":"and then 1 lot of 1."},{"Start":"01:18.615 ","End":"01:22.755","Text":"The final answer would be 512 plus 1,"},{"Start":"01:22.755 ","End":"01:24.585","Text":"which is 513,"},{"Start":"01:24.585 ","End":"01:27.530","Text":"so 201 in hexadecimal,"},{"Start":"01:27.530 ","End":"01:30.785","Text":"would be 513 in decimal."},{"Start":"01:30.785 ","End":"01:34.460","Text":"We\u0027re going to try and do the same process for this,"},{"Start":"01:34.460 ","End":"01:36.080","Text":"but the way we would do x,"},{"Start":"01:36.080 ","End":"01:37.700","Text":"we\u0027re obviously doing it in reverse."},{"Start":"01:37.700 ","End":"01:39.620","Text":"I made up a number here,"},{"Start":"01:39.620 ","End":"01:42.725","Text":"and showed you its decimal equivalent."},{"Start":"01:42.725 ","End":"01:46.685","Text":"What we actually want to do is take a decimal and turn it into a hexadecimal."},{"Start":"01:46.685 ","End":"01:49.340","Text":"I\u0027ll show you the process that we need to adopt there,"},{"Start":"01:49.340 ","End":"01:52.685","Text":"and I\u0027ll just clear this off before we start."},{"Start":"01:52.685 ","End":"01:58.070","Text":"What we need to do is we need to divide this number by 256."},{"Start":"01:58.070 ","End":"02:00.050","Text":"I know that the number doesn\u0027t need"},{"Start":"02:00.050 ","End":"02:03.995","Text":"more than 3 columns because if I had 1 in this column,"},{"Start":"02:03.995 ","End":"02:06.140","Text":"it will be 4,096, which is obviously"},{"Start":"02:06.140 ","End":"02:08.765","Text":"much bigger than the number we\u0027re trying to go for here."},{"Start":"02:08.765 ","End":"02:11.015","Text":"I\u0027m going to start with the 256 column,"},{"Start":"02:11.015 ","End":"02:17.035","Text":"because I know a few multiples of this will make more than enough for 1100."},{"Start":"02:17.035 ","End":"02:22.580","Text":"What I\u0027m going to do is I\u0027m going to take 1100 and into my calculator,"},{"Start":"02:22.580 ","End":"02:28.430","Text":"I\u0027m going to divide it by 256 and work out what the answer is."},{"Start":"02:28.430 ","End":"02:33.290","Text":"You\u0027ll find that it\u0027s 4.29687,"},{"Start":"02:33.290 ","End":"02:38.090","Text":"but I\u0027m not interested in the remainder part and the fractional part,"},{"Start":"02:38.090 ","End":"02:39.720","Text":"I just want to know the integer part."},{"Start":"02:39.720 ","End":"02:42.190","Text":"I\u0027m looking to do an integer division here,"},{"Start":"02:42.190 ","End":"02:44.590","Text":"so I\u0027m just interested in the fall."},{"Start":"02:44.590 ","End":"02:48.635","Text":"That gives me the figure I need to write down here,"},{"Start":"02:48.635 ","End":"02:52.065","Text":"so for lots of 256."},{"Start":"02:52.065 ","End":"02:57.950","Text":"Now, I also want to know what\u0027s left over to still make up the remaining columns,"},{"Start":"02:57.950 ","End":"03:02.840","Text":"and so the way I do that is it should be a button on your calculator"},{"Start":"03:02.840 ","End":"03:08.120","Text":"that allows you to work out the remainder when you divide 2 numbers, called modulo."},{"Start":"03:08.120 ","End":"03:11.950","Text":"If you did 1100 modulo 256,"},{"Start":"03:11.950 ","End":"03:17.600","Text":"you divide 256 into 1100 and you\u0027d get a remainder,"},{"Start":"03:17.600 ","End":"03:20.180","Text":"which should actually be 76."},{"Start":"03:20.180 ","End":"03:21.620","Text":"If you turn that into your calculator,"},{"Start":"03:21.620 ","End":"03:23.405","Text":"you\u0027ll find it\u0027s 76."},{"Start":"03:23.405 ","End":"03:28.310","Text":"My aim now is to reproduce 76 from these 2 columns,"},{"Start":"03:28.310 ","End":"03:30.710","Text":"and so how would I go ahead and do that?"},{"Start":"03:30.710 ","End":"03:33.740","Text":"Well, I\u0027ve got 16 columns and I\u0027ve got a 1 column,"},{"Start":"03:33.740 ","End":"03:38.270","Text":"so how many lots of 16 do I need for my 76?"},{"Start":"03:38.270 ","End":"03:40.535","Text":"Well, I could work that out by again doing"},{"Start":"03:40.535 ","End":"03:44.570","Text":"an integer division 76 divided by 16 this time,"},{"Start":"03:44.570 ","End":"03:53.925","Text":"and you\u0027ll find that that will give you the integer part of 76 divided by 16 is 4 again."},{"Start":"03:53.925 ","End":"03:56.885","Text":"If you were to do 76,"},{"Start":"03:56.885 ","End":"04:03.290","Text":"modulo 16, discovered that the remainder would be 12."},{"Start":"04:03.290 ","End":"04:05.035","Text":"This 4 here,"},{"Start":"04:05.035 ","End":"04:08.070","Text":"it needs to grow up into my 16s column,"},{"Start":"04:08.070 ","End":"04:11.000","Text":"and I\u0027ve just got to now find out this final part,"},{"Start":"04:11.000 ","End":"04:13.955","Text":"the 12 and put it in my 1s column."},{"Start":"04:13.955 ","End":"04:18.245","Text":"You\u0027ll remember that when we did our conversions before,"},{"Start":"04:18.245 ","End":"04:26.240","Text":"we noted down all of the numbers from 10-15 on the side of our paper"},{"Start":"04:26.240 ","End":"04:29.840","Text":"just to make it easier for ourselves and to make sure we didn\u0027t make"},{"Start":"04:29.840 ","End":"04:35.210","Text":"a silly mistake transposing 1 hexadecimal character for another."},{"Start":"04:35.210 ","End":"04:37.040","Text":"We go A to F like that."},{"Start":"04:37.040 ","End":"04:41.030","Text":"So 12 is what we\u0027re looking to create in that final column,"},{"Start":"04:41.030 ","End":"04:48.540","Text":"and 12 is the hexadecimal character C. That\u0027s us done 4,"},{"Start":"04:48.540 ","End":"04:53.515","Text":"4, C is the answer to this question."},{"Start":"04:53.515 ","End":"04:56.470","Text":"It\u0027s very straightforward if you have a calculator,"},{"Start":"04:56.470 ","End":"04:58.510","Text":"you work out how many columns you will need,"},{"Start":"04:58.510 ","End":"05:01.190","Text":"how many groups of 4-bits, essentially?"},{"Start":"05:01.190 ","End":"05:02.800","Text":"Once you\u0027ve worked that out,"},{"Start":"05:02.800 ","End":"05:09.550","Text":"you divide the value for the column into the number and take the remainder,"},{"Start":"05:09.550 ","End":"05:11.320","Text":"and then just keep doing"},{"Start":"05:11.320 ","End":"05:15.100","Text":"that process with a remainder until there\u0027s nothing left to make up,"},{"Start":"05:15.100 ","End":"05:17.845","Text":"and you will have your hexadecimal conversion."},{"Start":"05:17.845 ","End":"05:22.735","Text":"That\u0027s great, but what if we aren\u0027t able to use a calculator in the exam,"},{"Start":"05:22.735 ","End":"05:28.450","Text":"what methods can we use to come up with the hexadecimal conversion?"},{"Start":"05:28.450 ","End":"05:32.390","Text":"Well, actually what we can do is we can work"},{"Start":"05:32.390 ","End":"05:36.815","Text":"out all the multiples of 256 and similarly for 16."},{"Start":"05:36.815 ","End":"05:43.235","Text":"Using that, we could easily do the same operation we\u0027ve just done without a calculator."},{"Start":"05:43.235 ","End":"05:47.615","Text":"Again, I would recommend you write on the side of your paper the multiples,"},{"Start":"05:47.615 ","End":"05:53.065","Text":"so let\u0027s just write down the numbers from 256 upwards."},{"Start":"05:53.065 ","End":"05:55.020","Text":"You\u0027ve got 256,"},{"Start":"05:55.020 ","End":"05:57.690","Text":"add 256 to that you get 512,"},{"Start":"05:57.690 ","End":"06:01.785","Text":"add 256 to that, you get 768."},{"Start":"06:01.785 ","End":"06:05.550","Text":"Another 256, you get 124,"},{"Start":"06:05.550 ","End":"06:11.080","Text":"another 256, and you get 1280."},{"Start":"06:11.080 ","End":"06:12.890","Text":"I don\u0027t need to go any further than that,"},{"Start":"06:12.890 ","End":"06:17.120","Text":"because the number I\u0027m looking for 1100 is obviously,"},{"Start":"06:17.120 ","End":"06:18.200","Text":"less than 1280,"},{"Start":"06:18.200 ","End":"06:19.700","Text":"so I don\u0027t need to go any further."},{"Start":"06:19.700 ","End":"06:22.595","Text":"But let\u0027s write down what the multiples are."},{"Start":"06:22.595 ","End":"06:24.990","Text":"Just write down the numbers 1, 2,"},{"Start":"06:24.990 ","End":"06:27.875","Text":"3, 4, however many are going up to."},{"Start":"06:27.875 ","End":"06:34.330","Text":"We know that 4 lots of 256 is 1024,"},{"Start":"06:34.330 ","End":"06:37.325","Text":"so when we\u0027re going to do our conversion,"},{"Start":"06:37.325 ","End":"06:42.485","Text":"we are going to work across the number as we did before."},{"Start":"06:42.485 ","End":"06:47.675","Text":"Let\u0027s write the column values in red as we did previously."},{"Start":"06:47.675 ","End":"06:53.355","Text":"Let\u0027s say we know that that column is 496, that one\u0027s 256."},{"Start":"06:53.355 ","End":"06:55.730","Text":"This one is 16, and this is 1."},{"Start":"06:55.730 ","End":"06:57.920","Text":"We know we don\u0027t need to use this column,"},{"Start":"06:57.920 ","End":"07:00.995","Text":"so if you want, I\u0027ll put a 0 there just to indicate that."},{"Start":"07:00.995 ","End":"07:05.260","Text":"But we know that we will need 4 lots of 256,"},{"Start":"07:05.260 ","End":"07:08.230","Text":"so we\u0027ll write our 4 down here."},{"Start":"07:08.230 ","End":"07:11.060","Text":"We\u0027ve effectively done this division bar up"},{"Start":"07:11.060 ","End":"07:14.660","Text":"here simply by working out what the multiples of"},{"Start":"07:14.660 ","End":"07:22.280","Text":"256 and reading across to find that it\u0027s actually 4 is a multiple of 256,"},{"Start":"07:22.280 ","End":"07:23.885","Text":"that will give us 1024."},{"Start":"07:23.885 ","End":"07:25.550","Text":"Now we\u0027ve got our 1024,"},{"Start":"07:25.550 ","End":"07:27.735","Text":"we do as we did previously,"},{"Start":"07:27.735 ","End":"07:31.880","Text":"when we were doing decimal to binary conversions."},{"Start":"07:31.880 ","End":"07:34.865","Text":"We say, what was the number we started off with?"},{"Start":"07:34.865 ","End":"07:38.090","Text":"That was 1100."},{"Start":"07:38.090 ","End":"07:43.670","Text":"What have we now created that we don\u0027t need to make up anymore."},{"Start":"07:43.670 ","End":"07:50.075","Text":"That\u0027s 1024, if you were to subtract 1024 from 1100,"},{"Start":"07:50.075 ","End":"07:55.320","Text":"by whatever means, you would get 76 as the result."},{"Start":"07:55.320 ","End":"08:01.065","Text":"So 76 is what we\u0027ve got to make up from these remaining 2 columns."},{"Start":"08:01.065 ","End":"08:06.170","Text":"Again, we can work out a little table of multiples of 16,"},{"Start":"08:06.170 ","End":"08:08.630","Text":"and that\u0027s obviously, very straightforward."},{"Start":"08:08.630 ","End":"08:11.155","Text":"We just write down 16,"},{"Start":"08:11.155 ","End":"08:13.995","Text":"and then what 16 plus 16, 32,"},{"Start":"08:13.995 ","End":"08:17.385","Text":"the next one is going to be 48."},{"Start":"08:17.385 ","End":"08:28.185","Text":"Then we\u0027ve got 64 and to that we\u0027ve got 80 and to that we\u0027ve got 96."},{"Start":"08:28.185 ","End":"08:33.435","Text":"And then it\u0027d be 112, 128,"},{"Start":"08:33.435 ","End":"08:38.865","Text":"144, 160,176,"},{"Start":"08:38.865 ","End":"08:41.630","Text":"and I could just keep going for as long as I needed to."},{"Start":"08:41.630 ","End":"08:45.260","Text":"In this case, we\u0027re only looking to make the number 76."},{"Start":"08:45.260 ","End":"08:47.330","Text":"I don\u0027t need to go any further than 76,"},{"Start":"08:47.330 ","End":"08:53.030","Text":"but a useful little check-in is to just see if I\u0027ve got my multiples right."},{"Start":"08:53.030 ","End":"08:55.055","Text":"1, 2, 3,"},{"Start":"08:55.055 ","End":"08:56.720","Text":"4, 5,"},{"Start":"08:56.720 ","End":"08:58.340","Text":"6, 7,"},{"Start":"08:58.340 ","End":"08:59.640","Text":"8, 9,"},{"Start":"08:59.640 ","End":"09:01.050","Text":"when I get to 10,"},{"Start":"09:01.050 ","End":"09:04.800","Text":"obviously, know that 16 times 10 is just to stick a 0 on the end."},{"Start":"09:04.800 ","End":"09:06.650","Text":"I have indeed got 160,"},{"Start":"09:06.650 ","End":"09:08.045","Text":"here is the tenth number,"},{"Start":"09:08.045 ","End":"09:09.655","Text":"but let\u0027s not write it as 10,"},{"Start":"09:09.655 ","End":"09:13.400","Text":"let\u0027s write it as A and then B and so on."},{"Start":"09:13.400 ","End":"09:15.590","Text":"As a seller, I\u0027m going to go any further anyway,"},{"Start":"09:15.590 ","End":"09:18.740","Text":"but it\u0027s just useful to check that we\u0027ve"},{"Start":"09:18.740 ","End":"09:22.385","Text":"got the right number of multiples of 16 in the right order."},{"Start":"09:22.385 ","End":"09:24.680","Text":"Let\u0027s go back to what we were looking to create,"},{"Start":"09:24.680 ","End":"09:27.055","Text":"we were looking to find 76,"},{"Start":"09:27.055 ","End":"09:31.710","Text":"76 is not an exact power of 16,"},{"Start":"09:31.710 ","End":"09:38.555","Text":"so what we are going to find is somewhere between 64 and 80 is our number."},{"Start":"09:38.555 ","End":"09:40.895","Text":"That means 4,"},{"Start":"09:40.895 ","End":"09:42.572","Text":"so I\u0027m 16,"},{"Start":"09:42.572 ","End":"09:46.805","Text":"64, so we need 4 to go here,"},{"Start":"09:46.805 ","End":"09:54.305","Text":"and then if we subtract 64 from 76,"},{"Start":"09:54.305 ","End":"09:56.405","Text":"we know what we\u0027ve got left to make up,"},{"Start":"09:56.405 ","End":"09:57.995","Text":"which is 12,"},{"Start":"09:57.995 ","End":"10:01.040","Text":"and that will go into the final column here."},{"Start":"10:01.040 ","End":"10:04.120","Text":"We know already that 12 is C,"},{"Start":"10:04.120 ","End":"10:08.205","Text":"so we write our C here and we\u0027re done,"},{"Start":"10:08.205 ","End":"10:12.860","Text":"and it\u0027s the same result as we had when we use the calculator."},{"Start":"10:12.860 ","End":"10:14.975","Text":"We know we\u0027ve done the right job,"},{"Start":"10:14.975 ","End":"10:18.350","Text":"so that was just a way of doing a decimal to"},{"Start":"10:18.350 ","End":"10:23.195","Text":"hexadecimal conversion using a calculator or without a calculator,"},{"Start":"10:23.195 ","End":"10:27.590","Text":"just by looking at the powers of 256 or the powers of 16 and so on."},{"Start":"10:27.590 ","End":"10:34.400","Text":"Then, in fact, I should say multiples of 256 or multiples of 16,"},{"Start":"10:34.400 ","End":"10:37.130","Text":"and that works great."},{"Start":"10:37.130 ","End":"10:40.340","Text":"We could have also used the method that we did previously,"},{"Start":"10:40.340 ","End":"10:43.685","Text":"which is to have a 1 long binary number,"},{"Start":"10:43.685 ","End":"10:50.030","Text":"which we break up into 4-bit nibbles and just work out the binary first of all,"},{"Start":"10:50.030 ","End":"10:54.455","Text":"and then take each 4-bit nibble and convert that to hexadecimal."},{"Start":"10:54.455 ","End":"10:57.680","Text":"It\u0027s arguable which one would be quicker than the other?"},{"Start":"10:57.680 ","End":"10:59.870","Text":"But this is just another way of doing it."},{"Start":"10:59.870 ","End":"11:02.390","Text":"We\u0027ve had to look at that now and we\u0027ll do"},{"Start":"11:02.390 ","End":"11:06.450","Text":"a similar example in the next video. See you then."}],"ID":30083},{"Watched":false,"Name":"Exercise 4 part D","Duration":"7m 42s","ChapterTopicVideoID":25435,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.389","Text":"Hello, again. In this exercise,"},{"Start":"00:03.389 ","End":"00:07.200","Text":"we\u0027re going to continue on doing decimal to hexadecimal conversions."},{"Start":"00:07.200 ","End":"00:10.620","Text":"We\u0027ve got a fairly large and slightly strange looking number,"},{"Start":"00:10.620 ","End":"00:15.344","Text":"25432, doesn\u0027t look like it\u0027s particularly convenient."},{"Start":"00:15.344 ","End":"00:18.600","Text":"The first job we need to do is to work out"},{"Start":"00:18.600 ","End":"00:21.735","Text":"how many hexadecimal digits we\u0027re going to need."},{"Start":"00:21.735 ","End":"00:27.030","Text":"If it helps, then you can write out the positions,"},{"Start":"00:27.030 ","End":"00:29.385","Text":"the first one being 1,"},{"Start":"00:29.385 ","End":"00:36.450","Text":"the next one being 16, the next one being 256, and then 4096."},{"Start":"00:36.450 ","End":"00:38.480","Text":"We\u0027re looking at that number and we\u0027re saying,"},{"Start":"00:38.480 ","End":"00:42.064","Text":"can I do it with this number of columns?"},{"Start":"00:42.064 ","End":"00:44.000","Text":"The next one over by the way,"},{"Start":"00:44.000 ","End":"00:49.490","Text":"16 lots of 4096 would be 65536."},{"Start":"00:49.490 ","End":"00:53.810","Text":"That\u0027s clearly much bigger than 25432, so we don\u0027t need it."},{"Start":"00:53.810 ","End":"00:57.950","Text":"It looks like all we need is 4 columns for this particular number."},{"Start":"00:57.950 ","End":"01:01.295","Text":"I\u0027m assuming again that you have access to a calculator."},{"Start":"01:01.295 ","End":"01:07.055","Text":"If you do, then we start by looking at how many multiples of 4096 we need,"},{"Start":"01:07.055 ","End":"01:15.345","Text":"so we\u0027ll do an integer division of 25432 and we divide that by 4096,"},{"Start":"01:15.345 ","End":"01:25.545","Text":"and your calculator should tell you that what you need there is 6 lots of 4096,"},{"Start":"01:25.545 ","End":"01:29.570","Text":"so the digit we are going to put here is a 6,"},{"Start":"01:29.570 ","End":"01:33.370","Text":"a hexadecimal digit, and then we can proceed on,"},{"Start":"01:33.370 ","End":"01:35.195","Text":"but I need to know what\u0027s left to make up."},{"Start":"01:35.195 ","End":"01:38.245","Text":"The way I do that is I take the modulo,"},{"Start":"01:38.245 ","End":"01:43.670","Text":"4096 and that will give me what\u0027s left to make up,"},{"Start":"01:43.670 ","End":"01:46.880","Text":"and that is 856."},{"Start":"01:46.880 ","End":"01:49.685","Text":"If you type that into your calculator, you should find."},{"Start":"01:49.685 ","End":"01:53.510","Text":"We\u0027ve got to make 856 now from these remaining columns,"},{"Start":"01:53.510 ","End":"01:59.465","Text":"so 856 divided by 256."},{"Start":"01:59.465 ","End":"02:04.865","Text":"If you type that into your calculator you will find it\u0027s 3 and a remainder."},{"Start":"02:04.865 ","End":"02:10.295","Text":"We\u0027re going to write a 3 in this column because that\u0027s how many to 256s we need."},{"Start":"02:10.295 ","End":"02:15.545","Text":"We now need to work out what\u0027s left to make up in the remaining columns."},{"Start":"02:15.545 ","End":"02:21.390","Text":"We\u0027ll do 856 modulo 256 to see what\u0027s left."},{"Start":"02:21.390 ","End":"02:26.375","Text":"If you type that into your calculator you\u0027ll find it\u0027s 88."},{"Start":"02:26.375 ","End":"02:33.590","Text":"Now, what we want to do is take 88 and divide it by 16 and see how many we get."},{"Start":"02:33.590 ","End":"02:36.215","Text":"Again, that\u0027s an integer division."},{"Start":"02:36.215 ","End":"02:39.985","Text":"88 divided by 16 gives us 5."},{"Start":"02:39.985 ","End":"02:42.615","Text":"We need to write that 5 over here."},{"Start":"02:42.615 ","End":"02:44.495","Text":"We need 5 lots of 16."},{"Start":"02:44.495 ","End":"02:48.290","Text":"Then the last thing we need to do is work out the units column,"},{"Start":"02:48.290 ","End":"02:53.220","Text":"which we\u0027re going to get by taking the modulo."},{"Start":"02:53.220 ","End":"02:55.830","Text":"What you should find is,"},{"Start":"02:55.830 ","End":"02:57.915","Text":"that is 8,"},{"Start":"02:57.915 ","End":"03:04.145","Text":"so our remaining digit is 8 and we have completed"},{"Start":"03:04.145 ","End":"03:11.870","Text":"the conversion of 25432 decimal to hexadecimal,"},{"Start":"03:11.870 ","End":"03:14.615","Text":"which should give us 6358."},{"Start":"03:14.615 ","End":"03:18.365","Text":"As before, if you don\u0027t have a calculator,"},{"Start":"03:18.365 ","End":"03:21.410","Text":"we can still do what we need to do,"},{"Start":"03:21.410 ","End":"03:26.105","Text":"we just need to work out the multiples for each of the columns that we\u0027re looking at."},{"Start":"03:26.105 ","End":"03:31.660","Text":"If we start that for 4096,"},{"Start":"03:31.660 ","End":"03:35.355","Text":"the next one up would be 8192,"},{"Start":"03:35.355 ","End":"03:42.645","Text":"add 4096 to that and you\u0027d get 12288."},{"Start":"03:42.645 ","End":"03:44.820","Text":"Add 4096 to that,"},{"Start":"03:44.820 ","End":"03:48.990","Text":"you\u0027d get 16384,"},{"Start":"03:48.990 ","End":"03:54.615","Text":"add 4096 to that and you\u0027d get 20480,"},{"Start":"03:54.615 ","End":"03:57.025","Text":"add 4096 to that,"},{"Start":"03:57.025 ","End":"04:02.345","Text":"you would get 24576."},{"Start":"04:02.345 ","End":"04:08.780","Text":"We don\u0027t need to go any further because if we added another 4,000-ish onto that,"},{"Start":"04:08.780 ","End":"04:14.120","Text":"we\u0027d be up to the 28,000 mark and we know our number is 25,"},{"Start":"04:14.120 ","End":"04:15.710","Text":"so we don\u0027t need to go any further."},{"Start":"04:15.710 ","End":"04:18.215","Text":"If we write down the multiples next to it,"},{"Start":"04:18.215 ","End":"04:21.060","Text":"1, 2, 3, 4, 5, 6,"},{"Start":"04:22.300 ","End":"04:30.075","Text":"we know that if we were to divide this by 4096,"},{"Start":"04:30.075 ","End":"04:33.300","Text":"we wouldn\u0027t need to go beyond 6."},{"Start":"04:33.300 ","End":"04:38.140","Text":"Therefore, the first digit we\u0027re looking for in our answer would be 6."},{"Start":"04:38.140 ","End":"04:40.540","Text":"Let\u0027s write that down there."},{"Start":"04:40.540 ","End":"04:46.410","Text":"But we now have accounted for 24576 worth,"},{"Start":"04:46.410 ","End":"04:49.915","Text":"so we need to work out how much is left to make up."},{"Start":"04:49.915 ","End":"04:51.805","Text":"If I write that down here,"},{"Start":"04:51.805 ","End":"04:56.155","Text":"25432 is what we started with."},{"Start":"04:56.155 ","End":"05:00.925","Text":"We\u0027ve already got 24576 worth."},{"Start":"05:00.925 ","End":"05:07.500","Text":"What would be leftover if you subtract those 2 numbers out would be 856."},{"Start":"05:07.500 ","End":"05:12.300","Text":"So 856 is what we\u0027ve got to make up from the remaining columns."},{"Start":"05:12.300 ","End":"05:17.075","Text":"I\u0027m going to write out the column headings here just to make this clearer,"},{"Start":"05:17.075 ","End":"05:19.165","Text":"just as we\u0027ve done above."},{"Start":"05:19.165 ","End":"05:22.220","Text":"856 is what we\u0027ve got to make from these remaining columns."},{"Start":"05:22.220 ","End":"05:27.840","Text":"I\u0027ll need now to see how many times I\u0027ll need a 256."},{"Start":"05:27.840 ","End":"05:33.920","Text":"Effectively, I\u0027m going to need to do the same thing here as a table multiples of 256."},{"Start":"05:33.920 ","End":"05:37.025","Text":"If I write those out as far as I need to go,"},{"Start":"05:37.025 ","End":"05:41.930","Text":"we start with 256 and we\u0027ve got 512,"},{"Start":"05:41.930 ","End":"05:44.015","Text":"we\u0027ve got 768,"},{"Start":"05:44.015 ","End":"05:46.540","Text":"then we\u0027ve got 1024."},{"Start":"05:46.540 ","End":"05:49.130","Text":"Actually, I don\u0027t need to go any further than that because"},{"Start":"05:49.130 ","End":"05:52.070","Text":"856 is the number I\u0027m looking to make up."},{"Start":"05:52.070 ","End":"05:55.390","Text":"Let\u0027s just write down the multiples next to him."},{"Start":"05:55.390 ","End":"05:58.920","Text":"If I were to go to 4, it\u0027d be too big,"},{"Start":"05:58.920 ","End":"06:02.730","Text":"so the nearest value is 3."},{"Start":"06:02.730 ","End":"06:10.275","Text":"I\u0027ll get a 3 over here and subtract the 768,"},{"Start":"06:10.275 ","End":"06:12.645","Text":"see what\u0027s left to make up."},{"Start":"06:12.645 ","End":"06:19.090","Text":"If you do 856 minus 768 you will find it\u0027s 88."},{"Start":"06:19.090 ","End":"06:22.925","Text":"I\u0027ve got to make 88 from these remaining 2 columns."},{"Start":"06:22.925 ","End":"06:27.200","Text":"Once again, I can do multiples of 16."},{"Start":"06:27.200 ","End":"06:29.300","Text":"If I write those down here,"},{"Start":"06:29.300 ","End":"06:32.400","Text":"16, 32,"},{"Start":"06:32.400 ","End":"06:36.090","Text":"48, 64,"},{"Start":"06:36.090 ","End":"06:40.235","Text":"the next one\u0027s 80, next one is 96."},{"Start":"06:40.235 ","End":"06:42.020","Text":"But I only need to make 88,"},{"Start":"06:42.020 ","End":"06:44.150","Text":"so I don\u0027t need to go any further."},{"Start":"06:44.150 ","End":"06:46.859","Text":"These multiple 0, 1, 2, 3, 4, 5,"},{"Start":"06:46.859 ","End":"06:51.810","Text":"6, so the nearest 1 is 5."},{"Start":"06:51.810 ","End":"06:56.765","Text":"I\u0027m going to write my 5 here, and that was 80."},{"Start":"06:56.765 ","End":"07:04.220","Text":"Let\u0027s say that we\u0027ve accounted for 80 of the 88 and what\u0027s left to make up is just 8."},{"Start":"07:04.220 ","End":"07:06.770","Text":"I\u0027ve got only 1 column left and"},{"Start":"07:06.770 ","End":"07:09.785","Text":"the only value I need to write in there is what\u0027s left to make up,"},{"Start":"07:09.785 ","End":"07:12.065","Text":"which is 8, and I\u0027m done."},{"Start":"07:12.065 ","End":"07:18.445","Text":"It\u0027s obviously exactly the same as what we worked out when we had the calculator method."},{"Start":"07:18.445 ","End":"07:22.490","Text":"We\u0027ve once again converted from decimal to hexadecimal,"},{"Start":"07:22.490 ","End":"07:25.550","Text":"both with a calculator and without a calculator."},{"Start":"07:25.550 ","End":"07:27.590","Text":"Obviously, way easier with a calculator,"},{"Start":"07:27.590 ","End":"07:29.959","Text":"but it\u0027s not too bad without the calculator,"},{"Start":"07:29.959 ","End":"07:31.940","Text":"you work out the multiples."},{"Start":"07:31.940 ","End":"07:34.220","Text":"If it\u0027s not a massive multiple,"},{"Start":"07:34.220 ","End":"07:37.970","Text":"you don\u0027t have to go very far and then you write out the relevant column value."},{"Start":"07:37.970 ","End":"07:43.080","Text":"We\u0027ll do one more example like this in the next video."}],"ID":30084},{"Watched":false,"Name":"Exercise 4 part E","Duration":"5m 54s","ChapterTopicVideoID":25436,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"Welcome back everyone for this final part of"},{"Start":"00:03.600 ","End":"00:08.775","Text":"the exercise on decimal to hexadecimal conversion."},{"Start":"00:08.775 ","End":"00:11.385","Text":"We\u0027ve got a bigger number again,"},{"Start":"00:11.385 ","End":"00:15.030","Text":"and perhaps sometimes you might wonder how you\u0027re going"},{"Start":"00:15.030 ","End":"00:19.395","Text":"to know how many digits you\u0027re going to need, hexadecimal digits."},{"Start":"00:19.395 ","End":"00:21.090","Text":"We really need to solve,"},{"Start":"00:21.090 ","End":"00:24.630","Text":"have an understanding of powers of 16 here."},{"Start":"00:24.630 ","End":"00:32.250","Text":"If we write out the number 16^0, 16^1,"},{"Start":"00:32.270 ","End":"00:39.570","Text":"16^2, 16^3, and 16^4,"},{"Start":"00:39.570 ","End":"00:41.643","Text":"let\u0027s just start out with those,"},{"Start":"00:41.643 ","End":"00:44.930","Text":"we can type those numbers into our calculator and"},{"Start":"00:44.930 ","End":"00:48.740","Text":"work out what the various column values will be."},{"Start":"00:48.740 ","End":"00:50.615","Text":"This one\u0027s going to be 1."},{"Start":"00:50.615 ","End":"00:52.930","Text":"We know this one\u0027s going to be 16,"},{"Start":"00:52.930 ","End":"00:57.850","Text":"256, we\u0027ve seen all of these already, 4,096."},{"Start":"00:57.850 ","End":"00:59.825","Text":"You might remember from one of them."},{"Start":"00:59.825 ","End":"01:04.855","Text":"The next one up would be 65,536."},{"Start":"01:04.855 ","End":"01:07.700","Text":"It answers the question, when do we know to"},{"Start":"01:07.700 ","End":"01:11.255","Text":"stop while we look at the number that we are trying to make,"},{"Start":"01:11.255 ","End":"01:15.275","Text":"and we consider that if we went to an extra column,"},{"Start":"01:15.275 ","End":"01:16.610","Text":"would that be necessary,"},{"Start":"01:16.610 ","End":"01:17.660","Text":"would it not be necessary."},{"Start":"01:17.660 ","End":"01:21.485","Text":"In this case, if we went to 16^5,"},{"Start":"01:21.485 ","End":"01:25.700","Text":"that would actually be a very large number,"},{"Start":"01:25.700 ","End":"01:28.945","Text":"which works out to be this."},{"Start":"01:28.945 ","End":"01:32.690","Text":"That\u0027s obviously way larger than a number we are trying to make."},{"Start":"01:32.690 ","End":"01:35.270","Text":"In fact, if you look at this by inspection,"},{"Start":"01:35.270 ","End":"01:38.510","Text":"we can see that if we double this number here,"},{"Start":"01:38.510 ","End":"01:43.140","Text":"we\u0027d be in the right ballpark for this number."},{"Start":"01:43.760 ","End":"01:49.190","Text":"In fact, if I were to double this number,"},{"Start":"01:49.190 ","End":"01:54.635","Text":"I would get a number 131,072."},{"Start":"01:54.635 ","End":"01:58.860","Text":"That\u0027s obviously bigger than this number here."},{"Start":"01:58.860 ","End":"02:03.155","Text":"So, I know that I don\u0027t need to go any further than this column here."},{"Start":"02:03.155 ","End":"02:06.290","Text":"If I have the benefit of a calculator,"},{"Start":"02:06.290 ","End":"02:09.039","Text":"this is exactly as we\u0027ve done before,"},{"Start":"02:09.039 ","End":"02:13.620","Text":"we take the number 112,000,"},{"Start":"02:16.180 ","End":"02:20.075","Text":"we divide it by, in this case,"},{"Start":"02:20.075 ","End":"02:25.719","Text":"65,536, and we obtain this digit here."},{"Start":"02:25.719 ","End":"02:30.544","Text":"It would give us 1 as an integer part and then a remainder."},{"Start":"02:30.544 ","End":"02:35.400","Text":"The remainder obviously we find by taking 112,000,"},{"Start":"02:35.410 ","End":"02:41.555","Text":"and the Modulo of 65,536."},{"Start":"02:41.555 ","End":"02:44.420","Text":"If we put that into our calculator,"},{"Start":"02:44.420 ","End":"02:53.600","Text":"we get 46,464."},{"Start":"02:53.600 ","End":"02:54.860","Text":"We\u0027ve got our first digit,"},{"Start":"02:54.860 ","End":"02:59.650","Text":"which is 1, and that\u0027s going to go up here."},{"Start":"02:59.650 ","End":"03:03.885","Text":"Let\u0027s carry on then with 46,464."},{"Start":"03:03.885 ","End":"03:08.400","Text":"This time we\u0027re looking to the 4,096 column,"},{"Start":"03:08.400 ","End":"03:11.715","Text":"so we divide by 4,096."},{"Start":"03:11.715 ","End":"03:16.130","Text":"You will find that the integer division of that is 11."},{"Start":"03:16.130 ","End":"03:19.215","Text":"So 11, we can\u0027t write in one column,"},{"Start":"03:19.215 ","End":"03:23.280","Text":"that is of course the hexadecimal character B."},{"Start":"03:23.280 ","End":"03:26.010","Text":"Let\u0027s write B in this column,"},{"Start":"03:26.010 ","End":"03:30.500","Text":"and let\u0027s find out what we\u0027ve got left to make up by"},{"Start":"03:30.500 ","End":"03:37.265","Text":"taking 46,464 Modulo 496."},{"Start":"03:37.265 ","End":"03:43.325","Text":"That will give us a remainder of 1,408."},{"Start":"03:43.325 ","End":"03:46.790","Text":"Carrying on then, we\u0027ve got"},{"Start":"03:46.790 ","End":"03:53.585","Text":"1,408 which we now divide by 256 because that\u0027s the next column alone."},{"Start":"03:53.585 ","End":"03:57.260","Text":"That will give us 5."},{"Start":"03:57.260 ","End":"04:01.405","Text":"Let\u0027s record our 5 up here."},{"Start":"04:01.405 ","End":"04:04.050","Text":"Then we obviously need to find the modulo."},{"Start":"04:04.050 ","End":"04:08.970","Text":"1,408 Modulo"},{"Start":"04:08.970 ","End":"04:16.680","Text":"256 gives us 128,"},{"Start":"04:16.680 ","End":"04:20.655","Text":"and 128 is a nice convenient number, actually."},{"Start":"04:20.655 ","End":"04:24.875","Text":"If we find out by doing the next column,"},{"Start":"04:24.875 ","End":"04:31.845","Text":"128 divided by 16 is actually exactly 8."},{"Start":"04:31.845 ","End":"04:33.880","Text":"We\u0027re going to put the 8 up here."},{"Start":"04:33.880 ","End":"04:41.790","Text":"But if you want to prove that by doing 128 Modulo 16,"},{"Start":"04:41.790 ","End":"04:48.350","Text":"you should get 0 and that is our final digit up here."},{"Start":"04:48.350 ","End":"04:53.360","Text":"So, the answer to our question is 1, B, 5, 8,"},{"Start":"04:53.360 ","End":"05:01.640","Text":"0 in hexadecimal is the equivalent of 112,000 in decimal."},{"Start":"05:01.640 ","End":"05:04.190","Text":"Now, if you were unfortunate enough to get"},{"Start":"05:04.190 ","End":"05:07.260","Text":"this question and not be allowed to use a calculator,"},{"Start":"05:07.260 ","End":"05:09.800","Text":"you can obviously use the method that we did before."},{"Start":"05:09.800 ","End":"05:15.020","Text":"We\u0027d have to find multiples of 65,536 and go as far as we need to,"},{"Start":"05:15.020 ","End":"05:20.825","Text":"multiples of 4096 and likewise multiples of 256, and multiples of 16."},{"Start":"05:20.825 ","End":"05:24.335","Text":"Then using the method on the previous exercises,"},{"Start":"05:24.335 ","End":"05:27.680","Text":"we could work out what we need for each column"},{"Start":"05:27.680 ","End":"05:31.760","Text":"until we have done all the columns and we\u0027d have a result."},{"Start":"05:31.760 ","End":"05:32.960","Text":"I\u0027m not going to do that in"},{"Start":"05:32.960 ","End":"05:36.250","Text":"this particular video because we\u0027ve seen the examples in the previous too."},{"Start":"05:36.250 ","End":"05:41.539","Text":"I\u0027d actually be very surprised if you go on this long to do without a calculator,"},{"Start":"05:41.539 ","End":"05:43.175","Text":"but I guess it could happen."},{"Start":"05:43.175 ","End":"05:48.140","Text":"That\u0027s it for the decimal to hexadecimal conversion questions."},{"Start":"05:48.140 ","End":"05:54.570","Text":"We\u0027ll move on to a new exercise shortly. I\u0027ll see you in the next one."}],"ID":30085},{"Watched":false,"Name":"Exercise 5 parts A-B","Duration":"3m 19s","ChapterTopicVideoID":25438,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.430","Text":"In this question, we\u0027re being asked"},{"Start":"00:02.430 ","End":"00:06.555","Text":"when the particular representations might be most useful,"},{"Start":"00:06.555 ","End":"00:11.325","Text":"and the keyword here I think is to whom it might be useful,"},{"Start":"00:11.325 ","End":"00:17.070","Text":"and we are seeing here that the programmer is the person we\u0027re interested in."},{"Start":"00:17.070 ","End":"00:20.040","Text":"In this situation, we\u0027re really talking about"},{"Start":"00:20.040 ","End":"00:25.245","Text":"the programmer in contrast to the user of the program."},{"Start":"00:25.245 ","End":"00:31.275","Text":"User of the program would be more familiar or humans would be with the decimal system."},{"Start":"00:31.275 ","End":"00:37.595","Text":"So why am I a programmer who want to use a different representation than a normal user?"},{"Start":"00:37.595 ","End":"00:41.420","Text":"Well, we know that binary is more useful to the machine,"},{"Start":"00:41.420 ","End":"00:45.505","Text":"so to a CPU or to a circuit."},{"Start":"00:45.505 ","End":"00:48.770","Text":"That should form the first part of our answer."},{"Start":"00:48.770 ","End":"00:55.990","Text":"We\u0027re seeing here that we\u0027re dealing with a bit or group of bits,"},{"Start":"00:55.990 ","End":"01:01.190","Text":"and that\u0027s an appropriate situation for binary representation."},{"Start":"01:01.190 ","End":"01:05.255","Text":"We then adding to the answer an example,"},{"Start":"01:05.255 ","End":"01:10.144","Text":"a suitable example would be a CPU status register."},{"Start":"01:10.144 ","End":"01:11.870","Text":"Any register really in the CPU."},{"Start":"01:11.870 ","End":"01:17.060","Text":"But that\u0027s a particularly useful situation where you might want to use"},{"Start":"01:17.060 ","End":"01:23.035","Text":"bit representation or network address as well."},{"Start":"01:23.035 ","End":"01:26.110","Text":"Network addresses are often masked with"},{"Start":"01:26.110 ","End":"01:31.450","Text":"binary values to remove certain ranges from the address."},{"Start":"01:31.450 ","End":"01:35.110","Text":"That would be another classic situation where you might want to"},{"Start":"01:35.110 ","End":"01:39.055","Text":"use a binary representation rather than decimal or anything else."},{"Start":"01:39.055 ","End":"01:43.975","Text":"The next part of the question then leads onto us thinking about why"},{"Start":"01:43.975 ","End":"01:50.395","Text":"hexadecimal would be more suited in particular situation for the programmer again."},{"Start":"01:50.395 ","End":"01:55.000","Text":"If we think about one of the characteristics of hexadecimal is"},{"Start":"01:55.000 ","End":"01:59.455","Text":"that 4 bits can be represented by a single hexadecimal digit,"},{"Start":"01:59.455 ","End":"02:04.875","Text":"and that gives us a clue as to what we might want to put in our answer."},{"Start":"02:04.875 ","End":"02:06.610","Text":"In the first part of our answer,"},{"Start":"02:06.610 ","End":"02:12.385","Text":"we could talk about the fact that anytime we\u0027ve got a quite lengthy number,"},{"Start":"02:12.385 ","End":"02:16.284","Text":"it would be fairly inconvenient to write that out as binary."},{"Start":"02:16.284 ","End":"02:19.600","Text":"If the number could be grouped as nibbles,"},{"Start":"02:19.600 ","End":"02:24.130","Text":"then it\u0027s a situation where hexadecimal is particularly useful."},{"Start":"02:24.130 ","End":"02:27.295","Text":"We know that 4 bits is a nibble,"},{"Start":"02:27.295 ","End":"02:32.770","Text":"and therefore, a single hexadecimal digit can represent that nibble."},{"Start":"02:32.770 ","End":"02:37.060","Text":"An example would be 24 bit color value,"},{"Start":"02:37.060 ","End":"02:43.015","Text":"which would need 24 binary digits if we put it as a binary value."},{"Start":"02:43.015 ","End":"02:47.420","Text":"However, if we wrote it out as hexadecimal,"},{"Start":"02:47.420 ","End":"02:51.740","Text":"we just need 6 hexadecimal digits in order to represent"},{"Start":"02:51.740 ","End":"02:56.210","Text":"exactly the same number as a 24 bit binary value,"},{"Start":"02:56.210 ","End":"02:59.960","Text":"which would obviously be 24 binary digits much longer."},{"Start":"02:59.960 ","End":"03:03.545","Text":"There we can see the advantage really of hexadecimal."},{"Start":"03:03.545 ","End":"03:07.325","Text":"That is then the final answer for both parts of this question,"},{"Start":"03:07.325 ","End":"03:10.250","Text":"and in the final question in this topic,"},{"Start":"03:10.250 ","End":"03:14.540","Text":"we\u0027ll move on in the next video in looking at the implications"},{"Start":"03:14.540 ","End":"03:20.340","Text":"of different numbers being encoded in the different representations."}],"ID":30086},{"Watched":false,"Name":"Exercise 6","Duration":"5m 4s","ChapterTopicVideoID":25439,"CourseChapterTopicPlaylistID":217268,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.960","Text":"Hello, everyone. In this final exercise we are going to look at a question where it asks"},{"Start":"00:06.960 ","End":"00:09.930","Text":"us to compare 3 numbers in"},{"Start":"00:09.930 ","End":"00:13.860","Text":"different bases and state which is the largest of those numbers."},{"Start":"00:13.860 ","End":"00:15.345","Text":"One of them is in decimal,"},{"Start":"00:15.345 ","End":"00:16.875","Text":"one is in hexidecimal,"},{"Start":"00:16.875 ","End":"00:18.210","Text":"and one is in octal,"},{"Start":"00:18.210 ","End":"00:19.800","Text":"which you don\u0027t come across much,"},{"Start":"00:19.800 ","End":"00:22.275","Text":"but we\u0027ll cover it anyway."},{"Start":"00:22.275 ","End":"00:28.349","Text":"The simplest thing to do here is to put them all into a common base."},{"Start":"00:28.349 ","End":"00:33.245","Text":"I would suggest that given that one that we are all familiar with is decimal,"},{"Start":"00:33.245 ","End":"00:35.520","Text":"is we put them into the base of decimal."},{"Start":"00:35.520 ","End":"00:38.360","Text":"But we could put them into any base and then"},{"Start":"00:38.360 ","End":"00:41.915","Text":"compare them and we work out which one was the largest that way."},{"Start":"00:41.915 ","End":"00:43.790","Text":"So let\u0027s stick with decimals."},{"Start":"00:43.790 ","End":"00:46.790","Text":"We already know what 57 is in decimal."},{"Start":"00:46.790 ","End":"00:53.795","Text":"So let\u0027s go to 39 hexadecimal and work out what that is in decimal."},{"Start":"00:53.795 ","End":"00:57.020","Text":"We know that this first digit is"},{"Start":"00:57.020 ","End":"01:01.385","Text":"the ones column and then the column over is the 16ths column."},{"Start":"01:01.385 ","End":"01:07.595","Text":"Therefore, this number 39 in hexadecimal would be in decimal,"},{"Start":"01:07.595 ","End":"01:12.540","Text":"3 lots of 16 added onto 1 lot of 9,"},{"Start":"01:12.540 ","End":"01:17.455","Text":"and that would give us 57, 48 plus 9."},{"Start":"01:17.455 ","End":"01:25.040","Text":"Turns out that\u0027s exactly the same as the decimal number that we\u0027re comparing it to."},{"Start":"01:25.040 ","End":"01:26.435","Text":"So it\u0027s not bigger."},{"Start":"01:26.435 ","End":"01:29.630","Text":"It\u0027s the same as 57 decimal."},{"Start":"01:29.630 ","End":"01:31.780","Text":"So we know that this one is not the largest."},{"Start":"01:31.780 ","End":"01:33.230","Text":"Let\u0027s look at the last one now,"},{"Start":"01:33.230 ","End":"01:36.785","Text":"which is 71 in octal."},{"Start":"01:36.785 ","End":"01:45.730","Text":"Octal is a number base with only 8 available digits from 0-7."},{"Start":"01:45.730 ","End":"01:48.980","Text":"The reason it\u0027s called octal is"},{"Start":"01:48.980 ","End":"01:54.615","Text":"because 8 is the highest number of combinations obviously."},{"Start":"01:54.615 ","End":"01:57.075","Text":"So it\u0027s based on the number 8."},{"Start":"01:57.075 ","End":"02:01.770","Text":"If we want to convert an octal number into decimal,"},{"Start":"02:01.770 ","End":"02:07.835","Text":"we simply take the first digit and we add that to 8 times the second digit."},{"Start":"02:07.835 ","End":"02:10.265","Text":"So this one is worth 8,"},{"Start":"02:10.265 ","End":"02:11.900","Text":"this one is worth 1,"},{"Start":"02:11.900 ","End":"02:14.565","Text":"and if we put that into decimal,"},{"Start":"02:14.565 ","End":"02:17.745","Text":"we\u0027ll get 8 times 7 plus 1,"},{"Start":"02:17.745 ","End":"02:19.590","Text":"and that\u0027s 56 plus 1,"},{"Start":"02:19.590 ","End":"02:21.495","Text":"which, surprise, surprise,"},{"Start":"02:21.495 ","End":"02:25.455","Text":"is 57 in decimal."},{"Start":"02:25.455 ","End":"02:31.670","Text":"We have got 3 different numbers and it turns out they\u0027re all exactly the same number,"},{"Start":"02:31.670 ","End":"02:33.740","Text":"which is 57 decimal."},{"Start":"02:33.740 ","End":"02:37.940","Text":"So I suppose the point of this exercise would just be to"},{"Start":"02:37.940 ","End":"02:42.995","Text":"highlight that the underlying number is the important thing."},{"Start":"02:42.995 ","End":"02:47.090","Text":"How we represent it is up to us and it depends on"},{"Start":"02:47.090 ","End":"02:53.885","Text":"the circumstances which way or which number base we would choose to represent it as."},{"Start":"02:53.885 ","End":"02:57.274","Text":"The number base we haven\u0027t used here is binary."},{"Start":"02:57.274 ","End":"03:00.890","Text":"So if we wanted to, we could convert this into binary as well."},{"Start":"03:00.890 ","End":"03:02.435","Text":"I\u0027ll go ahead and do that."},{"Start":"03:02.435 ","End":"03:05.210","Text":"Let\u0027s take the decimal number and turn it into"},{"Start":"03:05.210 ","End":"03:08.855","Text":"binary in the fashion that we know already."},{"Start":"03:08.855 ","End":"03:11.780","Text":"We just put the bit values and then we"},{"Start":"03:11.780 ","End":"03:15.530","Text":"decide whether we need the digit or we don\u0027t need the digit."},{"Start":"03:15.530 ","End":"03:19.420","Text":"So it\u0027s going to be 48 plus 9,"},{"Start":"03:19.420 ","End":"03:23.065","Text":"which would be this bit pattern."},{"Start":"03:23.065 ","End":"03:29.959","Text":"That\u0027s also then going to be equivalent to 57 in decimal."},{"Start":"03:29.959 ","End":"03:33.470","Text":"So it doesn\u0027t matter how we represent it,"},{"Start":"03:33.470 ","End":"03:36.395","Text":"whether we represent it as a binary number,"},{"Start":"03:36.395 ","End":"03:39.650","Text":"an octal number, a hexadecimal number,"},{"Start":"03:39.650 ","End":"03:41.855","Text":"or a decimal number."},{"Start":"03:41.855 ","End":"03:43.460","Text":"They\u0027re all the same number."},{"Start":"03:43.460 ","End":"03:48.020","Text":"In this case, the representation depends on your circumstances. What\u0027s most useful."},{"Start":"03:48.020 ","End":"03:52.615","Text":"The machine is going to be storing in a circuit in"},{"Start":"03:52.615 ","End":"03:58.840","Text":"RAM this representation in binary bits of 1s and 0s."},{"Start":"03:58.840 ","End":"04:02.705","Text":"So that\u0027s the best representation to use if we\u0027re looking at the machine level."},{"Start":"04:02.705 ","End":"04:06.335","Text":"If we were looking at things in groups of 3 bits,"},{"Start":"04:06.335 ","End":"04:10.190","Text":"then octal would be a more sensible representation,"},{"Start":"04:10.190 ","End":"04:15.275","Text":"because octal uses groups of 3 bits for each of the octal digits."},{"Start":"04:15.275 ","End":"04:18.440","Text":"If we were doing it in groups of 4 bits,"},{"Start":"04:18.440 ","End":"04:22.220","Text":"then hexadecimal would be a more appropriate representation."},{"Start":"04:22.220 ","End":"04:24.080","Text":"So 4 bits being a nibble."},{"Start":"04:24.080 ","End":"04:26.630","Text":"If we want to look at things in terms of nibbles like we do,"},{"Start":"04:26.630 ","End":"04:29.360","Text":"for example, when we\u0027re talking about hexadecimal numbers,"},{"Start":"04:29.360 ","End":"04:32.495","Text":"RGB is often a good way of using hexadecimal,"},{"Start":"04:32.495 ","End":"04:37.130","Text":"because red, green, and blue values are 8 bits each generally."},{"Start":"04:37.130 ","End":"04:40.475","Text":"So you can divide that into 2 lots of 4 bits."},{"Start":"04:40.475 ","End":"04:44.615","Text":"So RGB tends to be a way that hexadecimal is used."},{"Start":"04:44.615 ","End":"04:48.440","Text":"But the number underlying it all is the same and"},{"Start":"04:48.440 ","End":"04:53.015","Text":"obviously humans would prefer to see it in the decimal form."},{"Start":"04:53.015 ","End":"04:56.900","Text":"That\u0027s it for all the exercises in this first chapter,"},{"Start":"04:56.900 ","End":"04:59.675","Text":"and we\u0027ll move on to build on"},{"Start":"04:59.675 ","End":"05:04.860","Text":"all the concepts we\u0027ve learned here in the next chapter. So I\u0027ll see you then."}],"ID":30087}],"Thumbnail":null,"ID":217268},{"Name":"Signed and twos complement","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Signed Numbers","Duration":"5m 5s","ChapterTopicVideoID":28715,"CourseChapterTopicPlaylistID":237670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.225","Text":"Hello and welcome to this video on signed numbers."},{"Start":"00:03.225 ","End":"00:04.470","Text":"By the end of this section,"},{"Start":"00:04.470 ","End":"00:08.160","Text":"you\u0027ll be able to distinguish between signed and unsigned integers,"},{"Start":"00:08.160 ","End":"00:12.795","Text":"interpret binary numbers in sign and magnitude or two\u0027s complement format,"},{"Start":"00:12.795 ","End":"00:17.003","Text":"convert a signed binary integer into its decimal equivalent,"},{"Start":"00:17.003 ","End":"00:21.730","Text":"and perform calculations in binary with signed numbers."},{"Start":"00:21.800 ","End":"00:25.934","Text":"Whilst developing our understanding of number systems previously,"},{"Start":"00:25.934 ","End":"00:31.573","Text":"the decimal values we used in examples were for simplicity sake, all fairly limited."},{"Start":"00:31.573 ","End":"00:33.230","Text":"They were whole numbers."},{"Start":"00:33.230 ","End":"00:38.045","Text":"We didn\u0027t use numbers with a fractional part or negative numbers."},{"Start":"00:38.045 ","End":"00:40.610","Text":"We\u0027ll continue to work with whole numbers for now,"},{"Start":"00:40.610 ","End":"00:44.450","Text":"but consider how to represent negative ones as well."},{"Start":"00:44.450 ","End":"00:49.955","Text":"We\u0027ve already been working with what\u0027s formerly known as an unsigned integer,"},{"Start":"00:49.955 ","End":"00:54.860","Text":"a number with no fractional part that\u0027s greater than or equal to 0,"},{"Start":"00:54.860 ","End":"00:58.580","Text":"whereas a signed integer is a number with"},{"Start":"00:58.580 ","End":"01:03.125","Text":"no fractional part that can be positive or negative."},{"Start":"01:03.125 ","End":"01:07.790","Text":"As we know, everything in a digital system is encoded as 1s and 0s,"},{"Start":"01:07.790 ","End":"01:14.307","Text":"so we need to come up with a way of encoding a signed decimal integer in binary."},{"Start":"01:14.307 ","End":"01:17.380","Text":"There are a few systems we can choose between."},{"Start":"01:17.380 ","End":"01:22.480","Text":"One is called sign-magnitude or sign and magnitude."},{"Start":"01:22.480 ","End":"01:25.870","Text":"It\u0027s a system of encoding a number in binary with"},{"Start":"01:25.870 ","End":"01:30.395","Text":"a positive and negative range using a sign bit."},{"Start":"01:30.395 ","End":"01:35.755","Text":"A sign bit of 0 is used to indicate positive numbers or 0 and"},{"Start":"01:35.755 ","End":"01:41.810","Text":"a sign bit of 1 is used to indicate negative numbers or 0."},{"Start":"01:41.810 ","End":"01:43.680","Text":"In sign and magnitude,"},{"Start":"01:43.680 ","End":"01:46.630","Text":"the leftmost bit is singled out as the sign bit,"},{"Start":"01:46.630 ","End":"01:53.080","Text":"the rest of the binary pattern represents the size or magnitude of the number."},{"Start":"01:53.080 ","End":"01:56.731","Text":"Let\u0027s say we were storing our number in a single byte."},{"Start":"01:56.731 ","End":"01:58.820","Text":"For an unsigned integer,"},{"Start":"01:58.820 ","End":"02:02.790","Text":"the bit pattern would represent the following."},{"Start":"02:02.830 ","End":"02:05.705","Text":"For an unsigned integer,"},{"Start":"02:05.705 ","End":"02:12.930","Text":"this bit pattern would be 1 times 2^7 plus 1 times 2^2,"},{"Start":"02:12.930 ","End":"02:18.100","Text":"which is 128 plus 4, which is 132."},{"Start":"02:18.100 ","End":"02:19.970","Text":"For a signed number,"},{"Start":"02:19.970 ","End":"02:25.490","Text":"we give up the most significant bit and make that a sign bit instead."},{"Start":"02:25.490 ","End":"02:31.502","Text":"A number is now 1 times 2^2, which is 4."},{"Start":"02:31.502 ","End":"02:36.230","Text":"But because the sign bit set this indicates the numbers negative,"},{"Start":"02:36.230 ","End":"02:41.060","Text":"so the binary bit pattern should actually be interpreted as minus 4."},{"Start":"02:41.060 ","End":"02:44.180","Text":"What we\u0027ve just seen is quite significant."},{"Start":"02:44.180 ","End":"02:48.560","Text":"What a binary value signifies is dependent on the context."},{"Start":"02:48.560 ","End":"02:52.430","Text":"In this case, the same binary pattern either indicates"},{"Start":"02:52.430 ","End":"02:59.090","Text":"132 if we interpret it as an unsigned 8-bit integer,"},{"Start":"02:59.090 ","End":"03:08.945","Text":"or minus 4 if we interpret it as a signed 8-bit integer using sign and magnitude format."},{"Start":"03:08.945 ","End":"03:12.350","Text":"Given another example of a different length,"},{"Start":"03:12.350 ","End":"03:17.760","Text":"7 bits, this one has the sign bit set to 0,"},{"Start":"03:17.760 ","End":"03:24.140","Text":"so it indicates a positive number and our job is simply to work out the magnitude,"},{"Start":"03:24.140 ","End":"03:27.245","Text":"which turns out to be 50."},{"Start":"03:27.245 ","End":"03:31.820","Text":"Negative integers can be represented in binary by giving up"},{"Start":"03:31.820 ","End":"03:37.729","Text":"the most significant binary digit value and replacing it with an indicator of the sign."},{"Start":"03:37.729 ","End":"03:40.550","Text":"This is a nice simple system and has"},{"Start":"03:40.550 ","End":"03:44.780","Text":"the advantage that the magnitude part of the binary value is"},{"Start":"03:44.780 ","End":"03:51.742","Text":"the same for our positive or negative number and only the sign bit changes."},{"Start":"03:51.742 ","End":"03:57.890","Text":"The downside of encoding integers as signed rather than unsigned is that the range of"},{"Start":"03:57.890 ","End":"04:00.410","Text":"numbers that can be represented with a given number of"},{"Start":"04:00.410 ","End":"04:04.675","Text":"bits is reduced when using signed numbers."},{"Start":"04:04.675 ","End":"04:12.800","Text":"With 8 bits, the largest number we could encode would be 255 if unsigned."},{"Start":"04:12.800 ","End":"04:18.680","Text":"But we\u0027d be given up half the range if you\u0027re using signed numbers in 8 bits,"},{"Start":"04:18.680 ","End":"04:23.690","Text":"because if 1 bit is being used for the sign then the remaining 7 bits"},{"Start":"04:23.690 ","End":"04:29.145","Text":"mean that the largest value that can be encoded is 127,"},{"Start":"04:29.145 ","End":"04:36.390","Text":"so 8 bits unsigned gives us a range of 0-255 whereas"},{"Start":"04:36.390 ","End":"04:44.535","Text":"8 bits signed gives us a range of minus 127 to positive 127."},{"Start":"04:44.535 ","End":"04:50.750","Text":"One quirk of sign and magnitude is that there are 2 representations of 0,"},{"Start":"04:50.750 ","End":"04:57.185","Text":"one in the negative half of the range and the other in the positive half."},{"Start":"04:57.185 ","End":"04:59.915","Text":"We\u0027ll pause there. In the next video,"},{"Start":"04:59.915 ","End":"05:03.155","Text":"we\u0027ll look at adding numbers in binary."},{"Start":"05:03.155 ","End":"05:06.360","Text":"Thanks for watching and see you soon."}],"ID":30244},{"Watched":false,"Name":"Binary Addition","Duration":"4m 50s","ChapterTopicVideoID":28712,"CourseChapterTopicPlaylistID":237670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.725","Text":"Hello, welcome back."},{"Start":"00:01.725 ","End":"00:04.680","Text":"We already know that the electronic circuits used to create"},{"Start":"00:04.680 ","End":"00:08.295","Text":"digital systems are suited to binary representation."},{"Start":"00:08.295 ","End":"00:13.245","Text":"When digital circuits are used to implement mathematical operations such as addition,"},{"Start":"00:13.245 ","End":"00:17.175","Text":"naturally, we\u0027d want that addition to be performed in binary."},{"Start":"00:17.175 ","End":"00:21.630","Text":"There\u0027s actually no difference in the procedure for adding binary numbers and"},{"Start":"00:21.630 ","End":"00:27.570","Text":"decimal numbers once you take into account the more limited set of symbols in binary."},{"Start":"00:27.570 ","End":"00:30.405","Text":"When adding two decimal numbers together,"},{"Start":"00:30.405 ","End":"00:33.000","Text":"you proceed from the least significant digits,"},{"Start":"00:33.000 ","End":"00:37.935","Text":"adding them together and recording the result in the row below the two digits."},{"Start":"00:37.935 ","End":"00:41.705","Text":"Where the addition can\u0027t be accommodated in a single column,"},{"Start":"00:41.705 ","End":"00:46.430","Text":"we carry over the most significant digit of the result into the next column."},{"Start":"00:46.430 ","End":"00:49.355","Text":"Here, because 3 plus 9 is 12,"},{"Start":"00:49.355 ","End":"00:51.125","Text":"and that involves two digits,"},{"Start":"00:51.125 ","End":"00:55.850","Text":"we record the 2 here and carry the 1 over to the next column."},{"Start":"00:55.850 ","End":"01:01.640","Text":"The final operation involves adding the two digits in this column and the carry value."},{"Start":"01:01.640 ","End":"01:05.270","Text":"The process is identical in binary,"},{"Start":"01:05.270 ","End":"01:08.075","Text":"0 plus 1 is just 1,"},{"Start":"01:08.075 ","End":"01:09.635","Text":"involves no carry,"},{"Start":"01:09.635 ","End":"01:12.290","Text":"1 plus 0 is the same."},{"Start":"01:12.290 ","End":"01:14.860","Text":"But 1 plus 1, which is 2,"},{"Start":"01:14.860 ","End":"01:20.060","Text":"does involve a carry because we can\u0027t write the symbol 2 as we\u0027re dealing in binary,"},{"Start":"01:20.060 ","End":"01:22.400","Text":"2 in binary is 1, 0,"},{"Start":"01:22.400 ","End":"01:23.750","Text":"and we can\u0027t write 1,"},{"Start":"01:23.750 ","End":"01:25.240","Text":"0 in a single column."},{"Start":"01:25.240 ","End":"01:30.180","Text":"We\u0027d write the 0 in this column and carry the 1 over."},{"Start":"01:30.180 ","End":"01:34.690","Text":"We then bring the 1 down into the next column,"},{"Start":"01:34.690 ","End":"01:37.105","Text":"because there\u0027s nothing else to add."},{"Start":"01:37.105 ","End":"01:42.710","Text":"Similarly, if we had 1 plus 1 plus another 1 from a carry to add,"},{"Start":"01:42.710 ","End":"01:44.465","Text":"we couldn\u0027t write 3."},{"Start":"01:44.465 ","End":"01:47.540","Text":"We\u0027d need to put 3 in binary, which is 1,"},{"Start":"01:47.540 ","End":"01:51.430","Text":"1, but we can\u0027t write that into a single column."},{"Start":"01:51.430 ","End":"01:57.560","Text":"We put a 1 in this column and carry over 1 to the next column."},{"Start":"01:57.560 ","End":"02:02.545","Text":"As before, bring that down as there\u0027s nothing else to add it to here."},{"Start":"02:02.545 ","End":"02:07.250","Text":"Let\u0027s now look at a simple addition of 2, 4-bit numbers."},{"Start":"02:07.250 ","End":"02:09.365","Text":"To keep it as simple as possible,"},{"Start":"02:09.365 ","End":"02:12.845","Text":"we\u0027ll make the numbers unsigned, for now."},{"Start":"02:12.845 ","End":"02:15.805","Text":"1 plus 0 would give us 1,"},{"Start":"02:15.805 ","End":"02:18.480","Text":"1 plus 1 would give us 1, 0."},{"Start":"02:18.480 ","End":"02:20.960","Text":"We write 0 in this first column,"},{"Start":"02:20.960 ","End":"02:22.775","Text":"carry a 1 across,"},{"Start":"02:22.775 ","End":"02:26.880","Text":"then 1 plus 0 plus the carried 1 would again give us 1,"},{"Start":"02:26.880 ","End":"02:31.280","Text":"0, so we write 0 in this column and carry the 1 across."},{"Start":"02:31.280 ","End":"02:33.995","Text":"There\u0027s nothing else to add in the next column,"},{"Start":"02:33.995 ","End":"02:37.460","Text":"so we just bring down the 1 that we carried across and we\u0027ve"},{"Start":"02:37.460 ","End":"02:41.740","Text":"got our final result of 1001."},{"Start":"02:41.740 ","End":"02:45.125","Text":"If we convert each line into decimal,"},{"Start":"02:45.125 ","End":"02:48.725","Text":"we can see if this addition gives us the result we expect."},{"Start":"02:48.725 ","End":"02:53.585","Text":"The first line would give us 4 plus 2 plus 1, which is 7,"},{"Start":"02:53.585 ","End":"02:56.545","Text":"the next line is just 2,"},{"Start":"02:56.545 ","End":"03:01.800","Text":"and the final line is 8 plus 1, which is 9."},{"Start":"03:01.800 ","End":"03:07.145","Text":"That\u0027s what we\u0027d expect the result to be if we added the decimal integers 7 and 2."},{"Start":"03:07.145 ","End":"03:10.085","Text":"The binary addition has worked."},{"Start":"03:10.085 ","End":"03:13.820","Text":"That was a binary addition of two unsigned numbers."},{"Start":"03:13.820 ","End":"03:17.125","Text":"With signed numbers in sign and magnitude format,"},{"Start":"03:17.125 ","End":"03:22.040","Text":"the addition is still straightforward when both numbers have the same sign."},{"Start":"03:22.040 ","End":"03:23.660","Text":"If they\u0027re both positive,"},{"Start":"03:23.660 ","End":"03:28.880","Text":"then you just add the magnitudes together like we\u0027ve done in this example."},{"Start":"03:28.880 ","End":"03:31.145","Text":"If they\u0027re both negative,"},{"Start":"03:31.145 ","End":"03:37.220","Text":"then you add the magnitudes together and make the sign bit of the result negative."},{"Start":"03:37.220 ","End":"03:39.750","Text":"Let\u0027s stick with 2, 4-bit numbers,"},{"Start":"03:39.750 ","End":"03:42.905","Text":"although 1 bit has now been given up to the sign bit,"},{"Start":"03:42.905 ","End":"03:45.185","Text":"which indicates in both cases,"},{"Start":"03:45.185 ","End":"03:47.875","Text":"these are negative numbers."},{"Start":"03:47.875 ","End":"03:51.660","Text":"1 plus 0 would give us 1, again, 1 plus 0,"},{"Start":"03:51.660 ","End":"03:52.875","Text":"and the next column is 1,"},{"Start":"03:52.875 ","End":"03:55.580","Text":"and also in the final column."},{"Start":"03:55.580 ","End":"03:58.670","Text":"The final job now is then to set the sign bit to"},{"Start":"03:58.670 ","End":"04:02.620","Text":"negative because both the numbers we added were negative."},{"Start":"04:02.620 ","End":"04:05.660","Text":"Let\u0027s check the values again in decimal."},{"Start":"04:05.660 ","End":"04:08.030","Text":"The first line is minus 3,"},{"Start":"04:08.030 ","End":"04:10.924","Text":"2 plus 1 with a sign bit of negative."},{"Start":"04:10.924 ","End":"04:15.750","Text":"The second is 4 with just the 4 column and a sign of negative,"},{"Start":"04:15.750 ","End":"04:19.065","Text":"and the result is 4 plus 2 plus 1,"},{"Start":"04:19.065 ","End":"04:20.610","Text":"which is minus 7,"},{"Start":"04:20.610 ","End":"04:24.490","Text":"given that the sign bit is negative."},{"Start":"04:24.530 ","End":"04:29.360","Text":"A nice simple system of using a bit to indicate the sign is straightforward"},{"Start":"04:29.360 ","End":"04:33.440","Text":"when adding a pair of numbers that are both positive or both negative."},{"Start":"04:33.440 ","End":"04:37.040","Text":"But if there\u0027s a mixture of positive and negative numbers,"},{"Start":"04:37.040 ","End":"04:39.250","Text":"it gets more complicated."},{"Start":"04:39.250 ","End":"04:42.815","Text":"Another system of encoding signed numbers is possible,"},{"Start":"04:42.815 ","End":"04:44.915","Text":"the one\u0027s complement form."},{"Start":"04:44.915 ","End":"04:47.870","Text":"We\u0027ll cover one\u0027s complement in the next video."},{"Start":"04:47.870 ","End":"04:51.540","Text":"See you shortly for that one. Thanks for watching."}],"ID":30241},{"Watched":false,"Name":"Ones Complement","Duration":"3m 25s","ChapterTopicVideoID":28713,"CourseChapterTopicPlaylistID":237670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"Hello, welcome back. In the previous video,"},{"Start":"00:03.570 ","End":"00:06.390","Text":"we discovered that the addition of sign and magnitude numbers was"},{"Start":"00:06.390 ","End":"00:09.930","Text":"straightforward if the sign of both numbers being added was the same,"},{"Start":"00:09.930 ","End":"00:11.940","Text":"but when the signs were different,"},{"Start":"00:11.940 ","End":"00:13.500","Text":"it got more complicated."},{"Start":"00:13.500 ","End":"00:17.835","Text":"An alternative is one\u0027s complement."},{"Start":"00:17.835 ","End":"00:22.800","Text":"This is a system of encoding a number in binary with a positive and negative range,"},{"Start":"00:22.800 ","End":"00:25.800","Text":"with the negative representation of a number being"},{"Start":"00:25.800 ","End":"00:30.045","Text":"the bitwise inversion of the positive representation."},{"Start":"00:30.045 ","End":"00:32.385","Text":"Bitwise inversion, simply put,"},{"Start":"00:32.385 ","End":"00:37.080","Text":"is to just flip each bit in the number to its opposite value."},{"Start":"00:37.080 ","End":"00:39.150","Text":"Where it was a 1 make it 0,"},{"Start":"00:39.150 ","End":"00:41.705","Text":"where it was a 0 make it a 1."},{"Start":"00:41.705 ","End":"00:48.290","Text":"Given this number, inverting each binary digit would give us the one\u0027s complement."},{"Start":"00:48.290 ","End":"00:50.479","Text":"As with sign and magnitude,"},{"Start":"00:50.479 ","End":"00:55.975","Text":"half the range will be used for negative numbers and half the range for positive."},{"Start":"00:55.975 ","End":"01:00.050","Text":"Positive numbers, 0-127 are the same as they"},{"Start":"01:00.050 ","End":"01:04.565","Text":"are for unsigned binary and sign and magnitude representation."},{"Start":"01:04.565 ","End":"01:09.349","Text":"But the negative representations are different to sign and magnitude."},{"Start":"01:09.349 ","End":"01:14.165","Text":"A pattern of all 1s indicates minus 0."},{"Start":"01:14.165 ","End":"01:18.515","Text":"Any number with the MSB as 1 will be a negative number."},{"Start":"01:18.515 ","End":"01:22.174","Text":"As soon as we see a 0 at the MSB,"},{"Start":"01:22.174 ","End":"01:24.660","Text":"we\u0027re in the positive range."},{"Start":"01:24.660 ","End":"01:27.635","Text":"Looking at another example in detail."},{"Start":"01:27.635 ","End":"01:31.160","Text":"Converting a number to a decimal value for a positive number,"},{"Start":"01:31.160 ","End":"01:33.470","Text":"we add up the column values whilst"},{"Start":"01:33.470 ","End":"01:36.935","Text":"ignoring the most significant bit which indicates the sign."},{"Start":"01:36.935 ","End":"01:40.025","Text":"As we can see for this positive number,"},{"Start":"01:40.025 ","End":"01:43.370","Text":"this gives the same bit pattern in one\u0027s complement as we\u0027d"},{"Start":"01:43.370 ","End":"01:47.600","Text":"have with both unsigned numbers and sign and magnitude."},{"Start":"01:47.600 ","End":"01:52.355","Text":"If we now convert the number into its inverse to get the one\u0027s complement,"},{"Start":"01:52.355 ","End":"01:54.095","Text":"we see this pattern."},{"Start":"01:54.095 ","End":"01:56.345","Text":"In the sign magnitude format,"},{"Start":"01:56.345 ","End":"02:02.015","Text":"we\u0027d expect positive and negative values to be the same apart from the sign bit."},{"Start":"02:02.015 ","End":"02:05.795","Text":"But here, we can see clearly that that\u0027s not the same."},{"Start":"02:05.795 ","End":"02:08.855","Text":"This would\u0027ve been the magnitude part and it\u0027s clearly different"},{"Start":"02:08.855 ","End":"02:12.395","Text":"from the magnitude part of the positive number."},{"Start":"02:12.395 ","End":"02:15.485","Text":"The inverse, as we add up the columns,"},{"Start":"02:15.485 ","End":"02:18.110","Text":"actually gives us 83."},{"Start":"02:18.110 ","End":"02:21.949","Text":"We clearly need a different process to calculate"},{"Start":"02:21.949 ","End":"02:26.515","Text":"the decimal value when we\u0027re dealing with one\u0027s complement numbers."},{"Start":"02:26.515 ","End":"02:31.460","Text":"We first take what would\u0027ve been the MSB value in an unsigned number,"},{"Start":"02:31.460 ","End":"02:33.020","Text":"in this case 2^7,"},{"Start":"02:33.020 ","End":"02:37.335","Text":"which is a 128 and we make it negative."},{"Start":"02:37.335 ","End":"02:42.920","Text":"To that, we add 1 and finally,"},{"Start":"02:42.920 ","End":"02:49.440","Text":"add the number we just calculated from the other 7 bits which gives us minus 44."},{"Start":"02:49.440 ","End":"02:52.595","Text":"This meets the definition of a one\u0027s complement number."},{"Start":"02:52.595 ","End":"02:57.590","Text":"A negative integer is the bitwise inverse of its positive counterpart."},{"Start":"02:57.590 ","End":"03:02.089","Text":"But converting it to decimals is a little trickier than sign and magnitude."},{"Start":"03:02.089 ","End":"03:03.980","Text":"Unlike sign and magnitude,"},{"Start":"03:03.980 ","End":"03:08.330","Text":"it also has complications when adding numbers."},{"Start":"03:08.330 ","End":"03:11.090","Text":"You guessed it. There\u0027s yet another system for"},{"Start":"03:11.090 ","End":"03:14.720","Text":"representing binary numbers called two\u0027s complement,"},{"Start":"03:14.720 ","End":"03:18.020","Text":"which happily allows for straightforward additions."},{"Start":"03:18.020 ","End":"03:21.005","Text":"We\u0027ll cover that in the next video."},{"Start":"03:21.005 ","End":"03:25.440","Text":"See you for that one shortly. Thanks for watching."}],"ID":30242},{"Watched":false,"Name":"Twos Complement","Duration":"8m 8s","ChapterTopicVideoID":28714,"CourseChapterTopicPlaylistID":237670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"Hello, again. Previously we,"},{"Start":"00:02.580 ","End":"00:07.050","Text":"heard how one\u0027s complement allowed us to represent a negative number in binary,"},{"Start":"00:07.050 ","End":"00:12.525","Text":"simply by performing a bitwise inversion on its positive counterpart."},{"Start":"00:12.525 ","End":"00:16.020","Text":"Two\u0027s complement is a system of encoding a number in binary with"},{"Start":"00:16.020 ","End":"00:22.050","Text":"a positive and negative range using a negative weighting for the most significant bit."},{"Start":"00:22.050 ","End":"00:24.615","Text":"For negative numbers in two\u0027s complement,"},{"Start":"00:24.615 ","End":"00:27.030","Text":"the most significant bit will be 1."},{"Start":"00:27.030 ","End":"00:29.760","Text":"If you\u0027re using 8 bits to encode our number,"},{"Start":"00:29.760 ","End":"00:34.945","Text":"this will give a weighting of minus 128 for that binary digit."},{"Start":"00:34.945 ","End":"00:37.385","Text":"For this specific bit pattern,"},{"Start":"00:37.385 ","End":"00:43.712","Text":"we get the decimal value of minus 79 because we\u0027re adding 49,"},{"Start":"00:43.712 ","End":"00:49.625","Text":"32 plus 16 plus 1, to minus 128."},{"Start":"00:49.625 ","End":"00:53.405","Text":"The system\u0027s closely related to one\u0027s complement."},{"Start":"00:53.405 ","End":"00:57.980","Text":"If we interpret the same binary pattern as a one\u0027s complement decimal number,"},{"Start":"00:57.980 ","End":"01:03.155","Text":"we\u0027ll see that there is a difference of 1 between the 2 interpretations."},{"Start":"01:03.155 ","End":"01:06.920","Text":"In fact, an 8-bit number in one\u0027s complement differs"},{"Start":"01:06.920 ","End":"01:10.640","Text":"very little from an 8-bit number in two\u0027s complement."},{"Start":"01:10.640 ","End":"01:15.100","Text":"For the positive half where the MSB is 0, it\u0027s the same."},{"Start":"01:15.100 ","End":"01:18.905","Text":"But for the negative half where the MSB is 1,"},{"Start":"01:18.905 ","End":"01:24.980","Text":"the two\u0027s complement value and the one\u0027s complement value differ by 1."},{"Start":"01:24.980 ","End":"01:28.880","Text":"There\u0027s also only a single 0 in two\u0027s complement,"},{"Start":"01:28.880 ","End":"01:36.815","Text":"so we get a slightly wider range of numbers between minus 128 and positive 127."},{"Start":"01:36.815 ","End":"01:40.880","Text":"We can convert a negative integer in decimal into"},{"Start":"01:40.880 ","End":"01:45.290","Text":"two\u0027s complement binary by performing the following steps."},{"Start":"01:45.290 ","End":"01:49.010","Text":"Turn the negative number into its positive binary equivalent,"},{"Start":"01:49.010 ","End":"01:50.720","Text":"pad it out with zeros,"},{"Start":"01:50.720 ","End":"01:53.840","Text":"invert a pattern, and then add 1."},{"Start":"01:53.840 ","End":"01:57.475","Text":"Let\u0027s work through an example, 8-bit number."},{"Start":"01:57.475 ","End":"02:00.715","Text":"First, we find the positive representation"},{"Start":"02:00.715 ","End":"02:04.059","Text":"of the negative number we\u0027re looking to encode."},{"Start":"02:04.059 ","End":"02:06.925","Text":"We\u0027re looking to encode minus 27,"},{"Start":"02:06.925 ","End":"02:10.035","Text":"what values do we need to encode?"},{"Start":"02:10.035 ","End":"02:12.410","Text":"We\u0027ll need 16,"},{"Start":"02:12.410 ","End":"02:14.300","Text":"so let\u0027s start there."},{"Start":"02:14.300 ","End":"02:19.025","Text":"We\u0027ll also need the 8 as there\u0027s 11 left to make up."},{"Start":"02:19.025 ","End":"02:24.465","Text":"Now, we need 3 more so 0 here, 1,"},{"Start":"02:24.465 ","End":"02:31.510","Text":"and 1, so 16 plus 8 plus 2 plus 1 gives us 27 as was our objective."},{"Start":"02:31.510 ","End":"02:37.340","Text":"The next step is to pad the number out with zeros on the left-hand side."},{"Start":"02:37.340 ","End":"02:44.315","Text":"Now, we get the one\u0027s complement of the binary number by inverting every bit."},{"Start":"02:44.315 ","End":"02:49.790","Text":"This, by the way, is why we had to pad out a number with zeros to the left."},{"Start":"02:49.790 ","End":"02:52.565","Text":"That\u0027s the inversion done."},{"Start":"02:52.565 ","End":"02:56.974","Text":"All that\u0027s left to do is add 1 to the result."},{"Start":"02:56.974 ","End":"03:02.020","Text":"We add 1 on the first column, we\u0027ll get 1."},{"Start":"03:02.020 ","End":"03:08.120","Text":"Next column. We\u0027re adding 0 and we will for all the subsequent columns,"},{"Start":"03:08.120 ","End":"03:15.600","Text":"so the number will stay the same in these columns because we\u0027re adding 0."},{"Start":"03:15.940 ","End":"03:22.210","Text":"The final result is our decimal number now encoded in two\u0027s complement binary."},{"Start":"03:22.210 ","End":"03:26.360","Text":"The advantage of two\u0027s complement over sign and magnitude and"},{"Start":"03:26.360 ","End":"03:30.965","Text":"one\u0027s complement is adding two\u0027s complement numbers is more straightforward."},{"Start":"03:30.965 ","End":"03:33.830","Text":"We don\u0027t have to apply any special rules depending on"},{"Start":"03:33.830 ","End":"03:37.160","Text":"whether the numbers are positive or negative or a mixture."},{"Start":"03:37.160 ","End":"03:39.320","Text":"Let\u0027s see an example of this where we add"},{"Start":"03:39.320 ","End":"03:44.255","Text":"a positive number to a negative number using 4 bits."},{"Start":"03:44.255 ","End":"03:46.940","Text":"1 plus 0 would be 1,"},{"Start":"03:46.940 ","End":"03:48.560","Text":"0 plus 1 would be 1,"},{"Start":"03:48.560 ","End":"03:50.450","Text":"1 plus 0, again, 1,"},{"Start":"03:50.450 ","End":"03:53.155","Text":"and 0 plus 1 would be 1."},{"Start":"03:53.155 ","End":"03:55.610","Text":"Let\u0027s check that that calculation gives us"},{"Start":"03:55.610 ","End":"03:59.225","Text":"the correct result by converting each line to decimal."},{"Start":"03:59.225 ","End":"04:03.665","Text":"The first line would give us 4 plus 1, which is 5."},{"Start":"04:03.665 ","End":"04:08.180","Text":"The second line would give us minus 6 because it\u0027s minus 8 plus"},{"Start":"04:08.180 ","End":"04:13.125","Text":"2 and the final line is minus 8 plus 7,"},{"Start":"04:13.125 ","End":"04:15.100","Text":"which will give us minus 1."},{"Start":"04:15.100 ","End":"04:18.470","Text":"If we had done a decimal addition of 5 to minus 6,"},{"Start":"04:18.470 ","End":"04:20.300","Text":"we would have got minus 1 as well,"},{"Start":"04:20.300 ","End":"04:26.270","Text":"so this binary addition of this pair of two\u0027s complement numbers seems to have worked."},{"Start":"04:26.270 ","End":"04:29.900","Text":"Let\u0027s try it again with 2 negative numbers."},{"Start":"04:29.900 ","End":"04:33.670","Text":"We know they\u0027re negative because the MSB in both bits is 1,"},{"Start":"04:33.670 ","End":"04:37.940","Text":"1 plus 1 is 1, 0,"},{"Start":"04:37.940 ","End":"04:41.290","Text":"so we put 0 in this column and carry 1 across,"},{"Start":"04:41.290 ","End":"04:44.460","Text":"0 plus 1 plus 1 is again 1,"},{"Start":"04:44.460 ","End":"04:47.625","Text":"0, so 0 in this column and 1 across,"},{"Start":"04:47.625 ","End":"04:51.420","Text":"1 plus 1 plus 1 would be 1, 1,"},{"Start":"04:51.420 ","End":"04:55.980","Text":"so 1 in this column and carry the 1 across and finally,"},{"Start":"04:55.980 ","End":"04:57.750","Text":"1 plus 1 plus 1 would, again,"},{"Start":"04:57.750 ","End":"05:03.259","Text":"be 1, 1, 1 in this column and we would carry the 1 across,"},{"Start":"05:03.259 ","End":"05:05.810","Text":"but it won\u0027t make a difference to the result,"},{"Start":"05:05.810 ","End":"05:08.615","Text":"so we\u0027ll ignore it here."},{"Start":"05:08.615 ","End":"05:14.115","Text":"Let\u0027s check that that calculation worked by converting each line to decimal, again."},{"Start":"05:14.115 ","End":"05:18.180","Text":"Minus 8 plus 5 is minus 3,"},{"Start":"05:18.180 ","End":"05:22.255","Text":"and minus 8 plus 7 is minus 1,"},{"Start":"05:22.255 ","End":"05:25.730","Text":"and minus 8 plus 4 will give us minus 4."},{"Start":"05:25.730 ","End":"05:27.530","Text":"We do the addition in decimal,"},{"Start":"05:27.530 ","End":"05:31.680","Text":"minus 3 plus minus 1 would indeed give us minus 4."},{"Start":"05:31.680 ","End":"05:37.584","Text":"Again, the binary addition has worked with our pair of two\u0027s complement numbers."},{"Start":"05:37.584 ","End":"05:41.920","Text":"Two\u0027s complement for its simplicity and adding numbers is therefore"},{"Start":"05:41.920 ","End":"05:47.285","Text":"the preferred encoding for signed binary numbers generally."},{"Start":"05:47.285 ","End":"05:51.550","Text":"Binary subtraction works in the same way as addition."},{"Start":"05:51.550 ","End":"05:55.505","Text":"If we took 1 away from 1, we\u0027d get 0."},{"Start":"05:55.505 ","End":"05:56.615","Text":"Again, in the next column,"},{"Start":"05:56.615 ","End":"05:58.655","Text":"1 minus 1 would give us 0,"},{"Start":"05:58.655 ","End":"06:00.890","Text":"1 minus 0 would just be 1,"},{"Start":"06:00.890 ","End":"06:04.385","Text":"and 0 minus 0 is just 0."},{"Start":"06:04.385 ","End":"06:07.130","Text":"If we were to convert each line to decimal,"},{"Start":"06:07.130 ","End":"06:09.665","Text":"we\u0027d see that the subtraction has worked."},{"Start":"06:09.665 ","End":"06:13.970","Text":"But we can actually do the subtraction using addition instead."},{"Start":"06:13.970 ","End":"06:20.720","Text":"We do this by turning the second number into a negative and then perform addition."},{"Start":"06:20.720 ","End":"06:28.545","Text":"We can convert 3 to minus 3 in two\u0027s complement by flipping the bits and adding 1."},{"Start":"06:28.545 ","End":"06:35.235","Text":"Now, we can add 7 and minus 3 instead of subtracting 3 from 7,"},{"Start":"06:35.235 ","End":"06:38.940","Text":"and we get 1, 0 for that first column."},{"Start":"06:38.940 ","End":"06:40.395","Text":"Carry the 1 across,"},{"Start":"06:40.395 ","End":"06:43.515","Text":"1 plus 0 plus 1 would give us 1, 0, again."},{"Start":"06:43.515 ","End":"06:47.460","Text":"Carry the 1 across, 1 plus 1 plus 1 would give us 1, 1,"},{"Start":"06:47.460 ","End":"06:51.240","Text":"so 1 in this column and 1 across and then finally,"},{"Start":"06:51.240 ","End":"06:54.600","Text":"1 plus 1 would give us 1, 0,"},{"Start":"06:54.600 ","End":"06:59.565","Text":"so 0 in this column and we discard the carried 1."},{"Start":"06:59.565 ","End":"07:01.860","Text":"That final line, 0, 1, 0,"},{"Start":"07:01.860 ","End":"07:05.500","Text":"0 would give us 4."},{"Start":"07:06.890 ","End":"07:09.950","Text":"Well, that\u0027s obviously the correct result, but you might ask,"},{"Start":"07:09.950 ","End":"07:14.930","Text":"why would you want to add a negative number instead of just subtracting?"},{"Start":"07:14.930 ","End":"07:18.650","Text":"Well, assuming we already have a digital circuit in place that can"},{"Start":"07:18.650 ","End":"07:22.655","Text":"convert a number into its negative two\u0027s complement equivalent,"},{"Start":"07:22.655 ","End":"07:25.550","Text":"we only need to create a digital circuit for"},{"Start":"07:25.550 ","End":"07:30.110","Text":"addition and we don\u0027t need 1 for subtraction."},{"Start":"07:30.110 ","End":"07:33.800","Text":"That potentially saves us money on components, on space,"},{"Start":"07:33.800 ","End":"07:39.380","Text":"on the circuit layout and electrical energy for the extra subtraction circuit."},{"Start":"07:39.380 ","End":"07:44.060","Text":"That\u0027s it on two\u0027s compliment and this topic in general,"},{"Start":"07:44.060 ","End":"07:49.340","Text":"so be sure to complete the exercises to consolidate your understanding."},{"Start":"07:49.340 ","End":"07:53.809","Text":"In this section, we learned how to distinguish between signed and unsigned integers,"},{"Start":"07:53.809 ","End":"07:58.040","Text":"interpret binary numbers in sign and magnitude or two\u0027s complement format."},{"Start":"07:58.040 ","End":"08:02.000","Text":"Convert a signed binary integer into its decimal equivalent."},{"Start":"08:02.000 ","End":"08:06.410","Text":"Perform calculations in binary with signed numbers."},{"Start":"08:06.410 ","End":"08:08.850","Text":"Thank you very much for watching."}],"ID":30243},{"Watched":false,"Name":"Exercise 1 parts A-C","Duration":"3m 28s","ChapterTopicVideoID":25523,"CourseChapterTopicPlaylistID":237670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"Hello everybody, in this question we\u0027ve been asked to"},{"Start":"00:03.000 ","End":"00:05.940","Text":"take a binary number and state its value in"},{"Start":"00:05.940 ","End":"00:11.775","Text":"decimal when it\u0027s an unsigned number or a sign number using sign and magnitude,"},{"Start":"00:11.775 ","End":"00:14.370","Text":"or a sign number using two\u0027s complement."},{"Start":"00:14.370 ","End":"00:15.960","Text":"Let\u0027s go ahead and do that."},{"Start":"00:15.960 ","End":"00:19.500","Text":"In part a, we simply need to write out a number,"},{"Start":"00:19.500 ","End":"00:21.900","Text":"1, 0, 1, 0,"},{"Start":"00:21.900 ","End":"00:23.490","Text":"1, 1, 0,"},{"Start":"00:23.490 ","End":"00:29.400","Text":"and its bit values above each bit so we\u0027ve got 1, 2, 4,"},{"Start":"00:29.400 ","End":"00:31.785","Text":"8, 16,"},{"Start":"00:31.785 ","End":"00:36.225","Text":"32, and we stop at 64 because there\u0027s no more bits."},{"Start":"00:36.225 ","End":"00:40.250","Text":"We\u0027ve done many exercises like this already hopefully so far,"},{"Start":"00:40.250 ","End":"00:44.210","Text":"we know the answer to this is simply by adding together,"},{"Start":"00:44.210 ","End":"00:51.900","Text":"the digits where there is a 1. so we have 64 plus 16 plus 4 plus 2,"},{"Start":"00:51.900 ","End":"00:54.585","Text":"and that gives us 86."},{"Start":"00:54.585 ","End":"00:57.830","Text":"There we go answer for part a, done."},{"Start":"00:57.830 ","End":"00:59.310","Text":"For an unsigned number,"},{"Start":"00:59.310 ","End":"01:02.340","Text":"this bit pattern is 86 in decimal."},{"Start":"01:02.340 ","End":"01:03.750","Text":"Now the second part,"},{"Start":"01:03.750 ","End":"01:07.805","Text":"part b we\u0027ve got a signed number using sign and magnitude."},{"Start":"01:07.805 ","End":"01:09.605","Text":"For sign and magnitude,"},{"Start":"01:09.605 ","End":"01:13.130","Text":"we\u0027re still going to write the number out as we have done previously."},{"Start":"01:13.130 ","End":"01:18.245","Text":"However, the column values will stop at different point."},{"Start":"01:18.245 ","End":"01:20.660","Text":"Let\u0027s have a look up where we would stop,"},{"Start":"01:20.660 ","End":"01:30.435","Text":"we go 1 2 4 8 16 32 and in sign and magnitude this final bit here is the sign bit."},{"Start":"01:30.435 ","End":"01:32.015","Text":"It doesn\u0027t have a value,"},{"Start":"01:32.015 ","End":"01:36.740","Text":"is just indicating the sign is it a positive number or is it a negative number?"},{"Start":"01:36.740 ","End":"01:43.960","Text":"In this case, the answer is going to be 16 plus 4 plus 2,"},{"Start":"01:43.960 ","End":"01:46.250","Text":"which is obviously 22."},{"Start":"01:46.250 ","End":"01:49.790","Text":"But now we\u0027re indicating by the sign bit this is a negative number,"},{"Start":"01:49.790 ","End":"01:52.765","Text":"so the answer would actually be minus 22."},{"Start":"01:52.765 ","End":"01:56.015","Text":"That is indicated by a sign bit of 1,"},{"Start":"01:56.015 ","End":"01:58.580","Text":"where 1 indicates a negative number,"},{"Start":"01:58.580 ","End":"02:01.250","Text":"0 indicates a positive number."},{"Start":"02:01.250 ","End":"02:03.950","Text":"That\u0027s great, we\u0027ve done part a and b now,"},{"Start":"02:03.950 ","End":"02:08.280","Text":"a final part is using two\u0027s complement format."},{"Start":"02:08.280 ","End":"02:11.875","Text":"Once again we\u0027ll write out a number itself."},{"Start":"02:11.875 ","End":"02:16.130","Text":"However the interpretation will be different once again,"},{"Start":"02:16.130 ","End":"02:22.820","Text":"in this case, the final column will take on a different meaning."},{"Start":"02:22.820 ","End":"02:24.895","Text":"We got 1, 2, 4, 8,"},{"Start":"02:24.895 ","End":"02:27.460","Text":"16, 32, as before."},{"Start":"02:27.460 ","End":"02:29.525","Text":"However, this last column now,"},{"Start":"02:29.525 ","End":"02:32.750","Text":"rather than being 64 as it is for an unsigned number,"},{"Start":"02:32.750 ","End":"02:34.550","Text":"it will now be minus 64."},{"Start":"02:34.550 ","End":"02:37.430","Text":"The implications on the result therefore,"},{"Start":"02:37.430 ","End":"02:40.430","Text":"will be that we have minus 64,"},{"Start":"02:40.430 ","End":"02:45.430","Text":"to which we add 16 and 4 and 2."},{"Start":"02:45.430 ","End":"02:50.415","Text":"That will give us a final answer of minus 42,"},{"Start":"02:50.415 ","End":"02:52.505","Text":"if you add all those together,"},{"Start":"02:52.505 ","End":"02:54.725","Text":"you will get a result of minus 42."},{"Start":"02:54.725 ","End":"02:58.370","Text":"We\u0027re now done we\u0027ve converted the binary numbers we were asked in"},{"Start":"02:58.370 ","End":"03:03.590","Text":"the original question into decimal, but using unsigned,"},{"Start":"03:03.590 ","End":"03:05.630","Text":"signed with sign and magnitude,"},{"Start":"03:05.630 ","End":"03:10.010","Text":"and signed with two\u0027s complement and we see we get 3 very different results,"},{"Start":"03:10.010 ","End":"03:14.750","Text":"86 if it is a unsigned number,"},{"Start":"03:14.750 ","End":"03:16.870","Text":"we have minus 22,"},{"Start":"03:16.870 ","End":"03:23.045","Text":"if it is a signed number but using sign and magnitude and we have minus 42,"},{"Start":"03:23.045 ","End":"03:26.690","Text":"if it\u0027s again a signed number but this time using two\u0027s complement."},{"Start":"03:26.690 ","End":"03:29.370","Text":"That\u0027s it, see you in the next 1."}],"ID":26329},{"Watched":false,"Name":"Exercise 2 parts A-C","Duration":"7m 37s","ChapterTopicVideoID":25512,"CourseChapterTopicPlaylistID":237670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.890","Text":"Hello again, everybody."},{"Start":"00:01.890 ","End":"00:06.085","Text":"In this question we\u0027ve been asked to take a binary number and, again,"},{"Start":"00:06.085 ","End":"00:08.940","Text":"state the value in decimal for an unsigned number,"},{"Start":"00:08.940 ","End":"00:12.900","Text":"a signed number using sign and magnitude and a signed number using two\u0027s complement."},{"Start":"00:12.900 ","End":"00:16.605","Text":"We\u0027ve seen this before, but we\u0027ve got a much longer number here now."},{"Start":"00:16.605 ","End":"00:18.570","Text":"We\u0027ll proceed as we did before,"},{"Start":"00:18.570 ","End":"00:22.529","Text":"but this time I suggest we break up into nibbles, the numbers."},{"Start":"00:22.529 ","End":"00:25.142","Text":"We\u0027ve got 12 bits exactly,"},{"Start":"00:25.142 ","End":"00:27.150","Text":"so I\u0027ll write those out as 3 nibbles."},{"Start":"00:27.150 ","End":"00:34.016","Text":"That\u0027s 1011, 0011,"},{"Start":"00:34.016 ","End":"00:36.540","Text":"and 0110, and then let\u0027s do as before,"},{"Start":"00:36.540 ","End":"00:38.460","Text":"the column headings for each of those bits"},{"Start":"00:38.460 ","End":"00:40.510","Text":"are the weightings whatever way you want to call it,"},{"Start":"00:40.510 ","End":"00:41.862","Text":"1, 2, 4,"},{"Start":"00:41.862 ","End":"00:44.923","Text":"8,16, 32, 64,"},{"Start":"00:44.923 ","End":"00:50.380","Text":"128, and then finally 256,"},{"Start":"00:50.380 ","End":"00:53.605","Text":"512, 1,024,"},{"Start":"00:53.605 ","End":"00:57.390","Text":"and that means that final bit is worth 2,048."},{"Start":"00:57.390 ","End":"01:01.040","Text":"We now just need to add these numbers together, however,"},{"Start":"01:01.040 ","End":"01:05.180","Text":"because this is quite a large set of numbers and multi digits,"},{"Start":"01:05.180 ","End":"01:09.035","Text":"and I\u0027m going to assume you don\u0027t have access to a calculator in the exam,"},{"Start":"01:09.035 ","End":"01:14.135","Text":"then let\u0027s stack those numbers on top of each other where everywhere where we have a 1,"},{"Start":"01:14.135 ","End":"01:18.770","Text":"so we start with 2,048 and we have a 512."},{"Start":"01:18.770 ","End":"01:22.760","Text":"Make sure you\u0027re very careful about aligning the digits here by the way,"},{"Start":"01:22.760 ","End":"01:28.465","Text":"very easy to make a mistake if you misalign the digits."},{"Start":"01:28.465 ","End":"01:32.820","Text":"You got 16, and then 4, and 2."},{"Start":"01:32.820 ","End":"01:35.195","Text":"We\u0027re going to add those together now."},{"Start":"01:35.195 ","End":"01:38.255","Text":"Let\u0027s go for it. 8 plus 2, 10."},{"Start":"01:38.255 ","End":"01:40.225","Text":"16, 18,"},{"Start":"01:40.225 ","End":"01:43.110","Text":"24, 28, 30."},{"Start":"01:43.110 ","End":"01:49.185","Text":"Let\u0027s put our 0 here and carry across the 3 to up here."},{"Start":"01:49.185 ","End":"01:51.790","Text":"Now we\u0027ve got 7 plus 1,"},{"Start":"01:51.790 ","End":"01:54.420","Text":"8, 13, 16,"},{"Start":"01:54.420 ","End":"01:57.210","Text":"17, so I put my 7 here,"},{"Start":"01:57.210 ","End":"02:00.300","Text":"and carry across my 1 to here."},{"Start":"02:00.300 ","End":"02:02.220","Text":"We\u0027ve got 1 plus 5,"},{"Start":"02:02.220 ","End":"02:04.305","Text":"6 plus 2 is 8,"},{"Start":"02:04.305 ","End":"02:05.760","Text":"so 8 here,"},{"Start":"02:05.760 ","End":"02:09.095","Text":"and then finally just bring down the 2, and there we go."},{"Start":"02:09.095 ","End":"02:10.460","Text":"We have our result."},{"Start":"02:10.460 ","End":"02:14.330","Text":"This bit pattern, if it represents an unsigned number,"},{"Start":"02:14.330 ","End":"02:16.910","Text":"is a 2,870 in decimal."},{"Start":"02:16.910 ","End":"02:18.305","Text":"We\u0027ve done the first part."},{"Start":"02:18.305 ","End":"02:22.700","Text":"Now, we already know that a sign number using sign and magnitude,"},{"Start":"02:22.700 ","End":"02:25.040","Text":"we have to give up one of the bits as the sign bit."},{"Start":"02:25.040 ","End":"02:29.240","Text":"It\u0027s going to be this bit here is now going to not be a number,"},{"Start":"02:29.240 ","End":"02:30.880","Text":"it\u0027s going to be a signed bit."},{"Start":"02:30.880 ","End":"02:35.210","Text":"We can use the previous answer to help us actually for the next part,"},{"Start":"02:35.210 ","End":"02:39.140","Text":"we can ignore this first digit now, we don\u0027t need it."},{"Start":"02:39.140 ","End":"02:41.195","Text":"The next one was 1,024 anyway."},{"Start":"02:41.195 ","End":"02:45.950","Text":"All we have to do now is to add all of these other numbers together where there is a 1."},{"Start":"02:45.950 ","End":"02:51.755","Text":"Let\u0027s go ahead and stack those as before on the right-hand side here."},{"Start":"02:51.755 ","End":"02:55.100","Text":"512, 256,"},{"Start":"02:55.100 ","End":"02:57.485","Text":"32, 16,"},{"Start":"02:57.485 ","End":"03:02.262","Text":"4 and 2, and so 8,"},{"Start":"03:02.262 ","End":"03:05.595","Text":"10, 16, 20, 2 is 22."},{"Start":"03:05.595 ","End":"03:09.150","Text":"2 there carry across 2,"},{"Start":"03:09.150 ","End":"03:11.850","Text":"3, 8, 11, 12."},{"Start":"03:11.850 ","End":"03:18.630","Text":"Let\u0027s put a 2 here and carry across the 1 goes up here,"},{"Start":"03:18.630 ","End":"03:21.870","Text":"we\u0027ve got 6 plus 2 is 8."},{"Start":"03:21.870 ","End":"03:25.105","Text":"Our final answer is 822."},{"Start":"03:25.105 ","End":"03:29.060","Text":"However, it\u0027s not our final answer in that this is"},{"Start":"03:29.060 ","End":"03:33.200","Text":"a signed magnitude number and we don\u0027t completely ignore this bit here,"},{"Start":"03:33.200 ","End":"03:35.060","Text":"which we said is not 2048, anymore."},{"Start":"03:35.060 ","End":"03:37.535","Text":"It\u0027s the sign bit, and because it\u0027s a 1,"},{"Start":"03:37.535 ","End":"03:39.566","Text":"this means that this is a negative number,"},{"Start":"03:39.566 ","End":"03:42.650","Text":"therefore, the result is minus 822."},{"Start":"03:42.650 ","End":"03:46.535","Text":"For the final part, I am going to write out the number again this time,"},{"Start":"03:46.535 ","End":"03:52.640","Text":"just to highlight the fact that the most significant bit has taken on a different value."},{"Start":"03:52.640 ","End":"03:56.210","Text":"It\u0027s not just indicating a negative or a positive number,"},{"Start":"03:56.210 ","End":"03:58.745","Text":"it\u0027s actually taking on a value."},{"Start":"03:58.745 ","End":"04:03.335","Text":"It\u0027s as before for the column values apart from the final column,"},{"Start":"04:03.335 ","End":"04:05.030","Text":"it\u0027s 1, 2, 4,"},{"Start":"04:05.030 ","End":"04:06.770","Text":"and 8, and by the way,"},{"Start":"04:06.770 ","End":"04:11.250","Text":"you should be getting used to these numbers and you can write them down even without"},{"Start":"04:11.250 ","End":"04:16.310","Text":"thinking as you become more and more familiar with powers of 2."},{"Start":"04:16.310 ","End":"04:19.580","Text":"But the last one here is now not 2,048,"},{"Start":"04:19.580 ","End":"04:22.280","Text":"it\u0027s minus 2,048,"},{"Start":"04:22.280 ","End":"04:23.945","Text":"just that 1 column."},{"Start":"04:23.945 ","End":"04:25.670","Text":"All the others are positive numbers."},{"Start":"04:25.670 ","End":"04:30.830","Text":"The implication for the answer then is that what we\u0027ve got here is"},{"Start":"04:30.830 ","End":"04:36.270","Text":"minus 2,048 plus whatever the rest of these bits are,"},{"Start":"04:36.270 ","End":"04:40.220","Text":"and we\u0027ve already worked that out in the previous answer, part b,"},{"Start":"04:40.220 ","End":"04:45.020","Text":"it was 822 when you added all these numbers together."},{"Start":"04:45.020 ","End":"04:50.160","Text":"What we want to do is we want to say minus 2,048 plus 822,"},{"Start":"04:50.160 ","End":"04:54.830","Text":"and that would be our result for part c. It\u0027s a little"},{"Start":"04:54.830 ","End":"04:59.690","Text":"bit harder to take a large negative number and add a positive number to it,"},{"Start":"04:59.690 ","End":"05:04.370","Text":"it\u0027s more intuitive to take a small number away from a larger number."},{"Start":"05:04.370 ","End":"05:07.580","Text":"We can do that. We can take 822 away"},{"Start":"05:07.580 ","End":"05:11.120","Text":"from 2,048 and then just change the sign at the end,"},{"Start":"05:11.120 ","End":"05:16.260","Text":"and that will be the same as doing minus 2,048 plus 822."},{"Start":"05:16.260 ","End":"05:19.005","Text":"Let me just show you that."},{"Start":"05:19.005 ","End":"05:24.675","Text":"If I put 2,048 not minus 2,048, but 2,048,"},{"Start":"05:24.675 ","End":"05:28.595","Text":"and then I subtract from that 822,"},{"Start":"05:28.595 ","End":"05:31.520","Text":"and then whatever the result is will change"},{"Start":"05:31.520 ","End":"05:35.585","Text":"the sign and it\u0027ll be the same as doing these operations here."},{"Start":"05:35.585 ","End":"05:38.165","Text":"Let\u0027s go ahead and do that."},{"Start":"05:38.165 ","End":"05:40.695","Text":"8 minus 2 is 6,"},{"Start":"05:40.695 ","End":"05:43.110","Text":"4 minus 2 is 2."},{"Start":"05:43.110 ","End":"05:49.125","Text":"Can\u0027t do 0 minus 8 so this becomes 1 and I bring a 1 across to here."},{"Start":"05:49.125 ","End":"05:52.665","Text":"10 minus 8 is 2,"},{"Start":"05:52.665 ","End":"05:53.970","Text":"and I\u0027ve got 1 here,"},{"Start":"05:53.970 ","End":"05:55.365","Text":"so I bring the 1 down."},{"Start":"05:55.365 ","End":"05:57.865","Text":"There is my answer 1,226."},{"Start":"05:57.865 ","End":"05:59.120","Text":"But as I said,"},{"Start":"05:59.120 ","End":"06:04.625","Text":"the result here is going to be minus 1,226 because this,"},{"Start":"06:04.625 ","End":"06:07.670","Text":"I did just as a nice convenience subtraction to"},{"Start":"06:07.670 ","End":"06:11.960","Text":"subtract the smaller number from the larger number,"},{"Start":"06:11.960 ","End":"06:13.610","Text":"it\u0027s the same as doing this operation,"},{"Start":"06:13.610 ","End":"06:15.080","Text":"but flip the sign at the end."},{"Start":"06:15.080 ","End":"06:18.590","Text":"It gives me minus 1,226."},{"Start":"06:18.590 ","End":"06:20.435","Text":"If you\u0027re unsure of that,"},{"Start":"06:20.435 ","End":"06:23.015","Text":"then if you look at the columns here,"},{"Start":"06:23.015 ","End":"06:25.310","Text":"you\u0027ve got minus 2,048,"},{"Start":"06:25.310 ","End":"06:26.645","Text":"and the rest of this,"},{"Start":"06:26.645 ","End":"06:30.530","Text":"you\u0027ve got 512 plus 256 plus some other numbers."},{"Start":"06:30.530 ","End":"06:36.660","Text":"You\u0027re looking at roughly 800 here being added on to minus 2,000."},{"Start":"06:36.660 ","End":"06:40.500","Text":"Minus 2,000 plus roughly 800 is roughly 1,200,"},{"Start":"06:40.500 ","End":"06:43.865","Text":"and you can see that numbers in that ballpark."},{"Start":"06:43.865 ","End":"06:45.725","Text":"It looks like we\u0027ve got it right."},{"Start":"06:45.725 ","End":"06:48.245","Text":"Final answer then."},{"Start":"06:48.245 ","End":"06:53.622","Text":"We\u0027ve been asked to work out what the pattern 1011, 0011,"},{"Start":"06:53.622 ","End":"06:58.130","Text":"0110 is as an unsigned number,"},{"Start":"06:58.130 ","End":"07:00.424","Text":"as a signed number using sign and magnitude,"},{"Start":"07:00.424 ","End":"07:02.825","Text":"and as a signed number using two\u0027s complement,"},{"Start":"07:02.825 ","End":"07:06.470","Text":"and once again, we see that they\u0027re very different results depending"},{"Start":"07:06.470 ","End":"07:10.505","Text":"on how we\u0027re interpreting them because it\u0027s"},{"Start":"07:10.505 ","End":"07:13.940","Text":"either indicating a positive digit"},{"Start":"07:13.940 ","End":"07:18.080","Text":"or we\u0027re using less of the bits to represent the number,"},{"Start":"07:18.080 ","End":"07:21.200","Text":"the magnitude, and then the final bit is used for the sign."},{"Start":"07:21.200 ","End":"07:23.930","Text":"Therefore, we got a very different result in part b and"},{"Start":"07:23.930 ","End":"07:27.005","Text":"then slightly different method for two\u0027s complement."},{"Start":"07:27.005 ","End":"07:29.210","Text":"The last bit does have a value,"},{"Start":"07:29.210 ","End":"07:30.830","Text":"but it\u0027s a negative value,"},{"Start":"07:30.830 ","End":"07:34.640","Text":"and then we add on all the positives to get a final result."},{"Start":"07:34.640 ","End":"07:38.370","Text":"There we go, thanks very much and I\u0027ll see you in the next one."}],"ID":26330},{"Watched":false,"Name":"Exercise 3 parts A-B","Duration":"3m 5s","ChapterTopicVideoID":25513,"CourseChapterTopicPlaylistID":237670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.185","Text":"Hello, again, everyone. In this question we\u0027ve been asked to convert"},{"Start":"00:04.185 ","End":"00:08.429","Text":"the decimal number minus 256 into binary."},{"Start":"00:08.429 ","End":"00:10.070","Text":"We\u0027re going to do that in 2 ways."},{"Start":"00:10.070 ","End":"00:13.095","Text":"In part A, we\u0027re going to do it using sign and magnitude,"},{"Start":"00:13.095 ","End":"00:16.245","Text":"and in part B, we\u0027re going to use two\u0027s complement."},{"Start":"00:16.245 ","End":"00:20.430","Text":"In previous exercises, we saw that sign and"},{"Start":"00:20.430 ","End":"00:23.760","Text":"magnitude representation involves a bit"},{"Start":"00:23.760 ","End":"00:26.810","Text":"which is reserved for the sign and the rest of it is the magnitude,"},{"Start":"00:26.810 ","End":"00:28.440","Text":"so the first thing we\u0027ve got to do is work out"},{"Start":"00:28.440 ","End":"00:31.260","Text":"the magnitude and how many bits we\u0027re going to need for that."},{"Start":"00:31.260 ","End":"00:34.500","Text":"Let\u0027s start by writing out the bit values."},{"Start":"00:34.500 ","End":"00:39.005","Text":"We keep going essentially until we don\u0027t need any more bits."},{"Start":"00:39.005 ","End":"00:43.320","Text":"Initially, writes out 2 nibbles and I need to keep going."},{"Start":"00:43.320 ","End":"00:45.290","Text":"There is an extra nibble here,"},{"Start":"00:45.290 ","End":"00:48.950","Text":"that I\u0027ll need to go into in order to express 256."},{"Start":"00:48.950 ","End":"00:50.240","Text":"I don\u0027t need to go any further."},{"Start":"00:50.240 ","End":"00:52.580","Text":"This turns out to be a very convenient number,"},{"Start":"00:52.580 ","End":"00:59.085","Text":"it\u0027s just going to be a single 1 and all of the other bits are going to be 0."},{"Start":"00:59.085 ","End":"01:03.875","Text":"That\u0027s the magnitude part of it worked out, we\u0027ve got 256."},{"Start":"01:03.875 ","End":"01:10.610","Text":"It\u0027s a simple case in sign and magnitude of setting the value for the sign bit."},{"Start":"01:10.610 ","End":"01:13.460","Text":"In this case, it\u0027s a negative number,"},{"Start":"01:13.460 ","End":"01:15.815","Text":"so the sign bit is going to need to be 1,"},{"Start":"01:15.815 ","End":"01:19.880","Text":"and we\u0027ve completed the conversion for part A,"},{"Start":"01:19.880 ","End":"01:23.130","Text":"and the answer is 110000 and four more 0s after that as well."},{"Start":"01:23.150 ","End":"01:27.365","Text":"There we go."},{"Start":"01:27.365 ","End":"01:32.270","Text":"That\u0027s minus 256 in sign and magnitude, right there."},{"Start":"01:32.270 ","End":"01:35.990","Text":"For two\u0027s complement, it\u0027s slightly different method,"},{"Start":"01:35.990 ","End":"01:40.760","Text":"but we still have to start by using the column values."},{"Start":"01:40.760 ","End":"01:44.052","Text":"We know now how many columns we need,"},{"Start":"01:44.052 ","End":"01:46.445","Text":"but let\u0027s write those out again."},{"Start":"01:46.445 ","End":"01:49.470","Text":"Now, there we go and we\u0027ll stop there."},{"Start":"01:49.470 ","End":"01:54.920","Text":"What we\u0027re looking for is minus 256 because this is two\u0027s compliment,"},{"Start":"01:54.920 ","End":"01:58.985","Text":"the most significant bit becomes a negative value."},{"Start":"01:58.985 ","End":"02:00.380","Text":"We don\u0027t need a sign bit."},{"Start":"02:00.380 ","End":"02:03.935","Text":"We just made this most significant bit a negative number."},{"Start":"02:03.935 ","End":"02:06.890","Text":"Once again, this is very convenient because the number we\u0027re"},{"Start":"02:06.890 ","End":"02:10.670","Text":"actually looking for in this case is minus 256."},{"Start":"02:10.670 ","End":"02:13.465","Text":"That\u0027s simply going to be a 1 there,"},{"Start":"02:13.465 ","End":"02:17.885","Text":"and all the other binary digits are going to be 0."},{"Start":"02:17.885 ","End":"02:24.215","Text":"We are done, and the final answer is 1 and four 0s,"},{"Start":"02:24.215 ","End":"02:27.075","Text":"followed by another four 0s."},{"Start":"02:27.075 ","End":"02:32.255","Text":"That is minus 256 in two\u0027s complement."},{"Start":"02:32.255 ","End":"02:37.820","Text":"Supposing this very simple example we\u0027re really seeing here is the number of bits"},{"Start":"02:37.820 ","End":"02:40.115","Text":"required might be different for"},{"Start":"02:40.115 ","End":"02:44.430","Text":"the same number in sign and magnitude and two\u0027s complement."},{"Start":"02:44.430 ","End":"02:46.820","Text":"In the next exercise we\u0027ll use"},{"Start":"02:46.820 ","End":"02:50.120","Text":"a slightly longer number and we\u0027ll see what effect that has."},{"Start":"02:50.120 ","End":"02:53.405","Text":"But because we\u0027re on a power of 2 boundary here,"},{"Start":"02:53.405 ","End":"02:55.475","Text":"if this was quite a convenient number,"},{"Start":"02:55.475 ","End":"02:58.730","Text":"and we only needed 9 bits for two\u0027s complement,"},{"Start":"02:58.730 ","End":"03:02.955","Text":"but we actually needed 10 bits for sign and magnitude."},{"Start":"03:02.955 ","End":"03:06.280","Text":"That\u0027s us done, I\u0027ll see you in the next one."}],"ID":26331},{"Watched":false,"Name":"Exercise 4 parts A-B","Duration":"7m 52s","ChapterTopicVideoID":25514,"CourseChapterTopicPlaylistID":237670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"Hey, everyone. In this question,"},{"Start":"00:02.970 ","End":"00:06.134","Text":"we\u0027ve been asked to convert a decimal number,"},{"Start":"00:06.134 ","End":"00:09.855","Text":"which is minus 1,394 into binary this time."},{"Start":"00:09.855 ","End":"00:12.560","Text":"Previously, we were looking at binary into decimal."},{"Start":"00:12.560 ","End":"00:14.130","Text":"The order is slightly different this time."},{"Start":"00:14.130 ","End":"00:19.080","Text":"In part a, we\u0027ve been asked to do two\u0027s complement and then part b, sign and magnitude."},{"Start":"00:19.080 ","End":"00:22.320","Text":"We were very lucky in that we had a quite convenient number last time."},{"Start":"00:22.320 ","End":"00:27.255","Text":"This is a longer number and it\u0027s not on a boundary of a power of 2."},{"Start":"00:27.255 ","End":"00:30.780","Text":"This is probably a bit more realistic on previous exercise."},{"Start":"00:30.780 ","End":"00:34.290","Text":"But we\u0027ll start by doing what we did before,"},{"Start":"00:34.290 ","End":"00:37.065","Text":"which is to work out how many columns we need."},{"Start":"00:37.065 ","End":"00:43.950","Text":"I\u0027m going to start out by writing the column values on the right-hand side,"},{"Start":"00:43.950 ","End":"00:48.440","Text":"and then working across 2 and stopping when we don\u0027t need any more columns."},{"Start":"00:48.440 ","End":"00:52.190","Text":"If we start with 1 and we basically just keep going"},{"Start":"00:52.190 ","End":"00:58.190","Text":"until we\u0027ve reached a number that is larger than the number we\u0027re looking to represent."},{"Start":"00:58.190 ","End":"01:02.850","Text":"256, 512, 1,024."},{"Start":"01:02.850 ","End":"01:04.838","Text":"We\u0027re now getting close,"},{"Start":"01:04.838 ","End":"01:07.820","Text":"and 2,048 is indeed larger"},{"Start":"01:07.820 ","End":"01:12.125","Text":"than the number that we\u0027re looking to represent, which is 1394."},{"Start":"01:12.125 ","End":"01:18.775","Text":"What I\u0027m going to do now is I\u0027m going to write out where I will need a 1 and obviously,"},{"Start":"01:18.775 ","End":"01:19.955","Text":"leave the others blank."},{"Start":"01:19.955 ","End":"01:22.235","Text":"Now this is a negative number."},{"Start":"01:22.235 ","End":"01:27.560","Text":"This most significant bit here in two\u0027s complement would be minus 2,048."},{"Start":"01:27.560 ","End":"01:30.050","Text":"But I\u0027m going to show you a method that, for the moment,"},{"Start":"01:30.050 ","End":"01:32.420","Text":"ignores that and we\u0027ll convert"},{"Start":"01:32.420 ","End":"01:36.610","Text":"the number first into one\u0027s complement and then into two\u0027s complement."},{"Start":"01:36.610 ","End":"01:39.420","Text":"For reasons that will become apparent,"},{"Start":"01:39.420 ","End":"01:41.280","Text":"it\u0027s a much easier method."},{"Start":"01:41.280 ","End":"01:44.556","Text":"First of all, let\u0027s try and represent the positive number,"},{"Start":"01:44.556 ","End":"01:47.330","Text":"1,394 in the bits that we have available."},{"Start":"01:47.330 ","End":"01:48.890","Text":"In order to do that,"},{"Start":"01:48.890 ","End":"01:50.510","Text":"I will need this column,"},{"Start":"01:50.510 ","End":"01:52.460","Text":"so I\u0027ll need 1,024."},{"Start":"01:52.460 ","End":"01:55.490","Text":"The number I originally started with,"},{"Start":"01:55.490 ","End":"01:58.985","Text":"I need to subtract 1,024 from it."},{"Start":"01:58.985 ","End":"02:03.205","Text":"That\u0027s 0 there, 7 there,"},{"Start":"02:03.205 ","End":"02:08.075","Text":"and 3 there, so I\u0027ve got 370 left to make up. I\u0027ll carry on."},{"Start":"02:08.075 ","End":"02:10.050","Text":"I don\u0027t need the 512 column,"},{"Start":"02:10.050 ","End":"02:11.555","Text":"so I put a 0 there."},{"Start":"02:11.555 ","End":"02:16.820","Text":"I do need the 256 column because 370 is greater than 256."},{"Start":"02:16.820 ","End":"02:18.185","Text":"I put a 1 there,"},{"Start":"02:18.185 ","End":"02:22.775","Text":"and now I have to subtract the 256 from 370."},{"Start":"02:22.775 ","End":"02:28.675","Text":"I\u0027ll need to borrow 1 from here because I can\u0027t subtract 6 from 0."},{"Start":"02:28.675 ","End":"02:31.860","Text":"Let\u0027s take 1 from here."},{"Start":"02:31.860 ","End":"02:34.150","Text":"10 minus 6 would be 4,"},{"Start":"02:34.150 ","End":"02:37.806","Text":"6 now minus 5 would be 1,"},{"Start":"02:37.806 ","End":"02:40.350","Text":"and 3 minus 2 is 1."},{"Start":"02:40.350 ","End":"02:41.970","Text":"Now I\u0027ve got 114."},{"Start":"02:41.970 ","End":"02:46.450","Text":"Carrying on. I don\u0027t need 128 because 114 is less than 128."},{"Start":"02:46.450 ","End":"02:52.273","Text":"I do need 64, but I\u0027m going to have to now subtract 64 from 114,"},{"Start":"02:52.273 ","End":"02:53.690","Text":"and you get the idea."},{"Start":"02:53.690 ","End":"02:57.005","Text":"You just carry on in this manner."},{"Start":"02:57.005 ","End":"03:00.860","Text":"50 is indeed greater than 32."},{"Start":"03:00.860 ","End":"03:03.140","Text":"I will need a 32."},{"Start":"03:03.140 ","End":"03:10.235","Text":"Therefore, subtract 32 from 50 to work out what I need the remaining bits. That\u0027s 18."},{"Start":"03:10.235 ","End":"03:12.965","Text":"Now I can carry on inspecting each bit and comparing it,"},{"Start":"03:12.965 ","End":"03:15.980","Text":"but I can actually see now because I\u0027ve very few bits"},{"Start":"03:15.980 ","End":"03:19.520","Text":"left that 18 can be made up of 16 and 2."},{"Start":"03:19.520 ","End":"03:22.650","Text":"I\u0027ll just write a 1 where 16 is,"},{"Start":"03:22.650 ","End":"03:25.070","Text":"and a 1 where 2 is,"},{"Start":"03:25.070 ","End":"03:27.140","Text":"the other bits are going to be 0."},{"Start":"03:27.140 ","End":"03:30.770","Text":"There\u0027s my bit pattern now for 1,394."},{"Start":"03:30.770 ","End":"03:33.860","Text":"You\u0027ll notice I\u0027ve ignored this most significant bit."},{"Start":"03:33.860 ","End":"03:35.510","Text":"I\u0027m assuming these are all positive."},{"Start":"03:35.510 ","End":"03:37.835","Text":"They\u0027re not because this is a two\u0027s complement number."},{"Start":"03:37.835 ","End":"03:39.500","Text":"This one would be a negative number."},{"Start":"03:39.500 ","End":"03:43.545","Text":"But for now, let\u0027s just assume it\u0027s positive and I\u0027m going to put a 0 there."},{"Start":"03:43.545 ","End":"03:48.545","Text":"This representation is positive 1,394."},{"Start":"03:48.545 ","End":"03:50.930","Text":"Let\u0027s get it into a negative number now,"},{"Start":"03:50.930 ","End":"03:52.970","Text":"and the method for that is to,"},{"Start":"03:52.970 ","End":"03:56.525","Text":"first of all, take the one\u0027s complement of all these bits."},{"Start":"03:56.525 ","End":"03:57.710","Text":"To do the one\u0027s complement,"},{"Start":"03:57.710 ","End":"04:01.745","Text":"you simply just flip all the bits to the opposite value."},{"Start":"04:01.745 ","End":"04:03.980","Text":"Where you had a 0, you turn it into a 1."},{"Start":"04:03.980 ","End":"04:05.960","Text":"Where you have a 1, you turn it into a 0."},{"Start":"04:05.960 ","End":"04:09.320","Text":"Let\u0027s do that. There we go."},{"Start":"04:09.320 ","End":"04:11.780","Text":"Now, to turn this into two\u0027s complement,"},{"Start":"04:11.780 ","End":"04:14.953","Text":"you simply add 1 to the number,"},{"Start":"04:14.953 ","End":"04:17.780","Text":"and addition we haven\u0027t really covered yet in the exercises,"},{"Start":"04:17.780 ","End":"04:20.150","Text":"but it\u0027s a simple enough process."},{"Start":"04:20.150 ","End":"04:22.925","Text":"It\u0027s the same as it is for a decimal number."},{"Start":"04:22.925 ","End":"04:25.670","Text":"You just add the columns together."},{"Start":"04:25.670 ","End":"04:27.117","Text":"If they fit in the column, great."},{"Start":"04:27.117 ","End":"04:29.600","Text":"If not, you carry over to the next column."},{"Start":"04:29.600 ","End":"04:32.900","Text":"What we\u0027re trying to add here is a number 1,"},{"Start":"04:32.900 ","End":"04:37.025","Text":"which if I write it out in full with all of these bits here,"},{"Start":"04:37.025 ","End":"04:39.830","Text":"would be represented in this way."},{"Start":"04:39.830 ","End":"04:47.405","Text":"I want to add these two numbers together and see if I can add the columns 1 plus 1 is 2."},{"Start":"04:47.405 ","End":"04:50.480","Text":"I can\u0027t write 2 because this is a binary number."},{"Start":"04:50.480 ","End":"04:51.800","Text":"I\u0027d have to write 1, 0,"},{"Start":"04:51.800 ","End":"04:53.895","Text":"which is the binary equivalent of 2."},{"Start":"04:53.895 ","End":"04:56.180","Text":"I can\u0027t even write those in 1 column,"},{"Start":"04:56.180 ","End":"04:59.600","Text":"so I write the 0 and I carry the 1 across."},{"Start":"04:59.600 ","End":"05:02.900","Text":"Now I had 1 plus 0 plus 0."},{"Start":"05:02.900 ","End":"05:04.390","Text":"That is just 1."},{"Start":"05:04.390 ","End":"05:07.710","Text":"I can write the 1 there and it fits in 1 column and then just carry on"},{"Start":"05:07.710 ","End":"05:11.450","Text":"across all of these other columns doing the same thing."},{"Start":"05:11.450 ","End":"05:14.150","Text":"1 plus 0 is 1, 1 plus 0 is 1,"},{"Start":"05:14.150 ","End":"05:21.030","Text":"0 plus 0 is 0, 0, 0, 1, 0, 1, 0, 1."},{"Start":"05:22.180 ","End":"05:26.150","Text":"I now have the number in two\u0027s complement."},{"Start":"05:26.150 ","End":"05:30.860","Text":"This is the representation of minus 1,394."},{"Start":"05:30.860 ","End":"05:33.095","Text":"Here\u0027s positive 1,394."},{"Start":"05:33.095 ","End":"05:35.600","Text":"I\u0027ve taken the one\u0027s complement, flipping all the bits,"},{"Start":"05:35.600 ","End":"05:38.930","Text":"and then adding 1 gives me the two\u0027s complement."},{"Start":"05:38.930 ","End":"05:43.055","Text":"I now have minus 1394 in this bit pattern here."},{"Start":"05:43.055 ","End":"05:44.930","Text":"If you look at the column values,"},{"Start":"05:44.930 ","End":"05:48.795","Text":"this would be minus 2,048."},{"Start":"05:48.795 ","End":"05:50.568","Text":"If I write it up here,"},{"Start":"05:50.568 ","End":"05:52.970","Text":"and this number now therefore,"},{"Start":"05:52.970 ","End":"05:57.075","Text":"becomes minus 2,048 with all of these bits added on."},{"Start":"05:57.075 ","End":"06:00.300","Text":"If you look at that, we\u0027ve got 500 there, 128 there,"},{"Start":"06:00.300 ","End":"06:06.140","Text":"so roughly 600, and we\u0027re starting off with minus 2,000 and adding 600 to it."},{"Start":"06:06.140 ","End":"06:09.420","Text":"You\u0027ll see that will give us minus 1,400 roughly,"},{"Start":"06:09.420 ","End":"06:10.800","Text":"which is where we\u0027re at."},{"Start":"06:10.800 ","End":"06:15.230","Text":"It looks like with just a very rough check that we\u0027ve got the right number there."},{"Start":"06:15.230 ","End":"06:17.840","Text":"We have done the first part,"},{"Start":"06:17.840 ","End":"06:20.975","Text":"and so let\u0027s just write it out formally the answer,"},{"Start":"06:20.975 ","End":"06:26.705","Text":"what is minus 1,394 in two\u0027s complement would be as so."},{"Start":"06:26.705 ","End":"06:32.415","Text":"That\u0027s 101010001110."},{"Start":"06:32.415 ","End":"06:36.930","Text":"That is minus 1,394 in two\u0027s complement."},{"Start":"06:36.930 ","End":"06:41.300","Text":"Part b, we\u0027ve been asked to convert it to sign and magnitude format."},{"Start":"06:41.300 ","End":"06:42.770","Text":"It\u0027s actually very straightforward."},{"Start":"06:42.770 ","End":"06:44.900","Text":"We already had 1,394,"},{"Start":"06:44.900 ","End":"06:46.730","Text":"which was this bit pattern here."},{"Start":"06:46.730 ","End":"06:49.880","Text":"All we need to do now is add a sign bit to it and we\u0027ll"},{"Start":"06:49.880 ","End":"06:53.795","Text":"have minus 1,394 in sign and magnitude."},{"Start":"06:53.795 ","End":"06:56.060","Text":"If I just look at the magnitude part of"},{"Start":"06:56.060 ","End":"06:59.695","Text":"this first number where we worked out positive 1,394,"},{"Start":"06:59.695 ","End":"07:01.560","Text":"I could just make use of that."},{"Start":"07:01.560 ","End":"07:09.120","Text":"It\u0027s 101 0111 0010."},{"Start":"07:09.120 ","End":"07:10.190","Text":"There was my initial number,"},{"Start":"07:10.190 ","End":"07:17.450","Text":"but I need a sign bit here because this is a negative number that is going to be here,"},{"Start":"07:17.450 ","End":"07:20.510","Text":"which is going to be a 1 for my sign bit."},{"Start":"07:20.510 ","End":"07:22.820","Text":"There is my final answer then."},{"Start":"07:22.820 ","End":"07:31.660","Text":"Nice and easy because I\u0027ve already done most of the work, 1101 0111 0010."},{"Start":"07:31.660 ","End":"07:38.164","Text":"You\u0027ll see the bit pattern for minus 1,394 in two\u0027s complement"},{"Start":"07:38.164 ","End":"07:44.705","Text":"looks radically different to the bit pattern for minus 1,394 in sign and magnitude,"},{"Start":"07:44.705 ","End":"07:49.025","Text":"but rest assured those are the correct representation."},{"Start":"07:49.025 ","End":"07:50.825","Text":"That\u0027s this exercise done."},{"Start":"07:50.825 ","End":"07:53.370","Text":"Thanks very much and I\u0027ll see you in the next one."}],"ID":26332},{"Watched":false,"Name":"Exercise 5 parts A-B","Duration":"4m 10s","ChapterTopicVideoID":25515,"CourseChapterTopicPlaylistID":237670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.590","Text":"Hey, welcome back everyone."},{"Start":"00:01.590 ","End":"00:04.410","Text":"In this question, we\u0027ve been asked to state the range of"},{"Start":"00:04.410 ","End":"00:08.970","Text":"a 12-bit number using two\u0027s complement and sign and magnitude."},{"Start":"00:08.970 ","End":"00:13.380","Text":"We\u0027re talking here about a signed number and there is a formula we"},{"Start":"00:13.380 ","End":"00:17.925","Text":"can use to work out a range and you simply have to memorize this for the exams."},{"Start":"00:17.925 ","End":"00:23.070","Text":"The formula is, so it\u0027s 2 to the n minus 1,"},{"Start":"00:23.070 ","End":"00:26.010","Text":"I should say on the negative side,"},{"Start":"00:26.010 ","End":"00:30.000","Text":"up to 2 to the n minus 1 on the positive side,"},{"Start":"00:30.000 ","End":"00:32.070","Text":"but we have to subtract 1 from that,"},{"Start":"00:32.070 ","End":"00:33.570","Text":"and I\u0027ll explain why in a moment."},{"Start":"00:33.570 ","End":"00:34.830","Text":"If you have a calculator,"},{"Start":"00:34.830 ","End":"00:39.105","Text":"this is a simple case of taking the n minus 1 part,"},{"Start":"00:39.105 ","End":"00:43.565","Text":"putting 2 to the power of that and then the same on the positive side."},{"Start":"00:43.565 ","End":"00:46.070","Text":"But if you don\u0027t have a calculator,"},{"Start":"00:46.070 ","End":"00:48.110","Text":"what we have to work out is,"},{"Start":"00:48.110 ","End":"00:51.350","Text":"how do we find out what 2^11 is."},{"Start":"00:51.350 ","End":"00:54.740","Text":"We can use the method we\u0027ve been using up until now to do that."},{"Start":"00:54.740 ","End":"00:57.230","Text":"Let\u0027s start writing out digits from"},{"Start":"00:57.230 ","End":"01:02.150","Text":"0-12 and 0 is where we start and that\u0027s important that we start at 0,"},{"Start":"01:02.150 ","End":"01:07.411","Text":"so don\u0027t forget to do that, 0,"},{"Start":"01:07.411 ","End":"01:08.614","Text":"1, 2, 3,"},{"Start":"01:08.614 ","End":"01:10.221","Text":"4, 5, 6, 7,"},{"Start":"01:10.221 ","End":"01:11.627","Text":"8, 9, 10,"},{"Start":"01:11.627 ","End":"01:13.340","Text":"11, and 12."},{"Start":"01:13.340 ","End":"01:15.320","Text":"Let me stress again,"},{"Start":"01:15.320 ","End":"01:17.060","Text":"you need to start from 0."},{"Start":"01:17.060 ","End":"01:22.440","Text":"Then what we need to do is right underneath starting with 1,"},{"Start":"01:22.440 ","End":"01:25.140","Text":"because 2^0 is 1,"},{"Start":"01:25.140 ","End":"01:26.680","Text":"that\u0027s really all you need to remember."},{"Start":"01:26.680 ","End":"01:29.260","Text":"Anything to the power of 0 is 1, obviously."},{"Start":"01:29.260 ","End":"01:32.755","Text":"Then we just keep moving across and doubling each time."},{"Start":"01:32.755 ","End":"01:36.935","Text":"So we have 2 there, 4, 8,1 6, 32, 64, 128, 256, 512,"},{"Start":"01:36.935 ","End":"01:37.940","Text":"1,024,"},{"Start":"01:44.630 ","End":"01:47.210","Text":"2,048,"},{"Start":"01:47.210 ","End":"01:49.990","Text":"and then finally 4,096."},{"Start":"01:49.990 ","End":"01:52.990","Text":"What we\u0027ve been asked to work out"},{"Start":"01:52.990 ","End":"01:56.470","Text":"is the range of a 12-bit number and we\u0027re going so therefore"},{"Start":"01:56.470 ","End":"02:03.330","Text":"2^11 minus 2^11 and therefore we\u0027re going to need this digit here, 2^11 there."},{"Start":"02:03.330 ","End":"02:04.760","Text":"That\u0027s the number that we want,"},{"Start":"02:04.760 ","End":"02:06.990","Text":"so it\u0027s 2,048, we know,"},{"Start":"02:06.990 ","End":"02:10.985","Text":"is 2^11 and that\u0027s the negative side then done,"},{"Start":"02:10.985 ","End":"02:12.560","Text":"for the positive side,"},{"Start":"02:12.560 ","End":"02:15.240","Text":"it\u0027s going to be 2,048,"},{"Start":"02:15.240 ","End":"02:19.670","Text":"what we\u0027re going to subtract from that 1 so the final answer will therefore"},{"Start":"02:19.670 ","End":"02:25.225","Text":"be minus 2,048 to 2,047,"},{"Start":"02:25.225 ","End":"02:27.935","Text":"and we\u0027re done with part A."},{"Start":"02:27.935 ","End":"02:29.430","Text":"For sign and magnitude,"},{"Start":"02:29.430 ","End":"02:32.865","Text":"the formula is slightly different, it\u0027s this,"},{"Start":"02:32.865 ","End":"02:34.455","Text":"2 to the n minus 1,"},{"Start":"02:34.455 ","End":"02:39.740","Text":"minus 1 and then the positive part is the same as two\u0027s complement."},{"Start":"02:39.740 ","End":"02:42.005","Text":"So it\u0027s actually symmetrical,"},{"Start":"02:42.005 ","End":"02:45.740","Text":"which actually easier to remember this formula and as the previous one,"},{"Start":"02:45.740 ","End":"02:48.210","Text":"because we are getting minus 2 to the n minus 1,"},{"Start":"02:48.210 ","End":"02:51.090","Text":"minus 1 up to 2 to the n minus 1, minus 1,"},{"Start":"02:51.090 ","End":"02:53.630","Text":"so it just looks a little bit neater to the eyes,"},{"Start":"02:53.630 ","End":"02:54.830","Text":"slightly easier to remember."},{"Start":"02:54.830 ","End":"02:57.515","Text":"So you might want to remember it this way and then"},{"Start":"02:57.515 ","End":"03:01.115","Text":"just know that the two\u0027s complement version is lop-sided."},{"Start":"03:01.115 ","End":"03:05.580","Text":"The negative half doesn\u0027t have minus 1 subtracted from it."},{"Start":"03:05.580 ","End":"03:09.049","Text":"In this case then we\u0027ve got the same number as before."},{"Start":"03:09.049 ","End":"03:12.110","Text":"We know 2^11 is 2,048,"},{"Start":"03:12.110 ","End":"03:16.655","Text":"so we simply need to write here 2,048 minus 1,"},{"Start":"03:16.655 ","End":"03:20.930","Text":"and it\u0027s going to go up to positive 2,048 minus 1,"},{"Start":"03:20.930 ","End":"03:22.985","Text":"so the final answer, therefore,"},{"Start":"03:22.985 ","End":"03:28.935","Text":"is minus 2,047 up to positive 2,047,"},{"Start":"03:28.935 ","End":"03:32.330","Text":"and we have now completed the question and what"},{"Start":"03:32.330 ","End":"03:35.810","Text":"we\u0027re seeing here is basically that a two\u0027s complement number has"},{"Start":"03:35.810 ","End":"03:39.770","Text":"a slightly larger range by 1 and the reason for that is"},{"Start":"03:39.770 ","End":"03:44.705","Text":"because when you are encoding a number as sign and magnitude,"},{"Start":"03:44.705 ","End":"03:46.220","Text":"which we are in part B,"},{"Start":"03:46.220 ","End":"03:50.840","Text":"you\u0027re seeing that the number 0 has two representations,"},{"Start":"03:50.840 ","End":"03:52.640","Text":"minus 0 and positive 0,"},{"Start":"03:52.640 ","End":"03:54.260","Text":"it sounds crazy, but obviously,"},{"Start":"03:54.260 ","End":"03:56.390","Text":"the magnitude part occupies some of"},{"Start":"03:56.390 ","End":"03:59.630","Text":"the number and the final most significant bit is the sign bit."},{"Start":"03:59.630 ","End":"04:03.110","Text":"So you can actually have a bit pattern of plus 0 and minus 0 and"},{"Start":"04:03.110 ","End":"04:06.710","Text":"that explains why you\u0027ve got 1 less than you have with two\u0027s complement."},{"Start":"04:06.710 ","End":"04:09.900","Text":"So that\u0027s it for this question. I\u0027ll see you for the next one."}],"ID":26333},{"Watched":false,"Name":"Exercise 6","Duration":"5m 4s","ChapterTopicVideoID":25516,"CourseChapterTopicPlaylistID":237670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.490","Text":"Hello, everyone. In this question we\u0027ve been asked to find out how many bits would be"},{"Start":"00:05.490 ","End":"00:11.580","Text":"required to store a decimal number minus 32,000 in two\u0027s complement representation."},{"Start":"00:11.580 ","End":"00:15.645","Text":"So this is really a disguised version of the range question,"},{"Start":"00:15.645 ","End":"00:19.710","Text":"and we\u0027re going to need to have a slightly different formula this time,"},{"Start":"00:19.710 ","End":"00:22.490","Text":"and that\u0027s to work out the number of bits required,"},{"Start":"00:22.490 ","End":"00:25.485","Text":"and that formula looks like this."},{"Start":"00:25.485 ","End":"00:32.220","Text":"The number of bits b is log base 2 of p,"},{"Start":"00:32.220 ","End":"00:33.930","Text":"the number we\u0027re trying to find out."},{"Start":"00:33.930 ","End":"00:36.818","Text":"The reason I\u0027ve called it p is,"},{"Start":"00:36.818 ","End":"00:42.030","Text":"for the moment, we are going to treat this as a positive number."},{"Start":"00:42.030 ","End":"00:46.125","Text":"So let\u0027s just look at 32,000 not minus 32,000."},{"Start":"00:46.125 ","End":"00:51.770","Text":"If you type this into your calculator log base 2 of 32,000,"},{"Start":"00:51.770 ","End":"00:54.155","Text":"you will get an answer of"},{"Start":"00:54.155 ","End":"01:01.560","Text":"14.96578 and some other digits which I\u0027m not going to write down,"},{"Start":"01:01.560 ","End":"01:05.510","Text":"because we are not going to include this fractional part."},{"Start":"01:05.510 ","End":"01:07.550","Text":"We\u0027re simply going to ignore it."},{"Start":"01:07.550 ","End":"01:09.327","Text":"The reason for that is,"},{"Start":"01:09.327 ","End":"01:11.300","Text":"whatever number you\u0027re trying to find out,"},{"Start":"01:11.300 ","End":"01:15.785","Text":"p, is going to fall generally between 2 powers of 2."},{"Start":"01:15.785 ","End":"01:18.155","Text":"If it was exactly 14,"},{"Start":"01:18.155 ","End":"01:23.700","Text":"you\u0027ll find that 2^14 is 16384,"},{"Start":"01:23.700 ","End":"01:26.370","Text":"if it was exactly 15, the next power up,"},{"Start":"01:26.370 ","End":"01:29.925","Text":"it would be 32,768,"},{"Start":"01:29.925 ","End":"01:33.980","Text":"and that number is generally going to fall somewhere in-between those 2."},{"Start":"01:33.980 ","End":"01:36.230","Text":"It might be exactly 16,384,"},{"Start":"01:36.230 ","End":"01:38.240","Text":"it might be exactly 32,768,"},{"Start":"01:38.240 ","End":"01:40.160","Text":"but the chances are it\u0027s going to be somewhere in-between."},{"Start":"01:40.160 ","End":"01:42.658","Text":"Hence, why we ignore the fractional part,"},{"Start":"01:42.658 ","End":"01:45.680","Text":"and we simply add one on to the end."},{"Start":"01:45.680 ","End":"01:50.495","Text":"Our final answer for b is going to be 15,"},{"Start":"01:50.495 ","End":"01:55.220","Text":"except it\u0027s not really our final answer because this is just for a positive number."},{"Start":"01:55.220 ","End":"01:57.260","Text":"In summary, for this first bit,"},{"Start":"01:57.260 ","End":"02:01.805","Text":"we just needed to work out how many bits do we need for 32,000,"},{"Start":"02:01.805 ","End":"02:03.930","Text":"and we\u0027ve worked out that it\u0027s 15,"},{"Start":"02:03.930 ","End":"02:07.625","Text":"by taking the log of 32,000 to base 2,"},{"Start":"02:07.625 ","End":"02:11.070","Text":"ignoring the fractional part by adding 1 on,"},{"Start":"02:11.070 ","End":"02:14.330","Text":"so that we fall somewhere within the range."},{"Start":"02:14.330 ","End":"02:17.087","Text":"That\u0027s the first part done."},{"Start":"02:17.087 ","End":"02:20.920","Text":"What we now need to do is to think about it in terms of a negative number."},{"Start":"02:20.920 ","End":"02:24.170","Text":"You\u0027ll recall that the formula for the range"},{"Start":"02:24.170 ","End":"02:27.740","Text":"of a two\u0027s complement number looks like this,"},{"Start":"02:27.740 ","End":"02:35.500","Text":"it\u0027s minus 2 to the n minus 1 up to positive 2 to the n minus 1,"},{"Start":"02:35.500 ","End":"02:37.335","Text":"with 1 taken away."},{"Start":"02:37.335 ","End":"02:41.780","Text":"That number n is the number of bits that we\u0027re going to need."},{"Start":"02:41.780 ","End":"02:46.475","Text":"We have already worked out to get 32,000 we need 15."},{"Start":"02:46.475 ","End":"02:53.285","Text":"If we were to write this out minus 2^15 up to 2^15 with 1 taken off,"},{"Start":"02:53.285 ","End":"03:03.043","Text":"you\u0027ll find that that range is minus 32,768 up to 32,767."},{"Start":"03:03.043 ","End":"03:05.510","Text":"So it is 15 that we need up here."},{"Start":"03:05.510 ","End":"03:08.840","Text":"Then what we need to do finally to work out how many bits we"},{"Start":"03:08.840 ","End":"03:12.360","Text":"need for a two\u0027s complement number is to simply say,"},{"Start":"03:12.360 ","End":"03:16.215","Text":"well, we know that n minus 1 is 15,"},{"Start":"03:16.215 ","End":"03:19.296","Text":"therefore, n must be 16."},{"Start":"03:19.296 ","End":"03:22.640","Text":"So we\u0027ve now worked out how many bits we need to"},{"Start":"03:22.640 ","End":"03:26.770","Text":"represent a number minus 32,000 in two\u0027s complement."},{"Start":"03:26.770 ","End":"03:30.365","Text":"Now, what if you don\u0027t have access to a calculator in the exam?"},{"Start":"03:30.365 ","End":"03:33.619","Text":"Well, we can still work out how many bits will be required,"},{"Start":"03:33.619 ","End":"03:36.320","Text":"but it\u0027s going to be a bit more long-winded."},{"Start":"03:36.320 ","End":"03:38.060","Text":"We\u0027ve done similar things previously,"},{"Start":"03:38.060 ","End":"03:44.510","Text":"so what we do is we start out by writing out the digit 1 and then moving to the left,"},{"Start":"03:44.510 ","End":"03:48.117","Text":"just doubling each time as we have done many times before."},{"Start":"03:48.117 ","End":"03:49.880","Text":"So eventually, we\u0027ll get to"},{"Start":"03:49.880 ","End":"03:53.990","Text":"a number that\u0027s larger than the number that we\u0027re looking for."},{"Start":"03:53.990 ","End":"03:59.350","Text":"But it\u0027s going to take us a while to get there, 2048,"},{"Start":"03:59.350 ","End":"04:06.245","Text":"4096, 8192 getting close now, 16384, 32768."},{"Start":"04:06.245 ","End":"04:10.715","Text":"So we\u0027ve now got a number that\u0027s larger than 32,000,"},{"Start":"04:10.715 ","End":"04:15.725","Text":"and, of course, in two\u0027s complement this last column is going to be a negative number,"},{"Start":"04:15.725 ","End":"04:22.533","Text":"but we will be able to express the range minus 32,768 up to positive 32,767,"},{"Start":"04:22.533 ","End":"04:26.000","Text":"because if you add all of these digits together and this bit is a 0,"},{"Start":"04:26.000 ","End":"04:27.965","Text":"it will be 32,767."},{"Start":"04:27.965 ","End":"04:34.430","Text":"So we now simply have to count the number of bits that we\u0027ve got that we\u0027ve counted out,"},{"Start":"04:34.430 ","End":"04:37.265","Text":"and it turns out to be 16,"},{"Start":"04:37.265 ","End":"04:42.087","Text":"and that\u0027s exactly what we calculated when we had a calculator."},{"Start":"04:42.087 ","End":"04:48.300","Text":"So we\u0027ve got our final answer,"},{"Start":"04:48.300 ","End":"04:50.600","Text":"the number of bits required is 16,"},{"Start":"04:50.600 ","End":"04:55.215","Text":"which we worked out here by just using accounting method,"},{"Start":"04:55.215 ","End":"05:00.974","Text":"and here we worked out using 2 formulae and a calculator."},{"Start":"05:00.974 ","End":"05:05.340","Text":"So there we go. That\u0027s your answer to this one. I\u0027ll see you in the next one."}],"ID":26334},{"Watched":false,"Name":"Exercise 7 part A","Duration":"3m 24s","ChapterTopicVideoID":25517,"CourseChapterTopicPlaylistID":237670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.759","Text":"Hello, again. In this video we are going to perform a binary addition."},{"Start":"00:05.759 ","End":"00:10.410","Text":"We\u0027ve been told to perform the addition with 12 bits in the operation"},{"Start":"00:10.410 ","End":"00:15.450","Text":"and to use unsigned numbers in each of these parts of the question."},{"Start":"00:15.450 ","End":"00:21.270","Text":"We start with an 8-bit unsigned number and we\u0027re adding to another 8-bit unsigned number."},{"Start":"00:21.270 ","End":"00:23.220","Text":"Given that it\u0027s in 12 bits,"},{"Start":"00:23.220 ","End":"00:28.095","Text":"we should start by padding 4 bits at the beginning of each number."},{"Start":"00:28.095 ","End":"00:30.180","Text":"Obviously, that doesn\u0027t make a difference to"},{"Start":"00:30.180 ","End":"00:32.910","Text":"the number itself because they are leading 0s."},{"Start":"00:32.910 ","End":"00:36.500","Text":"There\u0027s our first number and adding to that again,"},{"Start":"00:36.500 ","End":"00:41.480","Text":"we\u0027re going to pad the first 4 bits because it is only an 8-bit number that we\u0027ve been"},{"Start":"00:41.480 ","End":"00:46.940","Text":"given and align all the digits underneath for the second binary number."},{"Start":"00:46.940 ","End":"00:49.670","Text":"There we go. If we add those two together,"},{"Start":"00:49.670 ","End":"00:54.274","Text":"we use the same method for adding as we would do for decimal numbers,"},{"Start":"00:54.274 ","End":"00:56.546","Text":"just doing a columnar addition."},{"Start":"00:56.546 ","End":"00:59.700","Text":"Each time the number can be represented in 1 column,"},{"Start":"00:59.700 ","End":"01:01.730","Text":"we just write the number, if it cannot,"},{"Start":"01:01.730 ","End":"01:04.010","Text":"we carry over to the next column."},{"Start":"01:04.010 ","End":"01:08.060","Text":"Now it\u0027s worth writing down at the side of your paper when you\u0027re doing an exam,"},{"Start":"01:08.060 ","End":"01:10.550","Text":"what exactly all of the combinations are,"},{"Start":"01:10.550 ","End":"01:13.100","Text":"because it\u0027s very easy to get mixed up."},{"Start":"01:13.100 ","End":"01:17.250","Text":"For example 0 plus 1 is 1 as we know,"},{"Start":"01:17.250 ","End":"01:18.810","Text":"and so it\u0027s 1 plus 0,"},{"Start":"01:18.810 ","End":"01:21.570","Text":"but 1 plus 1 is 1,"},{"Start":"01:21.570 ","End":"01:24.005","Text":"0, we\u0027re obviously in binary."},{"Start":"01:24.005 ","End":"01:27.065","Text":"When we write it down, we need to write down as 1, 0."},{"Start":"01:27.065 ","End":"01:29.600","Text":"Similarly, if we have a carry bit involved,"},{"Start":"01:29.600 ","End":"01:32.420","Text":"1 plus 1 plus 1 is 1, 1."},{"Start":"01:32.420 ","End":"01:35.090","Text":"Now, it may seem obvious now, but as I say,"},{"Start":"01:35.090 ","End":"01:39.410","Text":"it\u0027s very easy for the brain to misfire and for you to make a silly mistake."},{"Start":"01:39.410 ","End":"01:42.590","Text":"Let\u0027s go through this particular calculation now."},{"Start":"01:42.590 ","End":"01:45.440","Text":"What we\u0027re looking at is 0 plus 0,"},{"Start":"01:45.440 ","End":"01:48.280","Text":"which we know is 0, 0 plus 1 is 1,"},{"Start":"01:48.280 ","End":"01:50.850","Text":"1 plus 0 is 1,"},{"Start":"01:50.850 ","End":"01:54.038","Text":"1 plus 0 is 1 again,"},{"Start":"01:54.038 ","End":"01:55.665","Text":"and 1 again,"},{"Start":"01:55.665 ","End":"02:02.480","Text":"as we keep moving in this direction and then all these remaining digits are 0 plus 0,"},{"Start":"02:02.480 ","End":"02:04.405","Text":"so we know they\u0027re all going to be 0s."},{"Start":"02:04.405 ","End":"02:06.360","Text":"I\u0027ll just fill those in 1 go."},{"Start":"02:06.360 ","End":"02:08.795","Text":"We now have a 12-bit number,"},{"Start":"02:08.795 ","End":"02:11.795","Text":"which is the answer to this question."},{"Start":"02:11.795 ","End":"02:13.900","Text":"The result is that."},{"Start":"02:13.900 ","End":"02:17.090","Text":"If we wanted to do a quick crosscheck here,"},{"Start":"02:17.090 ","End":"02:20.570","Text":"you could get a calculator out and work out,"},{"Start":"02:20.570 ","End":"02:22.655","Text":"if you are allowed a calculator that is,"},{"Start":"02:22.655 ","End":"02:25.715","Text":"or you could simply use the method that we have always done,"},{"Start":"02:25.715 ","End":"02:28.250","Text":"which is to label the columns."},{"Start":"02:28.250 ","End":"02:30.290","Text":"I don\u0027t need to go any further because these are all 0s,"},{"Start":"02:30.290 ","End":"02:32.950","Text":"but if we were to add together these columns,"},{"Start":"02:32.950 ","End":"02:37.165","Text":"you\u0027d find that this value comes out to be 172 in"},{"Start":"02:37.165 ","End":"02:43.190","Text":"decimal and the one below it is 64 plus 16 is 80, 82."},{"Start":"02:43.190 ","End":"02:47.615","Text":"If you add those two numbers together, you\u0027ll get 254."},{"Start":"02:47.615 ","End":"02:50.630","Text":"Actually, we can see this number here"},{"Start":"02:50.630 ","End":"02:54.920","Text":"is all of the bits in the 8-bit number apart from 1,"},{"Start":"02:54.920 ","End":"02:57.350","Text":"and that would actually give us 254."},{"Start":"02:57.350 ","End":"02:59.945","Text":"We know we\u0027ve done our calculation correctly."},{"Start":"02:59.945 ","End":"03:02.870","Text":"Now, be careful about writing this at the side"},{"Start":"03:02.870 ","End":"03:05.750","Text":"of your paper if you are using this as a cross-check."},{"Start":"03:05.750 ","End":"03:09.230","Text":"You do want to make it obvious that you have used binary for"},{"Start":"03:09.230 ","End":"03:11.570","Text":"the addition not cheated and gone"},{"Start":"03:11.570 ","End":"03:14.675","Text":"straight to a decimal number and another decimal number,"},{"Start":"03:14.675 ","End":"03:18.290","Text":"and then added them up to produce a final decimal number,"},{"Start":"03:18.290 ","End":"03:20.320","Text":"which you\u0027ve then converted into binary."},{"Start":"03:20.320 ","End":"03:25.890","Text":"You need to make clear that you answered this question by doing a binary addition."}],"ID":26335},{"Watched":false,"Name":"Exercise 7 part B","Duration":"3m 13s","ChapterTopicVideoID":25518,"CourseChapterTopicPlaylistID":237670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.320 ","End":"00:03.090","Text":"Continuing on with part B."},{"Start":"00:03.090 ","End":"00:04.500","Text":"Now, we\u0027ve, again,"},{"Start":"00:04.500 ","End":"00:07.185","Text":"got two 8-bit numbers."},{"Start":"00:07.185 ","End":"00:10.920","Text":"We will start by writing them out on top of each other."},{"Start":"00:10.920 ","End":"00:18.495","Text":"The first number is 1011010101."},{"Start":"00:18.495 ","End":"00:25.020","Text":"The number we\u0027re adding to it is 11011011."},{"Start":"00:25.020 ","End":"00:28.440","Text":"Now once again, this is a 12-bit number we\u0027ve been told in the question,"},{"Start":"00:28.440 ","End":"00:33.885","Text":"so we should pad the four 0s in front for both of those numbers."},{"Start":"00:33.885 ","End":"00:38.435","Text":"Let\u0027s go ahead and calculate out each of those columns,"},{"Start":"00:38.435 ","End":"00:41.915","Text":"1 plus 1 is going to be 1, 0."},{"Start":"00:41.915 ","End":"00:50.235","Text":"So 0 in this column and now we carry across 1 to this column here."},{"Start":"00:50.235 ","End":"00:54.785","Text":"Now, 1 plus 0 plus 1 gives us 1, 0 again."},{"Start":"00:54.785 ","End":"01:00.485","Text":"We write the 0 once again here and carry across a 1 up to here,"},{"Start":"01:00.485 ","End":"01:02.615","Text":"1 plus 1 plus 0,"},{"Start":"01:02.615 ","End":"01:04.850","Text":"once again, is 1, 0."},{"Start":"01:04.850 ","End":"01:08.690","Text":"So 0 here, carry across the 1 up to here,"},{"Start":"01:08.690 ","End":"01:10.910","Text":"1 plus 0 plus 1, again,"},{"Start":"01:10.910 ","End":"01:12.680","Text":"0 in this column,"},{"Start":"01:12.680 ","End":"01:15.800","Text":"1 carried across to here."},{"Start":"01:15.800 ","End":"01:18.500","Text":"This time we\u0027ve got 1 plus 1 plus 1."},{"Start":"01:18.500 ","End":"01:21.900","Text":"If we look at in a little 8 memoir on the side here,"},{"Start":"01:21.900 ","End":"01:24.590","Text":"1 plus 1 plus 1 is actually 1, 1."},{"Start":"01:24.590 ","End":"01:30.925","Text":"What we need to do is to write a 1 in this column and carry a 1 across to here."},{"Start":"01:30.925 ","End":"01:34.035","Text":"Now we have 1 plus 0 plus 0,"},{"Start":"01:34.035 ","End":"01:35.710","Text":"which is just 1."},{"Start":"01:35.710 ","End":"01:38.315","Text":"No carry to worry about this time,"},{"Start":"01:38.315 ","End":"01:40.353","Text":"1 plus 1 again is 1,"},{"Start":"01:40.353 ","End":"01:43.490","Text":"0, so 0 here."},{"Start":"01:43.490 ","End":"01:46.665","Text":"Carry across a 1,"},{"Start":"01:46.665 ","End":"01:49.980","Text":"and 1 plus 1 plus 1 is,"},{"Start":"01:49.980 ","End":"01:52.650","Text":"as we know again, 1, 1."},{"Start":"01:52.650 ","End":"01:56.175","Text":"We have a 1 in this column,"},{"Start":"01:56.175 ","End":"02:00.765","Text":"and we carry across a 1 to here."},{"Start":"02:00.765 ","End":"02:05.230","Text":"Now, 1 plus 0 plus 0 gives us 1."},{"Start":"02:05.230 ","End":"02:08.735","Text":"All the remaining columns are just 0,"},{"Start":"02:08.735 ","End":"02:11.865","Text":"so going to end up at 0."},{"Start":"02:11.865 ","End":"02:13.670","Text":"Here is our final answer."},{"Start":"02:13.670 ","End":"02:18.260","Text":"These two binary numbers added together will give us this number."},{"Start":"02:18.260 ","End":"02:21.785","Text":"Once again, we could check this if we wanted to."},{"Start":"02:21.785 ","End":"02:30.795","Text":"I\u0027m just going to tell you that this first number here works out to 213 and denary,"},{"Start":"02:30.795 ","End":"02:35.120","Text":"this number below it is 219."},{"Start":"02:35.120 ","End":"02:40.685","Text":"If you worked out the result by putting the column headings above each number,"},{"Start":"02:40.685 ","End":"02:45.495","Text":"you find that the result works out to be 432,"},{"Start":"02:45.495 ","End":"02:51.830","Text":"and that is indeed what we should expect if we add together 213 and 219."},{"Start":"02:51.830 ","End":"02:56.240","Text":"Once again, a cross-check here has proved that addition has worked correctly."},{"Start":"02:56.240 ","End":"02:58.670","Text":"Once again, I\u0027m going to caveat this by saying,"},{"Start":"02:58.670 ","End":"03:02.150","Text":"be careful to not give the impression that you\u0027ve just added"},{"Start":"03:02.150 ","End":"03:07.145","Text":"up to denary numbers and produce a denary response and then written out as binary."},{"Start":"03:07.145 ","End":"03:08.540","Text":"If you\u0027re going to do a cross-check,"},{"Start":"03:08.540 ","End":"03:13.500","Text":"do it somewhere else just to not give the impression that you\u0027ve cheated."}],"ID":26336},{"Watched":false,"Name":"Exercise 7 part C","Duration":"3m 5s","ChapterTopicVideoID":25519,"CourseChapterTopicPlaylistID":237670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"In the final part of the question, Part C,"},{"Start":"00:02.490 ","End":"00:03.585","Text":"we\u0027re going to add 2,"},{"Start":"00:03.585 ","End":"00:06.420","Text":"10-bit numbers this time together."},{"Start":"00:06.420 ","End":"00:08.220","Text":"We still are going to produce"},{"Start":"00:08.220 ","End":"00:12.390","Text":"a 12-bit result because we are doing all of the operations using 12 bits."},{"Start":"00:12.390 ","End":"00:14.895","Text":"Let\u0027s write out those 2 numbers,"},{"Start":"00:14.895 ","End":"00:23.565","Text":"starting with the one on top, which is 1100111011."},{"Start":"00:23.565 ","End":"00:33.120","Text":"Then we\u0027ve got 1000101010."},{"Start":"00:33.120 ","End":"00:35.190","Text":"This time, because we\u0027ve got 10 bits already,"},{"Start":"00:35.190 ","End":"00:40.190","Text":"we only need to pad with 2 0s at the front of each of those numbers."},{"Start":"00:40.190 ","End":"00:44.260","Text":"We can proceed to add together all the columns as before."},{"Start":"00:44.260 ","End":"00:46.920","Text":"1 plus 0 is 1,"},{"Start":"00:46.920 ","End":"00:48.855","Text":"there is no carry to deal with that."},{"Start":"00:48.855 ","End":"00:50.790","Text":"1 plus 1 is 1, 0,"},{"Start":"00:50.790 ","End":"00:52.430","Text":"so we put a 0 here,"},{"Start":"00:52.430 ","End":"00:54.970","Text":"carry across a 1 to here,"},{"Start":"00:54.970 ","End":"00:57.660","Text":"1 plus 0 plus 0 is 1,"},{"Start":"00:57.660 ","End":"01:00.060","Text":"1 plus 1 is 1, 0,"},{"Start":"01:00.060 ","End":"01:03.360","Text":"again, so 0 here carry across a 1."},{"Start":"01:03.360 ","End":"01:06.960","Text":"1 plus 1 plus 0 is again 1, 0.,"},{"Start":"01:06.960 ","End":"01:09.240","Text":"so 0 here,"},{"Start":"01:09.240 ","End":"01:11.940","Text":"carry a 1 across to here."},{"Start":"01:11.940 ","End":"01:15.750","Text":"1 plus 1 plus 1 is 1, 1."},{"Start":"01:15.750 ","End":"01:18.750","Text":"This time, we\u0027ve got a whole 1 here,"},{"Start":"01:18.750 ","End":"01:21.840","Text":"and we carry a 1 over to here."},{"Start":"01:21.840 ","End":"01:24.975","Text":"1 plus 0 plus 0 is 1,"},{"Start":"01:24.975 ","End":"01:27.060","Text":"0 plus 0 is 0,"},{"Start":"01:27.060 ","End":"01:29.055","Text":"1 plus 0 is 1,"},{"Start":"01:29.055 ","End":"01:31.530","Text":"1 plus 1 is 1, 0."},{"Start":"01:31.530 ","End":"01:33.090","Text":"So 0 here,"},{"Start":"01:33.090 ","End":"01:34.875","Text":"carry across the 1,"},{"Start":"01:34.875 ","End":"01:39.150","Text":"1 plus 0 plus 0 is a 1 here."},{"Start":"01:39.150 ","End":"01:43.630","Text":"Then finally, two 0s added together gives us a leading 0 there."},{"Start":"01:43.630 ","End":"01:48.060","Text":"We\u0027ve got our final result, which is this."},{"Start":"01:48.060 ","End":"01:51.825","Text":"This number is an 11-bit number this time."},{"Start":"01:51.825 ","End":"01:54.775","Text":"We\u0027ve added these two 10-bit numbers and produced an 11-bit number."},{"Start":"01:54.775 ","End":"01:57.280","Text":"If we were to do a cross check,"},{"Start":"01:57.280 ","End":"02:03.610","Text":"we would find that this first number here comes out to be 827 tannery."},{"Start":"02:03.610 ","End":"02:08.060","Text":"The second number comes out to be 554 tannery."},{"Start":"02:08.060 ","End":"02:13.995","Text":"This result, if you work it out, is 1381 tannery."},{"Start":"02:13.995 ","End":"02:15.780","Text":"These 2 numbers added together do,"},{"Start":"02:15.780 ","End":"02:18.350","Text":"in fact, produce 1381."},{"Start":"02:18.350 ","End":"02:21.410","Text":"Once again, we know we\u0027ve done the correct calculation. There we go."},{"Start":"02:21.410 ","End":"02:27.740","Text":"We\u0027ve done three additions in 12 bits using unsigned binary numbers here."},{"Start":"02:27.740 ","End":"02:29.870","Text":"In each case, we were dealing with"},{"Start":"02:29.870 ","End":"02:33.605","Text":"quite convenient numbers because there was enough space to deal with the result."},{"Start":"02:33.605 ","End":"02:35.825","Text":"The first one being two 8-bit numbers,"},{"Start":"02:35.825 ","End":"02:37.925","Text":"that produced an 8-bit number of results."},{"Start":"02:37.925 ","End":"02:40.610","Text":"The second one also was two 8-bit numbers,"},{"Start":"02:40.610 ","End":"02:42.965","Text":"but this time we produced a 9-bit result."},{"Start":"02:42.965 ","End":"02:47.465","Text":"The final one, two 10-bit numbers produced an 11-bit results."},{"Start":"02:47.465 ","End":"02:49.100","Text":"In the next exercise,"},{"Start":"02:49.100 ","End":"02:54.005","Text":"we\u0027ll see how we can use signed and unsigned numbers."},{"Start":"02:54.005 ","End":"02:56.345","Text":"If we use a signed number,"},{"Start":"02:56.345 ","End":"02:58.490","Text":"we can add it to another signed number,"},{"Start":"02:58.490 ","End":"03:01.385","Text":"and using that, we can do subtraction."},{"Start":"03:01.385 ","End":"03:05.790","Text":"We\u0027ll see that in the next question. See you then."}],"ID":26337},{"Watched":false,"Name":"Exercise 8 part A","Duration":"4m 40s","ChapterTopicVideoID":25520,"CourseChapterTopicPlaylistID":237670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.875","Text":"Welcome back everyone."},{"Start":"00:01.875 ","End":"00:03.120","Text":"In this question,"},{"Start":"00:03.120 ","End":"00:09.794","Text":"we\u0027ve been asked to subtract one number from another using two\u0027s complement and addition."},{"Start":"00:09.794 ","End":"00:14.130","Text":"The way we do that is we take the second number,"},{"Start":"00:14.130 ","End":"00:16.560","Text":"the one that we\u0027re going to subtract from the first number,"},{"Start":"00:16.560 ","End":"00:19.890","Text":"and we turn it into a negative representation using"},{"Start":"00:19.890 ","End":"00:22.980","Text":"two\u0027s complement and then we can simply add the two numbers"},{"Start":"00:22.980 ","End":"00:25.785","Text":"together and the result will be the same as"},{"Start":"00:25.785 ","End":"00:30.105","Text":"subtracting this number from that one because we turned this into a negative number."},{"Start":"00:30.105 ","End":"00:32.240","Text":"Let\u0027s see how that works."},{"Start":"00:32.240 ","End":"00:35.535","Text":"The first thing we\u0027ve got to do is to take the number that we\u0027re going to"},{"Start":"00:35.535 ","End":"00:38.690","Text":"subtract and pad it to the relevant number of bits,"},{"Start":"00:38.690 ","End":"00:40.010","Text":"in this case 8."},{"Start":"00:40.010 ","End":"00:45.170","Text":"Let\u0027s write the number out first of all, 11011."},{"Start":"00:45.170 ","End":"00:46.760","Text":"That\u0027s only 5 bits,"},{"Start":"00:46.760 ","End":"00:51.355","Text":"so we need three 0s in front of it so there\u0027s our number as 8 bits now."},{"Start":"00:51.355 ","End":"00:55.010","Text":"Now what we\u0027re going to do is we\u0027re going to take a one\u0027s complement."},{"Start":"00:55.010 ","End":"00:57.600","Text":"What that means is everywhere where there is 0,"},{"Start":"00:57.600 ","End":"00:59.100","Text":"we\u0027re going to flip it into a 1,"},{"Start":"00:59.100 ","End":"01:00.150","Text":"everywhere there\u0027s a 1,"},{"Start":"01:00.150 ","End":"01:01.965","Text":"we\u0027re going to flip it into 0."},{"Start":"01:01.965 ","End":"01:04.740","Text":"This number will look like this,"},{"Start":"01:04.740 ","End":"01:07.695","Text":"0 turns into 1 so it\u0027s this 1 and this 1,"},{"Start":"01:07.695 ","End":"01:11.000","Text":"this 1 turns into 0 as does this 1."},{"Start":"01:11.000 ","End":"01:12.733","Text":"This 0 turns into 1,"},{"Start":"01:12.733 ","End":"01:15.060","Text":"these 1s turn into 0s,"},{"Start":"01:15.060 ","End":"01:17.665","Text":"so we\u0027ve now got the one\u0027s complement,"},{"Start":"01:17.665 ","End":"01:20.330","Text":"and to get into two\u0027s complement now this number"},{"Start":"01:20.330 ","End":"01:23.030","Text":"and a negative representation of what we originally had,"},{"Start":"01:23.030 ","End":"01:24.725","Text":"we simply add 1 to it."},{"Start":"01:24.725 ","End":"01:28.850","Text":"The 1 goes there but I\u0027m going to put all zeros in so"},{"Start":"01:28.850 ","End":"01:33.305","Text":"that they will align up and so reads correctly."},{"Start":"01:33.305 ","End":"01:38.520","Text":"Let\u0027s go ahead and add 1 to the one\u0027s complement here,"},{"Start":"01:38.520 ","End":"01:40.965","Text":"and we\u0027ll get 1 there,"},{"Start":"01:40.965 ","End":"01:43.080","Text":"0 there, 1 there,"},{"Start":"01:43.080 ","End":"01:45.000","Text":"0 there, 0 there,"},{"Start":"01:45.000 ","End":"01:48.941","Text":"1 there, 1 there, and 1 there."},{"Start":"01:48.941 ","End":"01:52.430","Text":"There we have it. This number now is a"},{"Start":"01:52.430 ","End":"01:57.725","Text":"negative of this number and it turns out this number here is 27."},{"Start":"01:57.725 ","End":"02:02.474","Text":"If you work it out using a bit positions minus 1 to 8,"},{"Start":"02:02.474 ","End":"02:04.166","Text":"64, 32,"},{"Start":"02:04.166 ","End":"02:07.395","Text":"skip the 16, skip the 8,"},{"Start":"02:07.395 ","End":"02:09.620","Text":"add 4, skip the 2, add 1,"},{"Start":"02:09.620 ","End":"02:12.725","Text":"you\u0027ll find that that work out to be minus 27."},{"Start":"02:12.725 ","End":"02:14.165","Text":"Now we\u0027ve got our number,"},{"Start":"02:14.165 ","End":"02:16.610","Text":"we\u0027re going to take it across to"},{"Start":"02:16.610 ","End":"02:20.510","Text":"the other side and we\u0027re going to add together this number"},{"Start":"02:20.510 ","End":"02:23.590","Text":"and then the negative number we just worked out"},{"Start":"02:23.590 ","End":"02:27.200","Text":"and that will be the same as subtracting this number from that number."},{"Start":"02:27.200 ","End":"02:28.850","Text":"Let\u0027s have a look."},{"Start":"02:28.850 ","End":"02:34.790","Text":"Let\u0027s start out by writing the original number 1010111."},{"Start":"02:34.790 ","End":"02:36.575","Text":"You\u0027ll notice there\u0027s only 7 bits,"},{"Start":"02:36.575 ","End":"02:38.510","Text":"so we need to pad it with a 0."},{"Start":"02:38.510 ","End":"02:40.730","Text":"There\u0027s our first number and then a"},{"Start":"02:40.730 ","End":"02:43.400","Text":"second number which we turn into a negative we\u0027re going to"},{"Start":"02:43.400 ","End":"02:52.340","Text":"write below it 11100101 and we\u0027re going to add those two together,"},{"Start":"02:52.340 ","End":"02:57.560","Text":"which is the same as subtracting because we\u0027ve turned this into"},{"Start":"02:57.560 ","End":"03:03.240","Text":"a negative number and we know how to do addition now,"},{"Start":"03:03.240 ","End":"03:04.365","Text":"so let\u0027s do this."},{"Start":"03:04.365 ","End":"03:06.450","Text":"1 plus 1 is 1, 0,"},{"Start":"03:06.450 ","End":"03:10.727","Text":"so 0 here and 1 gets carried across to there."},{"Start":"03:10.727 ","End":"03:13.350","Text":"1 plus 1 plus 0 is again 1,"},{"Start":"03:13.350 ","End":"03:17.070","Text":"0 so 0 goes here and 1 gets carried across."},{"Start":"03:17.070 ","End":"03:20.640","Text":"1 plus 1 plus 1 is 1, 1,"},{"Start":"03:20.640 ","End":"03:22.755","Text":"so 1 in this column,"},{"Start":"03:22.755 ","End":"03:24.600","Text":"carry 1 across to here,"},{"Start":"03:24.600 ","End":"03:27.990","Text":"1 plus 0 plus 0 is 1."},{"Start":"03:27.990 ","End":"03:30.750","Text":"That\u0027s the first 4 bits dealt with now."},{"Start":"03:30.750 ","End":"03:33.435","Text":"1 plus 0 is 1,"},{"Start":"03:33.435 ","End":"03:35.350","Text":"0 plus 1 is 1,"},{"Start":"03:35.350 ","End":"03:37.050","Text":"1 plus 1 is 1, 0,"},{"Start":"03:37.050 ","End":"03:38.940","Text":"so 0 here,"},{"Start":"03:38.940 ","End":"03:41.055","Text":"carry the 1 across to here."},{"Start":"03:41.055 ","End":"03:43.050","Text":"1 plus 0 plus 1 is 1, 0,"},{"Start":"03:43.050 ","End":"03:46.255","Text":"so 0 there, carry 1 across."},{"Start":"03:46.255 ","End":"03:47.870","Text":"However, this will give us"},{"Start":"03:47.870 ","End":"03:51.290","Text":"a 9th bit if we were to bring this down will be 1 plus 0 plus 0,"},{"Start":"03:51.290 ","End":"03:53.270","Text":"which would give us a ninth bit down here."},{"Start":"03:53.270 ","End":"03:57.525","Text":"We discard that leftmost bit it\u0027s not needed for the result."},{"Start":"03:57.525 ","End":"04:02.960","Text":"What we\u0027re left with then our answer is 00111100,"},{"Start":"04:02.960 ","End":"04:09.300","Text":"and that, if we were to work out these bit patterns,"},{"Start":"04:09.300 ","End":"04:12.225","Text":"is actually in decimal."},{"Start":"04:12.225 ","End":"04:15.035","Text":"That number works out to be 60."},{"Start":"04:15.035 ","End":"04:18.320","Text":"If we look at how original number that we started with,"},{"Start":"04:18.320 ","End":"04:23.105","Text":"that number there turns out to be 87 in decimal,"},{"Start":"04:23.105 ","End":"04:26.195","Text":"and this number is 27,"},{"Start":"04:26.195 ","End":"04:29.725","Text":"but we turned it into minus 27 here."},{"Start":"04:29.725 ","End":"04:33.169","Text":"If you were to add 87 and minus 27,"},{"Start":"04:33.169 ","End":"04:40.980","Text":"you would indeed just end up with 60 so it looks like we\u0027ve done the first part right."}],"ID":26338},{"Watched":false,"Name":"Exercise 8 part B","Duration":"4m 26s","ChapterTopicVideoID":25521,"CourseChapterTopicPlaylistID":237670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.690","Text":"For Part B, we\u0027ve got a 7-bit number now being subtracted from our 8-bit number."},{"Start":"00:06.690 ","End":"00:09.600","Text":"Let\u0027s do exactly as we did before,"},{"Start":"00:09.600 ","End":"00:14.790","Text":"write out the number 1101001,"},{"Start":"00:14.790 ","End":"00:16.740","Text":"but we\u0027re going to have to pad it with 1,"},{"Start":"00:16.740 ","End":"00:19.290","Text":"0 at the beginning so that we\u0027ve now got 8 bits,"},{"Start":"00:19.290 ","End":"00:24.150","Text":"then we\u0027re going to take the one\u0027s compliment by flipping all the bits."},{"Start":"00:24.150 ","End":"00:26.610","Text":"This 0 becomes a 1,"},{"Start":"00:26.610 ","End":"00:28.440","Text":"these two 1 becomes 0,"},{"Start":"00:28.440 ","End":"00:30.105","Text":"this 0 becomes a 1,"},{"Start":"00:30.105 ","End":"00:32.160","Text":"0 here, 1 here,"},{"Start":"00:32.160 ","End":"00:34.305","Text":"1 here, and 0 here."},{"Start":"00:34.305 ","End":"00:37.485","Text":"We now have the one\u0027s complement of this number,"},{"Start":"00:37.485 ","End":"00:40.580","Text":"and then we\u0027re going to add 1 to it again as we did"},{"Start":"00:40.580 ","End":"00:44.480","Text":"before so that we have a number in two\u0027s compliment,"},{"Start":"00:44.480 ","End":"00:46.445","Text":"but now it\u0027ll be a negative number."},{"Start":"00:46.445 ","End":"00:49.640","Text":"We\u0027re adding together this number and 1,"},{"Start":"00:49.640 ","End":"00:51.875","Text":"we will get 0 plus 1 is 1,"},{"Start":"00:51.875 ","End":"00:54.890","Text":"1 here, 1 here, 0 here,"},{"Start":"00:54.890 ","End":"00:58.745","Text":"1 there, 0 here, 0 here, and 1 there."},{"Start":"00:58.745 ","End":"01:01.865","Text":"Now we have an 8-bit number,"},{"Start":"01:01.865 ","End":"01:06.020","Text":"which is the negative of this number."},{"Start":"01:06.020 ","End":"01:09.770","Text":"What you will find if you work this out,"},{"Start":"01:09.770 ","End":"01:15.650","Text":"this number here that we started with was 105,"},{"Start":"01:15.650 ","End":"01:20.065","Text":"and we\u0027ve now turned it into minus 105."},{"Start":"01:20.065 ","End":"01:25.445","Text":"Let\u0027s go ahead and add the 2 numbers together."},{"Start":"01:25.445 ","End":"01:29.210","Text":"This number and the negative number that we\u0027ve now made from here,"},{"Start":"01:29.210 ","End":"01:33.725","Text":"and we\u0027ll have our result of subtracting this number from this number."},{"Start":"01:33.725 ","End":"01:41.095","Text":"Here\u0027s our first number, 11101111,"},{"Start":"01:41.095 ","End":"01:44.570","Text":"and we\u0027re going to add to that the number that we\u0027re going to bring across"},{"Start":"01:44.570 ","End":"01:50.070","Text":"from here, which is 10010111."},{"Start":"01:50.680 ","End":"01:54.605","Text":"Let\u0027s go ahead and do that calculation."},{"Start":"01:54.605 ","End":"01:57.320","Text":"1 and 1 is 1, 0,"},{"Start":"01:57.320 ","End":"01:59.230","Text":"so 0 here,"},{"Start":"01:59.230 ","End":"02:02.805","Text":"and carry across a 1 to here."},{"Start":"02:02.805 ","End":"02:05.450","Text":"1 plus 1 plus 1 is 1,"},{"Start":"02:05.450 ","End":"02:07.050","Text":"1, so 1 here,"},{"Start":"02:07.050 ","End":"02:09.030","Text":"and 1 carried across here,"},{"Start":"02:09.030 ","End":"02:12.435","Text":"1 plus 1 plus 1 again is 1, 1,"},{"Start":"02:12.435 ","End":"02:16.230","Text":"so 1 here and carry across a 1 there,"},{"Start":"02:16.230 ","End":"02:19.040","Text":"1 plus 1 plus 0 is 1,"},{"Start":"02:19.040 ","End":"02:22.150","Text":"0, so 0 in this column,"},{"Start":"02:22.150 ","End":"02:24.545","Text":"carry 1 across to here,"},{"Start":"02:24.545 ","End":"02:26.960","Text":"1 plus 0 plus 0 is again 1, 0,"},{"Start":"02:26.960 ","End":"02:29.880","Text":"0 here, carry 1 here,"},{"Start":"02:29.880 ","End":"02:31.920","Text":"1 plus 1 plus 0 is 1, 0,"},{"Start":"02:31.920 ","End":"02:34.290","Text":"0 here, carry across here,"},{"Start":"02:34.290 ","End":"02:37.260","Text":"1 plus 1 plus 0 is 0, 1, 0 again,"},{"Start":"02:37.260 ","End":"02:40.305","Text":"and then finally we got 1 plus 1 plus 1,"},{"Start":"02:40.305 ","End":"02:41.790","Text":"which is 1, 1,"},{"Start":"02:41.790 ","End":"02:43.080","Text":"and 1 here,"},{"Start":"02:43.080 ","End":"02:47.615","Text":"and I would carry a 1 across there and then add that to two 0s below it."},{"Start":"02:47.615 ","End":"02:49.895","Text":"But again, that would give me a 9th bit,"},{"Start":"02:49.895 ","End":"02:54.155","Text":"and I discard this leftmost bit as I don\u0027t need it for the result."},{"Start":"02:54.155 ","End":"02:59.330","Text":"Now, you may have noticed that the number that we started out with here"},{"Start":"02:59.330 ","End":"03:05.465","Text":"that we were going to subtract this number from is already a negative number,"},{"Start":"03:05.465 ","End":"03:07.670","Text":"you\u0027ll notice, because the most significant bit,"},{"Start":"03:07.670 ","End":"03:09.155","Text":"and this is an 8 bit number,"},{"Start":"03:09.155 ","End":"03:12.895","Text":"is a 1, which would represent minus 128,"},{"Start":"03:12.895 ","End":"03:15.170","Text":"and then we\u0027d add all these common values together."},{"Start":"03:15.170 ","End":"03:20.000","Text":"We\u0027ve started with a negative number and we\u0027re adding a negative number now."},{"Start":"03:20.000 ","End":"03:22.670","Text":"It\u0027d be interesting to see if this works out,"},{"Start":"03:22.670 ","End":"03:25.340","Text":"does work out, but let me just show you why."},{"Start":"03:25.340 ","End":"03:28.960","Text":"What we\u0027ve got in this first number here then,"},{"Start":"03:28.960 ","End":"03:33.345","Text":"this turns out to be minus 17."},{"Start":"03:33.345 ","End":"03:35.825","Text":"This number now that we\u0027ve worked out,"},{"Start":"03:35.825 ","End":"03:38.315","Text":"we did start with plus 105,"},{"Start":"03:38.315 ","End":"03:45.425","Text":"it\u0027s now minus 105 because we wanted to subtract the second number from the first number."},{"Start":"03:45.425 ","End":"03:47.780","Text":"If you work out this bit pattern here,"},{"Start":"03:47.780 ","End":"03:53.805","Text":"it\u0027s actually minus 128 plus 4 plus 2,"},{"Start":"03:53.805 ","End":"03:57.390","Text":"and so that is minus 122,"},{"Start":"03:57.390 ","End":"04:06.745","Text":"and minus 17 with minus 105 added to it would actually give you minus 122."},{"Start":"04:06.745 ","End":"04:09.795","Text":"We know that this operation has worked,"},{"Start":"04:09.795 ","End":"04:15.365","Text":"and the reason that it works is because we\u0027re within the range of the available numbers."},{"Start":"04:15.365 ","End":"04:19.475","Text":"You\u0027ll recall that an 8-bit two\u0027s complement number goes from minus"},{"Start":"04:19.475 ","End":"04:26.280","Text":"128 to positive 127 because the result fits into that range, we\u0027re absolutely fine."}],"ID":26339},{"Watched":false,"Name":"Exercise 8 part C","Duration":"3m 54s","ChapterTopicVideoID":25522,"CourseChapterTopicPlaylistID":237670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:02.820","Text":"For the final part of the question,"},{"Start":"00:02.820 ","End":"00:05.700","Text":"we\u0027re going to subtract this number from this one."},{"Start":"00:05.700 ","End":"00:07.665","Text":"It\u0027s quite a small number here."},{"Start":"00:07.665 ","End":"00:12.660","Text":"We\u0027re going to subtract a larger number from it and see what effect that has."},{"Start":"00:12.660 ","End":"00:15.690","Text":"Starting with the number we\u0027re going to subtract,"},{"Start":"00:15.690 ","End":"00:18.780","Text":"we\u0027ve got to turn that into a negative number in two\u0027s complement."},{"Start":"00:18.780 ","End":"00:24.795","Text":"So first of all, let\u0027s write out the number 110101."},{"Start":"00:24.795 ","End":"00:28.350","Text":"We need to add that. It\u0027s got the correct number of bits."},{"Start":"00:28.350 ","End":"00:32.010","Text":"We\u0027ve got 6 bits, so we need to add 2 zeros at the beginning."},{"Start":"00:32.010 ","End":"00:35.190","Text":"Then we\u0027re going to take the one\u0027s complement."},{"Start":"00:35.190 ","End":"00:36.765","Text":"Let\u0027s flip all the bits."},{"Start":"00:36.765 ","End":"00:40.055","Text":"0 becomes 1, 1 becomes 0."},{"Start":"00:40.055 ","End":"00:43.970","Text":"Then to the number here that we\u0027ve got the one\u0027s complement,"},{"Start":"00:43.970 ","End":"00:47.625","Text":"we need to add 1 to turn it into two\u0027s complement."},{"Start":"00:47.625 ","End":"00:50.325","Text":"That will give us the following,"},{"Start":"00:50.325 ","End":"00:51.930","Text":"0 plus 1 is 1,"},{"Start":"00:51.930 ","End":"00:58.590","Text":"1 plus 0 is 1,"},{"Start":"00:58.590 ","End":"01:00.285","Text":"0, 1, 0, 0, 1, and 1."},{"Start":"01:00.285 ","End":"01:03.110","Text":"There we have it. This number is"},{"Start":"01:03.110 ","End":"01:08.675","Text":"the negative representation of the number here that we started with,"},{"Start":"01:08.675 ","End":"01:11.020","Text":"which is the 1 we want to subtract."},{"Start":"01:11.020 ","End":"01:13.370","Text":"We\u0027re going to carry this number across,"},{"Start":"01:13.370 ","End":"01:17.070","Text":"but let\u0027s write out the number here first."},{"Start":"01:17.070 ","End":"01:20.030","Text":"We\u0027ve got 0111,"},{"Start":"01:20.030 ","End":"01:22.550","Text":"however, we got a pad this to make it 8 bits."},{"Start":"01:22.550 ","End":"01:25.190","Text":"We need 4 0s in front of it."},{"Start":"01:25.190 ","End":"01:29.970","Text":"There we go and bring this number across the side."},{"Start":"01:29.970 ","End":"01:33.150","Text":"We have 11001011."},{"Start":"01:37.340 ","End":"01:41.680","Text":"What we end up with is 1 plus 1, 1,"},{"Start":"01:41.680 ","End":"01:46.080","Text":"0, so 0 here and 1 carried across to here."},{"Start":"01:46.080 ","End":"01:48.660","Text":"1 plus 1 plus 1 is 1,"},{"Start":"01:48.660 ","End":"01:50.070","Text":"1 so 1 here,"},{"Start":"01:50.070 ","End":"01:52.380","Text":"carry across 1 to here."},{"Start":"01:52.380 ","End":"01:54.340","Text":"1 plus 1 plus 0 is 1,"},{"Start":"01:54.340 ","End":"01:57.750","Text":"0 0 here, carry across a 1 to there."},{"Start":"01:57.750 ","End":"01:59.820","Text":"1 plus 0 plus 1 is 1,"},{"Start":"01:59.820 ","End":"02:02.355","Text":"0 again, so 0 here."},{"Start":"02:02.355 ","End":"02:03.900","Text":"Carry out 1 across to here,"},{"Start":"02:03.900 ","End":"02:06.795","Text":"1 plus 0 plus 0 is 1,"},{"Start":"02:06.795 ","End":"02:09.645","Text":"0 plus 0 is 0,"},{"Start":"02:09.645 ","End":"02:12.000","Text":"0 plus 1 is 1,"},{"Start":"02:12.000 ","End":"02:14.590","Text":"0 plus 1 is 1."},{"Start":"02:14.590 ","End":"02:16.880","Text":"There\u0027s nothing left. There is no carry over here,"},{"Start":"02:16.880 ","End":"02:18.665","Text":"so there\u0027s no bits to discard."},{"Start":"02:18.665 ","End":"02:21.695","Text":"Our final answer is this number here."},{"Start":"02:21.695 ","End":"02:26.315","Text":"Once again, I could check all of these numbers."},{"Start":"02:26.315 ","End":"02:28.490","Text":"I started off with this number here,"},{"Start":"02:28.490 ","End":"02:33.545","Text":"which is clearly 7 and the number here,"},{"Start":"02:33.545 ","End":"02:40.235","Text":"which we turn into a negative representation down here was minus 53."},{"Start":"02:40.235 ","End":"02:49.865","Text":"We have minus 46 plus 7 minus 5300 to it is indeed minus 46."},{"Start":"02:49.865 ","End":"02:56.930","Text":"In summary, then we\u0027ve been able to subtract a number from another number by"},{"Start":"02:56.930 ","End":"02:59.690","Text":"using addition simply by taking the number we want to"},{"Start":"02:59.690 ","End":"03:03.620","Text":"subtract and turning it into a negative number in two\u0027s complement."},{"Start":"03:03.620 ","End":"03:07.285","Text":"To do that, we put it into one\u0027s complement first and then add 1."},{"Start":"03:07.285 ","End":"03:10.955","Text":"We\u0027ve got in two\u0027s complement and then we can add the 2 numbers together."},{"Start":"03:10.955 ","End":"03:16.420","Text":"The result would be the same as subtracting this number from the first number."},{"Start":"03:16.420 ","End":"03:23.520","Text":"It works in every case where we have an extra digit on the left-hand side, we discard it."},{"Start":"03:23.520 ","End":"03:25.080","Text":"We did in the first 2 examples,"},{"Start":"03:25.080 ","End":"03:26.795","Text":"we did it in the 3rd example,"},{"Start":"03:26.795 ","End":"03:32.000","Text":"the result will always be correct given that it will fit into the range available."},{"Start":"03:32.000 ","End":"03:34.070","Text":"All these numbers have been carefully chosen"},{"Start":"03:34.070 ","End":"03:36.350","Text":"so that they do fall into the range available."},{"Start":"03:36.350 ","End":"03:39.965","Text":"Remember that the range for a two\u0027s complement number,"},{"Start":"03:39.965 ","End":"03:41.720","Text":"which is unsigned format,"},{"Start":"03:41.720 ","End":"03:47.030","Text":"would be minus 128 to positive 127."},{"Start":"03:47.030 ","End":"03:49.820","Text":"All of these results fit into that range,"},{"Start":"03:49.820 ","End":"03:51.455","Text":"hence, they\u0027re all good."},{"Start":"03:51.455 ","End":"03:55.000","Text":"Thanks very much for watching. See you for the next one."}],"ID":26340}],"Thumbnail":null,"ID":237670},{"Name":"Fixed and floating point","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1 parts A-D","Duration":"6m 30s","ChapterTopicVideoID":25508,"CourseChapterTopicPlaylistID":237671,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"Welcome back, everybody. In this question we\u0027ve"},{"Start":"00:02.700 ","End":"00:05.160","Text":"been asked to convert a decimal number which,"},{"Start":"00:05.160 ","End":"00:09.660","Text":"this time, contains a fractional part or a decimal part,"},{"Start":"00:09.660 ","End":"00:11.175","Text":"whichever way you want to say that,"},{"Start":"00:11.175 ","End":"00:15.240","Text":"and it\u0027s fairly straightforward because we\u0027ve been told we\u0027ve"},{"Start":"00:15.240 ","End":"00:19.425","Text":"got to use 8 bits with 3 bits as the fractional part,"},{"Start":"00:19.425 ","End":"00:22.620","Text":"and therefore, the rest is going to be the integer part."},{"Start":"00:22.620 ","End":"00:24.360","Text":"How do we go ahead and do this?"},{"Start":"00:24.360 ","End":"00:28.170","Text":"Well, if we want to know the mathematical basis for this,"},{"Start":"00:28.170 ","End":"00:32.550","Text":"the way that we\u0027d calculate powers for base 2 numbers is"},{"Start":"00:32.550 ","End":"00:38.580","Text":"we start at the position of 2^0,"},{"Start":"00:38.580 ","End":"00:41.115","Text":"and then we go 2^1,"},{"Start":"00:41.115 ","End":"00:45.980","Text":"2^2 and so on for as long as we need to go."},{"Start":"00:45.980 ","End":"00:50.420","Text":"In this case, if we use 5 bits for the integer part,"},{"Start":"00:50.420 ","End":"00:53.420","Text":"and 3 bits for the fractional part,"},{"Start":"00:53.420 ","End":"00:56.050","Text":"indices will go to negative numbers."},{"Start":"00:56.050 ","End":"00:59.625","Text":"Here it\u0027s going to be 2 to the power minus 1,"},{"Start":"00:59.625 ","End":"01:04.275","Text":"2 to the power minus 2 and then finally 2 to the power minus 3."},{"Start":"01:04.275 ","End":"01:07.513","Text":"These 3 parts are going to represent the fractional part,"},{"Start":"01:07.513 ","End":"01:10.670","Text":"these 5 bits are going to represent the integer part."},{"Start":"01:10.670 ","End":"01:13.070","Text":"If we work out what these are,"},{"Start":"01:13.070 ","End":"01:17.825","Text":"we should be familiar with the positive part on the integer here."},{"Start":"01:17.825 ","End":"01:19.760","Text":"2^0 is 1,"},{"Start":"01:19.760 ","End":"01:22.295","Text":"2^1 is 2,"},{"Start":"01:22.295 ","End":"01:24.780","Text":"and 4, and then 8."},{"Start":"01:24.780 ","End":"01:29.165","Text":"We know this already from all the conversions we\u0027ve done in previous exercises."},{"Start":"01:29.165 ","End":"01:30.980","Text":"That was our integer part,"},{"Start":"01:30.980 ","End":"01:32.770","Text":"but the fractional part,"},{"Start":"01:32.770 ","End":"01:35.525","Text":"if we imagine that we have a decimal place here,"},{"Start":"01:35.525 ","End":"01:40.625","Text":"the positions to the right of the decimal place are going to be fractions."},{"Start":"01:40.625 ","End":"01:44.110","Text":"The first 1, 2 to the power minus 1 is 1/2."},{"Start":"01:44.110 ","End":"01:46.600","Text":"1/2 of 1/2 is 1/4,"},{"Start":"01:46.600 ","End":"01:51.030","Text":"and 1/2 of 1/4 is 1/8."},{"Start":"01:51.030 ","End":"01:54.365","Text":"These are now our new column values for"},{"Start":"01:54.365 ","End":"01:59.285","Text":"the binary number we\u0027re going to form to create this decimal number."},{"Start":"01:59.285 ","End":"02:01.415","Text":"Let\u0027s have a look at that then."},{"Start":"02:01.415 ","End":"02:04.139","Text":"To make 27 straightforward,"},{"Start":"02:04.139 ","End":"02:08.810","Text":"we can pretty much do this without any real calculations on the side."},{"Start":"02:08.810 ","End":"02:12.770","Text":"16 plus 8 would give us 24."},{"Start":"02:12.770 ","End":"02:17.265","Text":"We need 3 more so we put 0 there and there\u0027s our 3."},{"Start":"02:17.265 ","End":"02:20.355","Text":"24 plus 2 is 26,"},{"Start":"02:20.355 ","End":"02:24.610","Text":"plus 1 is 27, so we\u0027ve made 27 of the 27.5."},{"Start":"02:24.610 ","End":"02:27.290","Text":"0.5 of course is 1/2,"},{"Start":"02:27.290 ","End":"02:29.250","Text":"so I\u0027ll also need this column,"},{"Start":"02:29.250 ","End":"02:31.760","Text":"the others can be 0s."},{"Start":"02:31.760 ","End":"02:34.235","Text":"I\u0027ve done part a."},{"Start":"02:34.235 ","End":"02:41.000","Text":"Here is the binary pattern that represents 27.5 as an integer if we are"},{"Start":"02:41.000 ","End":"02:48.485","Text":"dealing with an 8-bit number with 3 bits as the fractional part in fixed point."},{"Start":"02:48.485 ","End":"02:51.740","Text":"We\u0027ll move on to a thing called floating point later on,"},{"Start":"02:51.740 ","End":"02:54.110","Text":"but we need to understand how fixed point works"},{"Start":"02:54.110 ","End":"02:57.245","Text":"before and there are some scenarios where you do use fixed point."},{"Start":"02:57.245 ","End":"03:00.415","Text":"There\u0027s a special type of CPU called a DSP."},{"Start":"03:00.415 ","End":"03:05.235","Text":"Some DSPs use fixed point arithmetic for speed."},{"Start":"03:05.235 ","End":"03:07.790","Text":"The next one, we\u0027ll do it in exactly the same way,"},{"Start":"03:07.790 ","End":"03:09.710","Text":"this time I won\u0027t bother looking at the powers,"},{"Start":"03:09.710 ","End":"03:13.175","Text":"we\u0027ll just write out the column values again."},{"Start":"03:13.175 ","End":"03:18.335","Text":"Column values, because we\u0027ve got the same number of bits available,"},{"Start":"03:18.335 ","End":"03:21.605","Text":"8 bits, and 3 of them are going to be the fractional part."},{"Start":"03:21.605 ","End":"03:27.530","Text":"We just half each time rather than double each time when we\u0027re going to the right."},{"Start":"03:27.530 ","End":"03:30.500","Text":"We used to double each time going in this direction,"},{"Start":"03:30.500 ","End":"03:33.245","Text":"now we\u0027re going to halve each time when we go in this direction."},{"Start":"03:33.245 ","End":"03:37.295","Text":"That effectively, is where our decimal point is going to be."},{"Start":"03:37.295 ","End":"03:40.800","Text":"For this one, to make 30.25,"},{"Start":"03:40.800 ","End":"03:42.870","Text":"I need the 16,"},{"Start":"03:42.870 ","End":"03:46.050","Text":"I\u0027ll need the 8, so that\u0027s 24."},{"Start":"03:46.050 ","End":"03:49.050","Text":"If I have the 4, I\u0027ll have 28."},{"Start":"03:49.050 ","End":"03:53.480","Text":"I\u0027ll need the 2 then to make up 30. So there\u0027s my 30."},{"Start":"03:53.480 ","End":"03:54.740","Text":"If I put in a 0 there,"},{"Start":"03:54.740 ","End":"03:57.800","Text":"that\u0027s the integer part done, I\u0027ve got 30."},{"Start":"03:57.800 ","End":"04:01.640","Text":"0.25 is going to be made up with a single one of these columns,"},{"Start":"04:01.640 ","End":"04:05.810","Text":"which is this 1 because 0.25 is, of course, 1/4."},{"Start":"04:05.810 ","End":"04:09.735","Text":"That is part b done."},{"Start":"04:09.735 ","End":"04:13.345","Text":"Moving on to part c then, same fashion."},{"Start":"04:13.345 ","End":"04:16.548","Text":"Let\u0027s do the column headings again, 16, 8, 4, 2, 1,"},{"Start":"04:16.548 ","End":"04:23.745","Text":"1/2, 1/4, and 1/8."},{"Start":"04:23.745 ","End":"04:27.200","Text":"To get 15, we don\u0027t need the 16 columns,"},{"Start":"04:27.200 ","End":"04:28.490","Text":"so let\u0027s put 0 there."},{"Start":"04:28.490 ","End":"04:34.220","Text":"We do need the 8. From 8 we\u0027ll need another 7 to make 15."},{"Start":"04:34.220 ","End":"04:37.435","Text":"I\u0027m going to need all of these columns,"},{"Start":"04:37.435 ","End":"04:40.185","Text":"4 plus 2 plus 1,"},{"Start":"04:40.185 ","End":"04:43.070","Text":"then 0.75 is a little"},{"Start":"04:43.070 ","End":"04:45.800","Text":"bit trickier than the ones we\u0027ve had previously because it doesn\u0027t fall neatly."},{"Start":"04:45.800 ","End":"04:50.345","Text":"I\u0027ll need to add 2 of these columns together and of course it\u0027s 3/4."},{"Start":"04:50.345 ","End":"04:53.735","Text":"Another way of saying that is 6/8,"},{"Start":"04:53.735 ","End":"04:59.105","Text":"is really 1/2 plus 1/4 will do me here so the 1/2 and the 1/4,"},{"Start":"04:59.105 ","End":"05:01.295","Text":"and there\u0027s nothing in that column."},{"Start":"05:01.295 ","End":"05:03.950","Text":"Now c is done also."},{"Start":"05:03.950 ","End":"05:05.750","Text":"Then the final one."},{"Start":"05:05.750 ","End":"05:08.600","Text":"I\u0027ll, once again, write out the column headings,"},{"Start":"05:08.600 ","End":"05:11.540","Text":"but it\u0027s all the same for this particular question because"},{"Start":"05:11.540 ","End":"05:15.515","Text":"all 4 parts of the question are using the same format."},{"Start":"05:15.515 ","End":"05:19.195","Text":"Then we just need to make the integer part which is 9."},{"Start":"05:19.195 ","End":"05:22.625","Text":"I can see straight away that\u0027s going to involve 8 and 1."},{"Start":"05:22.625 ","End":"05:25.385","Text":"Everything else is going to be a 0."},{"Start":"05:25.385 ","End":"05:29.385","Text":"Then 0.375 is 1/8,"},{"Start":"05:29.385 ","End":"05:32.970","Text":"which is 0.125 plus 1/4."},{"Start":"05:32.970 ","End":"05:37.560","Text":"That would give me 0.375 so I want 1s there,"},{"Start":"05:37.560 ","End":"05:43.115","Text":"a 0 there, and I have now completed all 4 parts of the exercise."},{"Start":"05:43.115 ","End":"05:45.065","Text":"Just to summarize what we\u0027ve done here,"},{"Start":"05:45.065 ","End":"05:48.230","Text":"we did the same method as we\u0027ve always done for binary numbers,"},{"Start":"05:48.230 ","End":"05:50.705","Text":"which is we have going leftwards,"},{"Start":"05:50.705 ","End":"05:54.185","Text":"twice the previous column for each column heading."},{"Start":"05:54.185 ","End":"05:58.010","Text":"This time though, we decided that where the decimal place was,"},{"Start":"05:58.010 ","End":"06:02.225","Text":"we had to go half when we\u0027re going right of the decimal place."},{"Start":"06:02.225 ","End":"06:05.060","Text":"1/2 of 1 is 1/2,"},{"Start":"06:05.060 ","End":"06:07.655","Text":"1/2 of 1/2 is 1/4,"},{"Start":"06:07.655 ","End":"06:09.470","Text":"1/2 of 1/4 is 1/8."},{"Start":"06:09.470 ","End":"06:11.315","Text":"That\u0027s why we\u0027ve got these column headings,"},{"Start":"06:11.315 ","End":"06:15.180","Text":"and because all 4 parts of this question were the same format,"},{"Start":"06:15.180 ","End":"06:17.795","Text":"8 bits with 3 bits of the fractional part,"},{"Start":"06:17.795 ","End":"06:20.450","Text":"we have the same column headings for each part of the question."},{"Start":"06:20.450 ","End":"06:24.110","Text":"Then we just worked out the integer part and the fractional part."},{"Start":"06:24.110 ","End":"06:26.465","Text":"We did it. That\u0027s great."},{"Start":"06:26.465 ","End":"06:31.270","Text":"We\u0027ll do a slightly harder one in the next question. See you then."}],"ID":26323},{"Watched":false,"Name":"Exercise 2 parts A-C","Duration":"7m 16s","ChapterTopicVideoID":25509,"CourseChapterTopicPlaylistID":237671,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"Welcome back, everyone. In this next question,"},{"Start":"00:02.760 ","End":"00:05.610","Text":"we have been told to use 12 bits to"},{"Start":"00:05.610 ","End":"00:09.825","Text":"generate a bit pattern for the following decimal numbers."},{"Start":"00:09.825 ","End":"00:13.500","Text":"We haven\u0027t actually been told how many bits to use for the fractional part."},{"Start":"00:13.500 ","End":"00:16.470","Text":"So let\u0027s have a look at how we would go about doing that."},{"Start":"00:16.470 ","End":"00:20.670","Text":"Really, what you need to do is to work out how many bits you need for the range of"},{"Start":"00:20.670 ","End":"00:25.665","Text":"the integer part and how many bits you\u0027d need to re-express the fractional part."},{"Start":"00:25.665 ","End":"00:27.390","Text":"The first one is very straightforward."},{"Start":"00:27.390 ","End":"00:29.250","Text":"It\u0027s a very small integer,"},{"Start":"00:29.250 ","End":"00:34.050","Text":"and it\u0027s not going to require many columns for the fractional part either."},{"Start":"00:34.050 ","End":"00:37.905","Text":"I can just see that I\u0027ll only need 1 bit to express 1/2."},{"Start":"00:37.905 ","End":"00:43.740","Text":"I can pretty much choose to put my binary point anywhere in the number."},{"Start":"00:43.740 ","End":"00:45.240","Text":"But what I\u0027ll do is I\u0027ll stick to"},{"Start":"00:45.240 ","End":"00:48.375","Text":"the format we used in the previous part of the question."},{"Start":"00:48.375 ","End":"00:53.325","Text":"We\u0027ll use 3 bits for the fractional part. There we go."},{"Start":"00:53.325 ","End":"00:55.515","Text":"Then the rest of it will be the integer part."},{"Start":"00:55.515 ","End":"00:57.945","Text":"I\u0027m going to use 1, 2,"},{"Start":"00:57.945 ","End":"01:00.690","Text":"4, 8, 16,"},{"Start":"01:00.690 ","End":"01:02.940","Text":"32, 64,"},{"Start":"01:02.940 ","End":"01:06.705","Text":"128, and 256."},{"Start":"01:06.705 ","End":"01:10.415","Text":"There\u0027s 9 bits for the integer,"},{"Start":"01:10.415 ","End":"01:11.930","Text":"3 bits for the fraction."},{"Start":"01:11.930 ","End":"01:16.055","Text":"17.5 is very easy to work out."},{"Start":"01:16.055 ","End":"01:18.980","Text":"It\u0027s 16 plus 1 plus 1/2."},{"Start":"01:18.980 ","End":"01:23.025","Text":"Everything else is going to be a 0."},{"Start":"01:23.025 ","End":"01:26.880","Text":"I should really put the leading 0s in here as well because I\u0027ve been"},{"Start":"01:26.880 ","End":"01:31.140","Text":"asked to express it as a 12-bit representation."},{"Start":"01:31.140 ","End":"01:33.420","Text":"There\u0027s my 12-bit representation."},{"Start":"01:33.420 ","End":"01:35.265","Text":"I should say in the answer,"},{"Start":"01:35.265 ","End":"01:39.585","Text":"how many bits I\u0027ve used for the fractional part."},{"Start":"01:39.585 ","End":"01:42.128","Text":"That\u0027s straightforward. Now,"},{"Start":"01:42.128 ","End":"01:46.605","Text":"the next one is trickier because it\u0027s actually a negative number,"},{"Start":"01:46.605 ","End":"01:52.515","Text":"so we\u0027re now needing to use a signed representation and it\u0027s also got a fractional part."},{"Start":"01:52.515 ","End":"01:58.200","Text":"The easiest way to do this question is to actually ignore the sign for the moment,"},{"Start":"01:58.200 ","End":"02:01.695","Text":"make it 37.75 as a positive number,"},{"Start":"02:01.695 ","End":"02:05.175","Text":"and then turn that into a two\u0027s complement representation."},{"Start":"02:05.175 ","End":"02:06.360","Text":"I\u0027m going to assume we\u0027ve got"},{"Start":"02:06.360 ","End":"02:08.880","Text":"the same column headings as we\u0027ve done for the previous one,"},{"Start":"02:08.880 ","End":"02:11.655","Text":"because again, we\u0027ve got a small number and a small fractional part."},{"Start":"02:11.655 ","End":"02:16.350","Text":"Now I can make 0.75 fairly easily using just 2 fractional columns."},{"Start":"02:16.350 ","End":"02:18.593","Text":"I\u0027m going to use the same headings, but obviously,"},{"Start":"02:18.593 ","End":"02:24.615","Text":"the first column is going to be minus 256 rather than just 256."},{"Start":"02:24.615 ","End":"02:26.700","Text":"Let\u0027s just write out those column headings."},{"Start":"02:26.700 ","End":"02:28.590","Text":"The first one is minus 256."},{"Start":"02:28.590 ","End":"02:30.480","Text":"Everything else is a positive number,"},{"Start":"02:30.480 ","End":"02:32.445","Text":"so positive 128,"},{"Start":"02:32.445 ","End":"02:34.815","Text":"positive 64, and so on."},{"Start":"02:34.815 ","End":"02:39.600","Text":"Then our fractional parts are also positive numbers as well."},{"Start":"02:39.600 ","End":"02:43.560","Text":"To make 37.75 as a positive number,"},{"Start":"02:43.560 ","End":"02:46.605","Text":"I\u0027m going to write that in black because it\u0027s not my final answer."},{"Start":"02:46.605 ","End":"02:49.395","Text":"Let\u0027s do 32 plus 5,"},{"Start":"02:49.395 ","End":"02:51.240","Text":"that will give us a 37,"},{"Start":"02:51.240 ","End":"02:54.735","Text":"and 0.75 is going to be 1/2 and 1/4."},{"Start":"02:54.735 ","End":"02:57.660","Text":"Everything else is a 0."},{"Start":"02:57.660 ","End":"02:58.830","Text":"Again, really important,"},{"Start":"02:58.830 ","End":"03:00.675","Text":"I put all my 0s in here."},{"Start":"03:00.675 ","End":"03:02.670","Text":"I got 4 bits there,"},{"Start":"03:02.670 ","End":"03:04.920","Text":"5 bits there, and 3 bits there."},{"Start":"03:04.920 ","End":"03:07.890","Text":"Now, this is just to make it clear,"},{"Start":"03:07.890 ","End":"03:14.220","Text":"positive 37.75, but I\u0027ve been asked to make minus 37.75."},{"Start":"03:14.220 ","End":"03:18.015","Text":"Let\u0027s take the one\u0027s complement of that number."},{"Start":"03:18.015 ","End":"03:22.905","Text":"Remember one\u0027s compliment is just to flip all the bits to the opposite. There we go."},{"Start":"03:22.905 ","End":"03:25.770","Text":"I\u0027ve taken the one\u0027s complement and then I\u0027m going"},{"Start":"03:25.770 ","End":"03:28.680","Text":"to add 1 to that to turn it into the two\u0027s compliment,"},{"Start":"03:28.680 ","End":"03:30.405","Text":"and I\u0027ll have my final answer."},{"Start":"03:30.405 ","End":"03:33.960","Text":"Just looking at it, I can see I\u0027ve got a 1 in this column here."},{"Start":"03:33.960 ","End":"03:35.895","Text":"If I add 1 to 1,"},{"Start":"03:35.895 ","End":"03:39.330","Text":"it\u0027s going to be 2, so I need a 2."},{"Start":"03:39.330 ","End":"03:41.520","Text":"I\u0027ll put 1 there and a 0 there,"},{"Start":"03:41.520 ","End":"03:44.175","Text":"otherwise the rest of the numbers is going to be exactly the same."},{"Start":"03:44.175 ","End":"03:45.975","Text":"Let\u0027s write that out."},{"Start":"03:45.975 ","End":"03:48.030","Text":"If I add 1 to here,"},{"Start":"03:48.030 ","End":"03:54.000","Text":"this 1 becomes a 0 and I get 1 over here because 1 plus 1,"},{"Start":"03:54.000 ","End":"03:56.265","Text":"remember is 1, 0."},{"Start":"03:56.265 ","End":"04:00.480","Text":"The rest of it is going to be exactly the same as above."},{"Start":"04:00.480 ","End":"04:03.225","Text":"There\u0027s my final answer."},{"Start":"04:03.225 ","End":"04:08.310","Text":"I would need to indicate how many bits I\u0027ve used for the fractional part and"},{"Start":"04:08.310 ","End":"04:14.670","Text":"that it\u0027s a signed number with 9 bits for the integer part."},{"Start":"04:14.670 ","End":"04:17.580","Text":"Great, we\u0027ve done the first 2 parts."},{"Start":"04:17.580 ","End":"04:18.810","Text":"The final part,"},{"Start":"04:18.810 ","End":"04:21.930","Text":"we need to really work out how many bits I need for the fractional part."},{"Start":"04:21.930 ","End":"04:24.150","Text":"I\u0027m just going to guess because I can see by"},{"Start":"04:24.150 ","End":"04:27.885","Text":"sight that I only need 2 bits or 3 bits here."},{"Start":"04:27.885 ","End":"04:29.580","Text":"I\u0027m not sure about this one."},{"Start":"04:29.580 ","End":"04:32.040","Text":"So what you need to do is you need to take the"},{"Start":"04:32.040 ","End":"04:34.020","Text":"reciprocal of the fractional part and"},{"Start":"04:34.020 ","End":"04:36.750","Text":"from that you should be able to work out what you need."},{"Start":"04:36.750 ","End":"04:42.200","Text":"If I take the reciprocal of 0.0625,"},{"Start":"04:42.200 ","End":"04:46.425","Text":"reciprocal is obviously 1 over the number,"},{"Start":"04:46.425 ","End":"04:49.905","Text":"and you actually will find it\u0027s 16."},{"Start":"04:49.905 ","End":"04:53.760","Text":"The column I\u0027m looking for is 1/16 column."},{"Start":"04:53.760 ","End":"04:56.190","Text":"I had 1/2,1/4 and 1/8,"},{"Start":"04:56.190 ","End":"04:57.720","Text":"I\u0027ll now need 1/16 column."},{"Start":"04:57.720 ","End":"05:01.110","Text":"That means I\u0027ll need 4 bits for the fractional part."},{"Start":"05:01.110 ","End":"05:05.850","Text":"Let\u0027s revise what we had previously as 2 bits,"},{"Start":"05:05.850 ","End":"05:09.450","Text":"a third bit and now here\u0027s my fourth bit."},{"Start":"05:09.450 ","End":"05:15.345","Text":"That means now I\u0027ve only got 8 bits left for the integer part."},{"Start":"05:15.345 ","End":"05:17.325","Text":"I\u0027ll write those out."},{"Start":"05:17.325 ","End":"05:20.970","Text":"It\u0027s 4 bits and here\u0027s the rest of it."},{"Start":"05:20.970 ","End":"05:24.090","Text":"8 bits in total and 4 bits are fractional."},{"Start":"05:24.090 ","End":"05:25.940","Text":"I\u0027ve got my 12 bits used up."},{"Start":"05:25.940 ","End":"05:30.140","Text":"I\u0027m going to assume now that this is a unsigned number otherwise,"},{"Start":"05:30.140 ","End":"05:31.415","Text":"this is not possible."},{"Start":"05:31.415 ","End":"05:33.935","Text":"It would have to be an unsigned number."},{"Start":"05:33.935 ","End":"05:40.685","Text":"253, I can make using all of the columns except for the 2,"},{"Start":"05:40.685 ","End":"05:44.450","Text":"because if I have 1s in every place here it would be 255,"},{"Start":"05:44.450 ","End":"05:47.255","Text":"2 less than that is 253."},{"Start":"05:47.255 ","End":"05:50.090","Text":"You can use whatever method you like to work that out."},{"Start":"05:50.090 ","End":"05:51.575","Text":"I\u0027ll just tell you right now,"},{"Start":"05:51.575 ","End":"05:57.365","Text":"it\u0027s 253 is all of the columns except for the 2 column."},{"Start":"05:57.365 ","End":"06:00.670","Text":"Then to make a 0.0625,"},{"Start":"06:00.670 ","End":"06:04.980","Text":"everything else is a 0 apart from the 1/16th column."},{"Start":"06:04.980 ","End":"06:07.620","Text":"We\u0027ve got part C now."},{"Start":"06:07.620 ","End":"06:12.560","Text":"Just to summarize, we need to work out how many bits we"},{"Start":"06:12.560 ","End":"06:15.080","Text":"need for the integer part of"},{"Start":"06:15.080 ","End":"06:18.665","Text":"the number and then how many bits we need for the fractional part of the number."},{"Start":"06:18.665 ","End":"06:24.165","Text":"We also need to say whether it\u0027s a signed number or an unsigned number."},{"Start":"06:24.165 ","End":"06:25.955","Text":"In this final answer,"},{"Start":"06:25.955 ","End":"06:34.120","Text":"it was unsigned and assuming 4-bits for fractional part."},{"Start":"06:34.120 ","End":"06:36.335","Text":"In the second one,"},{"Start":"06:36.335 ","End":"06:43.815","Text":"it was a signed number using 3 bits for fractional part."},{"Start":"06:43.815 ","End":"06:45.650","Text":"The first one we could have done a number of"},{"Start":"06:45.650 ","End":"06:48.110","Text":"ways to be honest because it\u0027s such a simple number."},{"Start":"06:48.110 ","End":"06:51.320","Text":"But just does go to show that choose a format"},{"Start":"06:51.320 ","End":"06:54.650","Text":"as in how many bits should I use for the fractional part?"},{"Start":"06:54.650 ","End":"06:57.095","Text":"How many bits should I use for the integer part?"},{"Start":"06:57.095 ","End":"07:00.350","Text":"Should I assume that it\u0027s a signed number or an unsigned number?"},{"Start":"07:00.350 ","End":"07:04.760","Text":"All of those factors would affect your choice of format."},{"Start":"07:04.760 ","End":"07:08.840","Text":"In this question, because they didn\u0027t give us the format,"},{"Start":"07:08.840 ","End":"07:10.490","Text":"we had to make a choice."},{"Start":"07:10.490 ","End":"07:14.060","Text":"That\u0027s what made this question a little bit trickier than the first one."},{"Start":"07:14.060 ","End":"07:17.040","Text":"There we go. I\u0027ll see you in the next one."}],"ID":26324},{"Watched":false,"Name":"Exercise 3 parts A-B","Duration":"5m 50s","ChapterTopicVideoID":25510,"CourseChapterTopicPlaylistID":237671,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.295","Text":"Hello everyone. Welcome back."},{"Start":"00:02.295 ","End":"00:05.295","Text":"In this question we\u0027ve been asked to take"},{"Start":"00:05.295 ","End":"00:10.800","Text":"a binary number in 2\u0027s complement and convert it into a denary number."},{"Start":"00:10.800 ","End":"00:16.154","Text":"We\u0027ve been told that the number has a 5-bit mantissa and a 3-bit exponent."},{"Start":"00:16.154 ","End":"00:18.615","Text":"Let\u0027s have a look at what that means."},{"Start":"00:18.615 ","End":"00:23.130","Text":"Hopefully, the question has been written out so that the mantissa has"},{"Start":"00:23.130 ","End":"00:27.750","Text":"actually got the 5-bits and a gap and the 3-bits to make it easy for us to work out."},{"Start":"00:27.750 ","End":"00:30.765","Text":"They might tell us in the question"},{"Start":"00:30.765 ","End":"00:34.830","Text":"whether it\u0027s the most significant 5-bits or the least significant 3-bits."},{"Start":"00:34.830 ","End":"00:37.005","Text":"But here they\u0027ve actually drawn it with a gap."},{"Start":"00:37.005 ","End":"00:39.659","Text":"Let\u0027s draw out the mantissa."},{"Start":"00:39.659 ","End":"00:40.980","Text":"First of all, it\u0027s 0,"},{"Start":"00:40.980 ","End":"00:43.260","Text":"1, 0, 1,"},{"Start":"00:43.260 ","End":"00:47.460","Text":"1, and then the exponent is 0, 0, 1."},{"Start":"00:47.460 ","End":"00:50.235","Text":"We need to know what the bit values are,"},{"Start":"00:50.235 ","End":"00:51.435","Text":"and it\u0027s going to be,"},{"Start":"00:51.435 ","End":"00:53.850","Text":"let\u0027s start with the exponent 1, 2,"},{"Start":"00:53.850 ","End":"00:58.860","Text":"and this is a signed number because these 2\u0027s complement it\u0027s minus 4,"},{"Start":"00:58.860 ","End":"01:00.885","Text":"is that final column."},{"Start":"01:00.885 ","End":"01:05.045","Text":"The exponent\u0027s value is 1, nice and easy."},{"Start":"01:05.045 ","End":"01:07.250","Text":"We won\u0027t work out what the column values are yet."},{"Start":"01:07.250 ","End":"01:10.415","Text":"There\u0027s no point for the mantissa until we\u0027ve shifted it."},{"Start":"01:10.415 ","End":"01:13.520","Text":"But what we should draw in is the binary point."},{"Start":"01:13.520 ","End":"01:16.130","Text":"The binary point is always going to be after"},{"Start":"01:16.130 ","End":"01:20.270","Text":"the first binary digit by standard and convention."},{"Start":"01:20.270 ","End":"01:23.660","Text":"Let\u0027s now shift our binary point by 1 place."},{"Start":"01:23.660 ","End":"01:27.125","Text":"I\u0027m going to write out exactly the same number again,"},{"Start":"01:27.125 ","End":"01:30.110","Text":"but without the binary point this time and we\u0027re going to shift it"},{"Start":"01:30.110 ","End":"01:34.430","Text":"1 place to the right because this is a positive number and therefore,"},{"Start":"01:34.430 ","End":"01:39.410","Text":"we\u0027re making a bigger number and therefore we shift it to the right."},{"Start":"01:39.410 ","End":"01:43.190","Text":"That\u0027s what I need to do and I\u0027ve got my answer now."},{"Start":"01:43.190 ","End":"01:48.260","Text":"To translate that into binary what I\u0027ll need to do is to write the column values,"},{"Start":"01:48.260 ","End":"01:49.835","Text":"so that\u0027s going to be 1,"},{"Start":"01:49.835 ","End":"01:53.340","Text":"and this is going to be not 2 but minus 2,"},{"Start":"01:53.340 ","End":"01:57.450","Text":"and this will be a 1/2,"},{"Start":"01:57.450 ","End":"02:00.360","Text":"1/4, and an 1/8,"},{"Start":"02:00.360 ","End":"02:07.189","Text":"so what we end up with as our final answer which is going to be 1"},{"Start":"02:07.189 ","End":"02:15.705","Text":"plus 1/4 plus an 1/8, which is 1.375."},{"Start":"02:15.705 ","End":"02:18.105","Text":"That\u0027s it. We\u0027re done for the first part."},{"Start":"02:18.105 ","End":"02:21.625","Text":"There\u0027s our answer in decimal."},{"Start":"02:21.625 ","End":"02:26.360","Text":"If we wanted to check whether what we\u0027ve done is correct we could look at"},{"Start":"02:26.360 ","End":"02:31.485","Text":"this number here and work that out in binary."},{"Start":"02:31.485 ","End":"02:37.020","Text":"This would actually be 0.1011."},{"Start":"02:37.260 ","End":"02:45.140","Text":"If you look at the column values for each of these digits you\u0027d be looking at 1/2 here,"},{"Start":"02:45.140 ","End":"02:47.780","Text":"1/4 here, 1/8 here,"},{"Start":"02:47.780 ","End":"02:50.840","Text":"and 1/16 for that final bit."},{"Start":"02:50.840 ","End":"02:54.320","Text":"What you\u0027d end up with is"},{"Start":"02:54.320 ","End":"03:03.180","Text":"11/16 and 11/16 is actually as a decimal,"},{"Start":"03:03.180 ","End":"03:08.630","Text":"0.6875 and the exponent is 2."},{"Start":"03:08.630 ","End":"03:13.900","Text":"But remember what we have to do with that is we have to say 2^1,"},{"Start":"03:13.900 ","End":"03:15.600","Text":"so the exponent is 1,"},{"Start":"03:15.600 ","End":"03:18.330","Text":"and we take 2^1 which gives us 2."},{"Start":"03:18.330 ","End":"03:24.555","Text":"If you multiply 0.6875 by 2 you will actually get 1.375,"},{"Start":"03:24.555 ","End":"03:28.965","Text":"so we know we\u0027ve done the correct thing for this first part of the question."},{"Start":"03:28.965 ","End":"03:35.585","Text":"Let\u0027s follow that up with the second question then see another example of how this works."},{"Start":"03:35.585 ","End":"03:38.165","Text":"Again, we write out our mantissa."},{"Start":"03:38.165 ","End":"03:40.115","Text":"Our mantissa is 1,"},{"Start":"03:40.115 ","End":"03:42.370","Text":"0, 1, 0, 1,"},{"Start":"03:42.370 ","End":"03:50.825","Text":"and we imply that the binary point is here and our exponent is 0, 1, 0."},{"Start":"03:50.825 ","End":"03:52.835","Text":"Now let\u0027s work out what that is."},{"Start":"03:52.835 ","End":"03:56.670","Text":"Again, 1 2 minus 4 here,"},{"Start":"03:56.670 ","End":"04:02.345","Text":"so its 2 is going to be the number of places we\u0027re going to shift our binary point by."},{"Start":"04:02.345 ","End":"04:07.670","Text":"Let\u0027s write our number out again with no binary point and we\u0027re going to shift it."},{"Start":"04:07.670 ","End":"04:10.830","Text":"We\u0027re going to shift it 2 places this time,"},{"Start":"04:10.830 ","End":"04:12.630","Text":"so it\u0027s going to end up here."},{"Start":"04:12.630 ","End":"04:16.640","Text":"Now when we write in our column values we\u0027ll find that"},{"Start":"04:16.640 ","End":"04:22.070","Text":"this number is a negative number because that most significant bit is set."},{"Start":"04:22.070 ","End":"04:24.755","Text":"We\u0027ve got minus 4 plus 1,"},{"Start":"04:24.755 ","End":"04:27.440","Text":"so obviously minus 3,"},{"Start":"04:27.440 ","End":"04:33.410","Text":"and we\u0027re going to add to that the fractional part which is just 1/4 because that\u0027s what"},{"Start":"04:33.410 ","End":"04:36.410","Text":"this final column is representing and"},{"Start":"04:36.410 ","End":"04:42.110","Text":"that obviously gives us the final result minus 2.75."},{"Start":"04:42.110 ","End":"04:44.765","Text":"Once again, we are done."},{"Start":"04:44.765 ","End":"04:48.250","Text":"We could check this once again here."},{"Start":"04:48.250 ","End":"04:52.130","Text":"If we were to work out what the column values were for there,"},{"Start":"04:52.130 ","End":"04:58.880","Text":"you get minus 1 first and then you got 1/4 there and 1/16 there,"},{"Start":"04:58.880 ","End":"05:04.330","Text":"so that works out to be minus 1 plus"},{"Start":"05:04.330 ","End":"05:11.470","Text":"5/16 and that will give you minus 0.6875."},{"Start":"05:11.470 ","End":"05:18.650","Text":"But the exponent here is 2 to the power 2 because this value is 2."},{"Start":"05:18.650 ","End":"05:21.725","Text":"That will be the same as multiplying it by 4."},{"Start":"05:21.725 ","End":"05:29.015","Text":"If you take minus 0.6875 and multiply it by 4 you will find that it\u0027s minus 2.75."},{"Start":"05:29.015 ","End":"05:32.075","Text":"Once again, we\u0027ve got the right answer."},{"Start":"05:32.075 ","End":"05:37.695","Text":"Our final answer for part a was 1.375,"},{"Start":"05:37.695 ","End":"05:43.020","Text":"final answer for part b was minus 2.75,"},{"Start":"05:43.020 ","End":"05:46.745","Text":"and in the next video we\u0027ll see the same thing again,"},{"Start":"05:46.745 ","End":"05:51.550","Text":"but with 12-bit number. See you then."}],"ID":26325},{"Watched":false,"Name":"Exercise 4 parts A-C","Duration":"8m 37s","ChapterTopicVideoID":25511,"CourseChapterTopicPlaylistID":237671,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.805","Text":"Hello, again, everyone. In this question,"},{"Start":"00:02.805 ","End":"00:06.180","Text":"we\u0027ve been asked to take a number which is in"},{"Start":"00:06.180 ","End":"00:11.543","Text":"2\u0027s complement encoding and it\u0027s got an 8-bit mantissa and a 4-bit exponent,"},{"Start":"00:11.543 ","End":"00:14.310","Text":"and to convert it into denary."},{"Start":"00:14.310 ","End":"00:22.193","Text":"Just as before, let\u0027s start out by writing out the mantissa 00100000,"},{"Start":"00:22.193 ","End":"00:25.845","Text":"and then the exponent which is 1011."},{"Start":"00:25.845 ","End":"00:30.560","Text":"Now let\u0027s work out what the exponent is."},{"Start":"00:30.560 ","End":"00:34.700","Text":"This is a 2\u0027s complement signed number,"},{"Start":"00:34.700 ","End":"00:37.790","Text":"so it\u0027s actually minus 8 plus 3,"},{"Start":"00:37.790 ","End":"00:40.168","Text":"which is minus 5,"},{"Start":"00:40.168 ","End":"00:49.655","Text":"and we\u0027ve got to now take our mantissa and shift the binary point, 5 places."},{"Start":"00:49.655 ","End":"00:55.999","Text":"But this time, it\u0027s in the left-hand direction because we\u0027ve got a negative exponent."},{"Start":"00:55.999 ","End":"00:59.105","Text":"Let\u0027s write the number out again,"},{"Start":"00:59.105 ","End":"01:00.590","Text":"and in front of it,"},{"Start":"01:00.590 ","End":"01:07.020","Text":"let\u0027s write 5 0s because we\u0027ve got 5 places to shift, and there we go."},{"Start":"01:07.020 ","End":"01:11.015","Text":"You\u0027ll recall that the implied binary point is always here."},{"Start":"01:11.015 ","End":"01:12.935","Text":"We\u0027re going to shift from there,"},{"Start":"01:12.935 ","End":"01:15.093","Text":"5 places to the left, 1,"},{"Start":"01:15.093 ","End":"01:17.450","Text":"2, 3, 4, 5,"},{"Start":"01:17.450 ","End":"01:19.738","Text":"so it\u0027s going to end up over here,"},{"Start":"01:19.738 ","End":"01:21.695","Text":"and there\u0027s our number in binary now,"},{"Start":"01:21.695 ","End":"01:26.210","Text":"and it\u0027s a simple case of taking that and converting it into decimal."},{"Start":"01:26.210 ","End":"01:29.345","Text":"We can do that by taking negative powers,"},{"Start":"01:29.345 ","End":"01:31.925","Text":"or we can do our old friend, the column values."},{"Start":"01:31.925 ","End":"01:34.175","Text":"I\u0027ll just show you the column values."},{"Start":"01:34.175 ","End":"01:38.000","Text":"We know that this is 1 and this is going to be a"},{"Start":"01:38.000 ","End":"01:42.320","Text":"1/2 because it\u0027s to the right of the decimal of the binary point,"},{"Start":"01:42.320 ","End":"01:45.130","Text":"then it\u0027s going to be halving each time,"},{"Start":"01:45.130 ","End":"01:48.110","Text":"so we get an 1/8,"},{"Start":"01:48.110 ","End":"01:51.755","Text":"1/16, 1/32, 1/64,"},{"Start":"01:51.755 ","End":"01:54.650","Text":"and 1/128, and that\u0027s the moment where we\u0027ve actually got"},{"Start":"01:54.650 ","End":"01:59.210","Text":"a value so our answer is, for part a,"},{"Start":"01:59.210 ","End":"02:08.590","Text":"1/128, which as a decimal number, is 0.0078125."},{"Start":"02:08.590 ","End":"02:12.485","Text":"There we go. That\u0027s the answer for part a."},{"Start":"02:12.485 ","End":"02:16.250","Text":"For part b, we\u0027re going to do it in exactly the same way."},{"Start":"02:16.250 ","End":"02:19.970","Text":"This time, we\u0027ve got a negative mantissa."},{"Start":"02:19.970 ","End":"02:23.375","Text":"You can spot that because the first digit is a 1,"},{"Start":"02:23.375 ","End":"02:26.230","Text":"and this is a signed number, remember."},{"Start":"02:26.230 ","End":"02:30.665","Text":"There\u0027s our mantissa, and there is our exponent."},{"Start":"02:30.665 ","End":"02:34.680","Text":"We have to work out the value of the exponent first."},{"Start":"02:34.680 ","End":"02:37.445","Text":"Just writing those values out for you."},{"Start":"02:37.445 ","End":"02:41.015","Text":"You\u0027ve got 6 and that\u0027s a positive number."},{"Start":"02:41.015 ","End":"02:42.695","Text":"This is a straightforward one."},{"Start":"02:42.695 ","End":"02:47.675","Text":"We\u0027re shifting the binary point 6 places to the right,"},{"Start":"02:47.675 ","End":"02:52.325","Text":"and we just simply write out a number again and move the binary point."},{"Start":"02:52.325 ","End":"02:54.685","Text":"The implied binary point was here,"},{"Start":"02:54.685 ","End":"02:56.908","Text":"and we\u0027re going to shift 1,"},{"Start":"02:56.908 ","End":"02:58.822","Text":"2, 3, 4,"},{"Start":"02:58.822 ","End":"03:01.604","Text":"5, 6 places,"},{"Start":"03:01.604 ","End":"03:03.795","Text":"so it ends up there,"},{"Start":"03:03.795 ","End":"03:07.040","Text":"and what we\u0027re looking for now is to make up"},{"Start":"03:07.040 ","End":"03:13.320","Text":"binary columns and add them all together and produce a decimal end of it,"},{"Start":"03:13.730 ","End":"03:16.024","Text":"1, 2, 4, 8, 16,"},{"Start":"03:16.024 ","End":"03:26.285","Text":"32 and this last column is minus 64 because we\u0027re dealing with a 2\u0027s complement number."},{"Start":"03:26.285 ","End":"03:29.090","Text":"If we add all these together,"},{"Start":"03:29.090 ","End":"03:35.390","Text":"we will get minus 10 plus a fractional part,"},{"Start":"03:35.390 ","End":"03:36.860","Text":"which I\u0027ve written out here,"},{"Start":"03:36.860 ","End":"03:39.410","Text":"which is a 1/2,"},{"Start":"03:39.410 ","End":"03:43.340","Text":"so minus 10 plus a 1/2 is our final result,"},{"Start":"03:43.340 ","End":"03:47.475","Text":"which is obviously minus 9.5."},{"Start":"03:47.475 ","End":"03:49.640","Text":"Another fairly straightforward one,"},{"Start":"03:49.640 ","End":"03:53.030","Text":"but this time we had a negative mantissa"},{"Start":"03:53.030 ","End":"03:56.800","Text":"and a positive exponent which made a negative number,"},{"Start":"03:56.800 ","End":"04:00.410","Text":"but the negative number was getting larger because we were"},{"Start":"04:00.410 ","End":"04:04.670","Text":"shifting it in the rightward direction of the binary point,"},{"Start":"04:04.670 ","End":"04:06.095","Text":"so we\u0027re making a bigger number."},{"Start":"04:06.095 ","End":"04:10.085","Text":"The final one is slightly different, slightly trickier."},{"Start":"04:10.085 ","End":"04:12.065","Text":"Let\u0027s approach that one then."},{"Start":"04:12.065 ","End":"04:13.585","Text":"Let\u0027s write it now again,"},{"Start":"04:13.585 ","End":"04:23.300","Text":"10110000 and the exponent is 1100."},{"Start":"04:23.300 ","End":"04:31.305","Text":"This time we\u0027re dealing with an exponent of minus 4 because minus 8 plus 4,"},{"Start":"04:31.305 ","End":"04:37.310","Text":"so we\u0027ve got to shift our binary point 4 places to the left."},{"Start":"04:37.310 ","End":"04:41.150","Text":"Now because we\u0027ve got a negative number here and we"},{"Start":"04:41.150 ","End":"04:45.475","Text":"can tell that because the left-most bit is a 1,"},{"Start":"04:45.475 ","End":"04:47.715","Text":"we\u0027ve got to preserve the sign."},{"Start":"04:47.715 ","End":"04:52.166","Text":"This time, whereas we were out 4 0s or 5 0s in part a,"},{"Start":"04:52.166 ","End":"04:54.410","Text":"this time we\u0027re going to write out 4,"},{"Start":"04:54.410 ","End":"04:56.765","Text":"1s in front of the number."},{"Start":"04:56.765 ","End":"05:01.440","Text":"If we start with the original mantissa 1, 0, 1, 1, 0, 0, 0, 0,"},{"Start":"05:01.520 ","End":"05:07.670","Text":"and this time put 4 1s in front because we\u0027re recognizing that this"},{"Start":"05:07.670 ","End":"05:13.190","Text":"is a negative number from the first binary digit and we\u0027re going to preserve the sign."},{"Start":"05:13.190 ","End":"05:18.850","Text":"Therefore, we shift in 1s rather than 0s as we did with a positive number."},{"Start":"05:18.850 ","End":"05:23.246","Text":"The implied binary point was there shifting it 1,"},{"Start":"05:23.246 ","End":"05:26.630","Text":"2, 3, 4 places, puts it there."},{"Start":"05:26.630 ","End":"05:28.220","Text":"Now we\u0027ve got our number,"},{"Start":"05:28.220 ","End":"05:31.955","Text":"and it\u0027s just a case of converting that into decimal,"},{"Start":"05:31.955 ","End":"05:39.257","Text":"and it might be easier in this case to use powers."},{"Start":"05:39.257 ","End":"05:41.945","Text":"It\u0027s up to you how you do this."},{"Start":"05:41.945 ","End":"05:43.280","Text":"You could do it the previous method."},{"Start":"05:43.280 ","End":"05:49.010","Text":"This first binary digit is going to be 2^minus 1."},{"Start":"05:49.010 ","End":"05:51.980","Text":"Actually, let\u0027s get the colors consistent."},{"Start":"05:51.980 ","End":"05:55.160","Text":"Let me use red here, 2^minus 1."},{"Start":"05:55.160 ","End":"05:58.480","Text":"This next one is 2^minus 2,"},{"Start":"05:58.480 ","End":"06:01.495","Text":"2^minus 3, and so on."},{"Start":"06:01.495 ","End":"06:04.925","Text":"Then we\u0027re going to get finally to 2^minus 7."},{"Start":"06:04.925 ","End":"06:08.180","Text":"It\u0027s exactly the same as doing what we did before."},{"Start":"06:08.180 ","End":"06:09.680","Text":"If you have access to a calculator,"},{"Start":"06:09.680 ","End":"06:11.266","Text":"you can do that,"},{"Start":"06:11.266 ","End":"06:14.840","Text":"or if you prefer the previous method, it\u0027s 1/2,"},{"Start":"06:14.840 ","End":"06:19.085","Text":"1/4, 1/8, 1/16,"},{"Start":"06:19.085 ","End":"06:24.580","Text":"1/32, 1/64, and 1/128."},{"Start":"06:24.580 ","End":"06:27.740","Text":"If we add all those fractions together,"},{"Start":"06:27.740 ","End":"06:35.580","Text":"what you\u0027ll find you\u0027ll get is a 109/128."},{"Start":"06:35.580 ","End":"06:37.095","Text":"That\u0027s the positive part,"},{"Start":"06:37.095 ","End":"06:43.055","Text":"but we\u0027re going to add minus 1 to that because we\u0027ve got this minus 1 over here."},{"Start":"06:43.055 ","End":"06:47.115","Text":"That\u0027s the most significant bit here."},{"Start":"06:47.115 ","End":"06:49.950","Text":"If you work that out,"},{"Start":"06:49.950 ","End":"06:53.615","Text":"and we can put this to a fraction form again,"},{"Start":"06:53.615 ","End":"06:58.270","Text":"you\u0027ll find that that\u0027s minus 19/128,"},{"Start":"06:58.630 ","End":"07:03.410","Text":"which in decimal would give you minus"},{"Start":"07:03.410 ","End":"07:12.255","Text":"0.0390625 and that\u0027s our final answer."},{"Start":"07:12.255 ","End":"07:17.215","Text":"If you wanted to write this out in slightly different way as we\u0027ve done previously,"},{"Start":"07:17.215 ","End":"07:20.720","Text":"you could take this number here and you can"},{"Start":"07:20.720 ","End":"07:25.100","Text":"multiply it by 2 to the power whatever the exponent is,"},{"Start":"07:25.100 ","End":"07:28.255","Text":"which in this case was minus 4."},{"Start":"07:28.255 ","End":"07:31.040","Text":"If you\u0027d taken minus 5/8,"},{"Start":"07:31.040 ","End":"07:36.520","Text":"which is what this is giving you here and multiplied it by 2^minus 4,"},{"Start":"07:36.520 ","End":"07:39.643","Text":"you would find you get exactly the same number as you get here,"},{"Start":"07:39.643 ","End":"07:41.360","Text":"so we know we\u0027ve done the right thing."},{"Start":"07:41.360 ","End":"07:46.280","Text":"There we go. This is quite long question, 3 part question."},{"Start":"07:46.280 ","End":"07:47.945","Text":"What we\u0027ve seen is,"},{"Start":"07:47.945 ","End":"07:50.390","Text":"when you have a number that is"},{"Start":"07:50.390 ","End":"07:55.340","Text":"a positive mantissa and you\u0027re shifting it in a negative direction,"},{"Start":"07:55.340 ","End":"07:59.590","Text":"you\u0027re making a smaller positive number and you put zeros in front of it."},{"Start":"07:59.590 ","End":"08:03.950","Text":"In part b, we had a negative number,"},{"Start":"08:03.950 ","End":"08:08.375","Text":"but we were shifting it in the right hand direction because we had a positive exponent,"},{"Start":"08:08.375 ","End":"08:10.940","Text":"we\u0027re just making a bigger negative number,"},{"Start":"08:10.940 ","End":"08:13.085","Text":"and that\u0027s why we ended up with the answer."},{"Start":"08:13.085 ","End":"08:14.225","Text":"But the final one,"},{"Start":"08:14.225 ","End":"08:21.049","Text":"we had a negative mantissa and a negative exponent which is making a negative number,"},{"Start":"08:21.049 ","End":"08:23.390","Text":"preserve its sign, it\u0027s still going to be negative,"},{"Start":"08:23.390 ","End":"08:24.950","Text":"but it\u0027s going to be a smaller number,"},{"Start":"08:24.950 ","End":"08:27.305","Text":"meaning it\u0027s going to be closer to 0"},{"Start":"08:27.305 ","End":"08:30.440","Text":"because of the way that we\u0027re shifting the binary point."},{"Start":"08:30.440 ","End":"08:33.800","Text":"That\u0027s why we ended up with very small number here."},{"Start":"08:33.800 ","End":"08:38.490","Text":"That\u0027s us done for this exercise. I\u0027ll see you in the next one."}],"ID":26326},{"Watched":false,"Name":"Exercise 5 parts A-C","Duration":"7m 43s","ChapterTopicVideoID":25506,"CourseChapterTopicPlaylistID":237671,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.775","Text":"Hi again everyone. In this question,"},{"Start":"00:02.775 ","End":"00:06.930","Text":"we\u0027ve been asked to take our set of decimal numbers,"},{"Start":"00:06.930 ","End":"00:10.840","Text":"and they\u0027re to be expressed using sign and magnitude this time"},{"Start":"00:10.840 ","End":"00:15.100","Text":"for the mantissa and two\u0027s complement for the exponent,"},{"Start":"00:15.100 ","End":"00:18.625","Text":"and then we will have to have a sign bit right at the beginning,"},{"Start":"00:18.625 ","End":"00:20.480","Text":"because this is a sign and magnitude,"},{"Start":"00:20.480 ","End":"00:25.030","Text":"this followed by the exponent and followed by 11-bits for the mantissa,"},{"Start":"00:25.030 ","End":"00:28.030","Text":"so let\u0027s go ahead and do this."},{"Start":"00:28.030 ","End":"00:31.955","Text":"I\u0027ll start with trying to convert this number into a binary number,"},{"Start":"00:31.955 ","End":"00:34.120","Text":"and that\u0027s pretty straightforward,"},{"Start":"00:34.120 ","End":"00:35.800","Text":"is a fixed point binary number."},{"Start":"00:35.800 ","End":"00:39.945","Text":"I keep going until I get to a column that\u0027s bigger than what I need,"},{"Start":"00:39.945 ","End":"00:41.140","Text":"so 16 will be bigger,"},{"Start":"00:41.140 ","End":"00:42.745","Text":"so I don\u0027t need to go that far."},{"Start":"00:42.745 ","End":"00:48.370","Text":"Then I\u0027ll go in this direction and I can see from this number,"},{"Start":"00:48.370 ","End":"00:50.995","Text":"I don\u0027t need to go beyond 1/8,"},{"Start":"00:50.995 ","End":"00:54.250","Text":"so 13 is what I need to make up."},{"Start":"00:54.250 ","End":"00:56.260","Text":"To get to 13,"},{"Start":"00:56.260 ","End":"00:59.510","Text":"I need 8 plus 4 will give me 12,"},{"Start":"00:59.510 ","End":"01:00.840","Text":"I don\u0027t need the 2,"},{"Start":"01:00.840 ","End":"01:02.685","Text":"but I need the one that\u0027s 13,"},{"Start":"01:02.685 ","End":"01:04.245","Text":"8 plus 4 plus 1,"},{"Start":"01:04.245 ","End":"01:07.715","Text":"and then 0.375 is actually 3/8."},{"Start":"01:07.715 ","End":"01:12.715","Text":"I can work that out by multiplying this by 8 and I should get 3."},{"Start":"01:12.715 ","End":"01:15.735","Text":"3/8 would mean nothing there,"},{"Start":"01:15.735 ","End":"01:18.085","Text":"but 1 there and 1 there."},{"Start":"01:18.085 ","End":"01:20.270","Text":"I\u0027ve got my mantissa now,"},{"Start":"01:20.270 ","End":"01:22.825","Text":"but as you will remember,"},{"Start":"01:22.825 ","End":"01:25.225","Text":"we always have a binary point here."},{"Start":"01:25.225 ","End":"01:28.460","Text":"That\u0027s where the binary point when we store binary numbers."},{"Start":"01:28.460 ","End":"01:30.860","Text":"We\u0027re implying that the binary point is there."},{"Start":"01:30.860 ","End":"01:35.375","Text":"I have to shift my binary point from where it is to over here,"},{"Start":"01:35.375 ","End":"01:37.442","Text":"and that would involve me shifting at 1,"},{"Start":"01:37.442 ","End":"01:39.650","Text":"2, 3 places."},{"Start":"01:39.650 ","End":"01:44.480","Text":"That is what I\u0027m looking to make my exponent, is 3,"},{"Start":"01:44.480 ","End":"01:51.935","Text":"and to get an exponent of 3 using my exponent stored in two\u0027s complement,"},{"Start":"01:51.935 ","End":"01:54.950","Text":"I would have the following column headings,"},{"Start":"01:54.950 ","End":"02:00.645","Text":"and so 3 would be the 2 and the 1 column and 0s elsewhere."},{"Start":"02:00.645 ","End":"02:02.370","Text":"I\u0027ve done the experiment,"},{"Start":"02:02.370 ","End":"02:04.050","Text":"I\u0027ve got my mantissa,"},{"Start":"02:04.050 ","End":"02:06.110","Text":"all I need to do is inspect the sign bit,"},{"Start":"02:06.110 ","End":"02:07.565","Text":"this is a positive number,"},{"Start":"02:07.565 ","End":"02:11.014","Text":"so the sign bit stays as 0,"},{"Start":"02:11.014 ","End":"02:13.250","Text":"but I do need to include it in this format,"},{"Start":"02:13.250 ","End":"02:18.136","Text":"so 0 for the sign bit, 0, 0, 1,"},{"Start":"02:18.136 ","End":"02:19.650","Text":"1 for the exponent,"},{"Start":"02:19.650 ","End":"02:22.368","Text":"and then 1, 1, 0, 1, 0, 1,"},{"Start":"02:22.368 ","End":"02:26.990","Text":"1 for the mantissa."},{"Start":"02:26.990 ","End":"02:29.192","Text":"However, my mantissa is only 1, 2,"},{"Start":"02:29.192 ","End":"02:31.735","Text":"3, 4, 5, 6,7-bits,"},{"Start":"02:31.735 ","End":"02:34.105","Text":"so it was supposed to be 11-bits,"},{"Start":"02:34.105 ","End":"02:39.035","Text":"so I need to pad it out with 4 more zeros at the end."},{"Start":"02:39.035 ","End":"02:43.505","Text":"I now have my final answer in this format here with a sign bit,"},{"Start":"02:43.505 ","End":"02:48.125","Text":"4-bits for the exponent and 11-bits for the mantissa."},{"Start":"02:48.125 ","End":"02:50.435","Text":"Let\u0027s have a go at part b then,"},{"Start":"02:50.435 ","End":"02:51.950","Text":"going to be similar,"},{"Start":"02:51.950 ","End":"02:54.259","Text":"but this time we\u0027ve got a negative mantissa."},{"Start":"02:54.259 ","End":"02:57.455","Text":"But for the moment we\u0027ll just work out the binary value"},{"Start":"02:57.455 ","End":"03:01.625","Text":"of the mantissa and then worry about the sign later."},{"Start":"03:01.625 ","End":"03:03.710","Text":"If I go to 16, that\u0027s fine,"},{"Start":"03:03.710 ","End":"03:05.480","Text":"but if I go to 32 it\u0027s bigger than this,"},{"Start":"03:05.480 ","End":"03:06.950","Text":"so I don\u0027t need to go any further."},{"Start":"03:06.950 ","End":"03:09.440","Text":"0.25 is obviously a quarter,"},{"Start":"03:09.440 ","End":"03:13.820","Text":"so I only need 2 columns over here and write out my number"},{"Start":"03:13.820 ","End":"03:19.605","Text":"then 31.25,16 plus 8 would give me 24,"},{"Start":"03:19.605 ","End":"03:22.410","Text":"plus 4 is 28, plus 2 is 30,"},{"Start":"03:22.410 ","End":"03:23.700","Text":"plus 1 is 31,"},{"Start":"03:23.700 ","End":"03:25.515","Text":"so I need all the ones there,"},{"Start":"03:25.515 ","End":"03:27.525","Text":"a 0 here,"},{"Start":"03:27.525 ","End":"03:29.790","Text":"and I\u0027ll need a 1 for my quarter."},{"Start":"03:29.790 ","End":"03:32.155","Text":"I\u0027ve got my mantissa again."},{"Start":"03:32.155 ","End":"03:34.250","Text":"I need to shift"},{"Start":"03:34.250 ","End":"03:38.765","Text":"the binary point to here because that\u0027s where we always imply the binary point is,"},{"Start":"03:38.765 ","End":"03:40.635","Text":"to do that I\u0027d have to go 1, 2, 3,"},{"Start":"03:40.635 ","End":"03:47.120","Text":"4 places, so my exponent is 4 this time,"},{"Start":"03:47.120 ","End":"03:52.830","Text":"and in binary, using 4-bits two\u0027s complement,"},{"Start":"03:52.830 ","End":"03:55.815","Text":"that would be just as far as."},{"Start":"03:55.815 ","End":"03:57.630","Text":"There\u0027s my exponent,"},{"Start":"03:57.630 ","End":"03:59.535","Text":"so that part is done again,"},{"Start":"03:59.535 ","End":"04:04.040","Text":"and now the only difference with the previous one is I\u0027ve got a negative number,"},{"Start":"04:04.040 ","End":"04:10.280","Text":"so I need to have a sign bit that\u0027s a 1 to indicate a negative number."},{"Start":"04:10.280 ","End":"04:13.625","Text":"Then I have my exponent,"},{"Start":"04:13.625 ","End":"04:17.090","Text":"which is 0, 1, 0,"},{"Start":"04:17.090 ","End":"04:21.800","Text":"0 and then I have my mantissa."},{"Start":"04:21.800 ","End":"04:23.500","Text":"Again, I\u0027ve got 1, 2,"},{"Start":"04:23.500 ","End":"04:25.635","Text":"3, 4,5, 6,"},{"Start":"04:25.635 ","End":"04:29.020","Text":"7 here, and I need to write that in 3,"},{"Start":"04:29.020 ","End":"04:32.720","Text":"4, 5, 1, 0, 1, there\u0027s my 7-bits."},{"Start":"04:32.720 ","End":"04:34.580","Text":"But I\u0027m supposed to have 11,"},{"Start":"04:34.580 ","End":"04:39.185","Text":"so I have to pad it by putting 4 zeros at the end,"},{"Start":"04:39.185 ","End":"04:41.705","Text":"and so there is part b done."},{"Start":"04:41.705 ","End":"04:43.780","Text":"Moving on to the final one,"},{"Start":"04:43.780 ","End":"04:48.915","Text":"Part C. This time we got minus 92.9375,"},{"Start":"04:48.915 ","End":"04:52.270","Text":"so just go with our columns,"},{"Start":"04:54.860 ","End":"04:57.070","Text":"1, 2, 4, 8, 16, 32,"},{"Start":"04:57.070 ","End":"04:58.130","Text":"64, if I carried on,"},{"Start":"04:58.130 ","End":"05:01.775","Text":"it would be a 128, but that\u0027s bigger, so I don\u0027t need that."},{"Start":"05:01.775 ","End":"05:04.775","Text":"To make 92,"},{"Start":"05:04.775 ","End":"05:09.570","Text":"I\u0027ll need 64, don\u0027t need the 32 that make it too big,"},{"Start":"05:09.570 ","End":"05:13.635","Text":"16 would make it 80 plus 8 will make it 88,"},{"Start":"05:13.635 ","End":"05:17.570","Text":"plus 4 would make it 92 exactly so I don\u0027t need anything else."},{"Start":"05:17.570 ","End":"05:19.345","Text":"There\u0027s my 92,"},{"Start":"05:19.345 ","End":"05:22.780","Text":"and then the 9.9375,"},{"Start":"05:22.780 ","End":"05:26.660","Text":"I don\u0027t know how many columns I need to go over,"},{"Start":"05:26.660 ","End":"05:28.610","Text":"but let\u0027s write out."},{"Start":"05:28.610 ","End":"05:32.000","Text":"That set, and let\u0027s stop say at 16."},{"Start":"05:32.000 ","End":"05:33.725","Text":"Let\u0027s try out 16."},{"Start":"05:33.725 ","End":"05:38.750","Text":"If we were to multiply this number by 16 and get a whole number,"},{"Start":"05:38.750 ","End":"05:40.685","Text":"then we know that we don\u0027t need to go any further."},{"Start":"05:40.685 ","End":"05:41.840","Text":"If we don\u0027t get a whole number,"},{"Start":"05:41.840 ","End":"05:43.160","Text":"then we need some more columns,"},{"Start":"05:43.160 ","End":"05:46.130","Text":"but I can just tell you that actually 16 will do it."},{"Start":"05:46.130 ","End":"05:48.320","Text":"If you multiply 16 by this,"},{"Start":"05:48.320 ","End":"05:49.700","Text":"you\u0027ll get 15,"},{"Start":"05:49.700 ","End":"05:53.960","Text":"so what we\u0027re trying to make up here is 15 sixteenths,"},{"Start":"05:53.960 ","End":"05:57.200","Text":"so that will be, these 3 columns will need a 1,"},{"Start":"05:57.200 ","End":"06:00.345","Text":"and I don\u0027t need the last one."},{"Start":"06:00.345 ","End":"06:03.120","Text":"Let\u0027s put our 1\u0027s in here,"},{"Start":"06:03.120 ","End":"06:05.430","Text":"then there\u0027ll be 15 sixteenths,"},{"Start":"06:05.430 ","End":"06:10.925","Text":"and I\u0027ve got my number now and that\u0027s me done with my mantissa,"},{"Start":"06:10.925 ","End":"06:14.940","Text":"I now need to work out where do I put my binary point,"},{"Start":"06:14.940 ","End":"06:17.165","Text":"and it\u0027s obviously going to be here."},{"Start":"06:17.165 ","End":"06:18.710","Text":"How many places do I have to shift it?"},{"Start":"06:18.710 ","End":"06:21.456","Text":"By 1, 2, 3, 4, 5,"},{"Start":"06:21.456 ","End":"06:26.640","Text":"6 places, so my exponent is 6,"},{"Start":"06:26.640 ","End":"06:32.160","Text":"and with my columns I need 4 and 2,"},{"Start":"06:32.160 ","End":"06:33.525","Text":"and I think there,"},{"Start":"06:33.525 ","End":"06:35.670","Text":"that\u0027s the exponent,"},{"Start":"06:35.670 ","End":"06:37.810","Text":"so I\u0027m done now,"},{"Start":"06:37.810 ","End":"06:39.695","Text":"I\u0027ve got a negative number,"},{"Start":"06:39.695 ","End":"06:41.900","Text":"so I need a sign bit of 1."},{"Start":"06:41.900 ","End":"06:44.420","Text":"I\u0027ve got an exponent of that,"},{"Start":"06:44.420 ","End":"06:52.646","Text":"0,1,1,0 and I\u0027ve got my mantissa of 1,"},{"Start":"06:52.646 ","End":"06:53.777","Text":"0, 1, 1, 1, 0, 0, 1, 1, 1,"},{"Start":"06:53.777 ","End":"07:00.785","Text":"0 and actually this time around,"},{"Start":"07:00.785 ","End":"07:02.465","Text":"I\u0027ve got 1, 2, 3, 4, 5,"},{"Start":"07:02.465 ","End":"07:04.810","Text":"6, 7, 8, 9, 10, 11."},{"Start":"07:04.810 ","End":"07:06.770","Text":"Magically I\u0027ve got 11,"},{"Start":"07:06.770 ","End":"07:08.750","Text":"so there\u0027s no padding necessary."},{"Start":"07:08.750 ","End":"07:11.530","Text":"This is a 16-bit number,"},{"Start":"07:11.530 ","End":"07:15.665","Text":"and just the format that we\u0027ve been asked for at the beginning."},{"Start":"07:15.665 ","End":"07:18.110","Text":"We\u0027re done with all 3 parts,"},{"Start":"07:18.110 ","End":"07:24.230","Text":"and this is a format you do see actually this is the closest to the real way"},{"Start":"07:24.230 ","End":"07:27.410","Text":"that floating point numbers are stored in a format"},{"Start":"07:27.410 ","End":"07:31.250","Text":"called I triple E that uses 32-bits not 16,"},{"Start":"07:31.250 ","End":"07:33.365","Text":"but it\u0027s a very similar principle,"},{"Start":"07:33.365 ","End":"07:36.680","Text":"so that\u0027s why it was worth doing this exercise to"},{"Start":"07:36.680 ","End":"07:40.880","Text":"understand how it would be stored for real on a real system."},{"Start":"07:40.880 ","End":"07:44.550","Text":"Great, so I\u0027ll see you in the next exercise."}],"ID":26327},{"Watched":false,"Name":"Exercise 6 parts A-C","Duration":"7m 56s","ChapterTopicVideoID":25507,"CourseChapterTopicPlaylistID":237671,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.175","Text":"Hello, welcome back everyone."},{"Start":"00:02.175 ","End":"00:05.370","Text":"In this question, we\u0027ve been asked to normalize"},{"Start":"00:05.370 ","End":"00:09.510","Text":"a binary number which is in 2\u0027s complement,"},{"Start":"00:09.510 ","End":"00:15.825","Text":"and uses 16 bits with the least significant 4 bits representing the exponent."},{"Start":"00:15.825 ","End":"00:22.050","Text":"The first thing I\u0027m going to do is mark-off the exponent which is there."},{"Start":"00:22.050 ","End":"00:23.625","Text":"If I work this out,"},{"Start":"00:23.625 ","End":"00:30.200","Text":"the 2\u0027s complement values for the column headings will be minus 8,"},{"Start":"00:30.200 ","End":"00:32.265","Text":"4, 2, and 1,"},{"Start":"00:32.265 ","End":"00:37.785","Text":"and so therefore my exponent will be 5."},{"Start":"00:37.785 ","End":"00:39.870","Text":"There\u0027s my exponent to start with."},{"Start":"00:39.870 ","End":"00:42.200","Text":"Now, to do normalization,"},{"Start":"00:42.200 ","End":"00:46.490","Text":"what you\u0027re looking for is you\u0027re looking for a particular pattern at the beginning."},{"Start":"00:46.490 ","End":"00:48.380","Text":"If you see a 0,"},{"Start":"00:48.380 ","End":"00:52.950","Text":"0 as the first 2 bits or a 1,"},{"Start":"00:52.950 ","End":"00:55.561","Text":"1 as the first 2 bits,"},{"Start":"00:55.561 ","End":"00:58.850","Text":"you know that the number has not been normalized."},{"Start":"00:58.850 ","End":"01:02.370","Text":"What we\u0027re looking to find I should say,"},{"Start":"01:02.370 ","End":"01:06.370","Text":"is either 0 and a 1 or a 1 and a 0,"},{"Start":"01:06.370 ","End":"01:10.655","Text":"that would indicate that it has been stored in normalized format."},{"Start":"01:10.655 ","End":"01:12.580","Text":"Coming back to question a,"},{"Start":"01:12.580 ","End":"01:14.990","Text":"we can see that this is not normalized,"},{"Start":"01:14.990 ","End":"01:19.790","Text":"there\u0027s 2 consecutive 0s at the beginning and the next 2 numbers happen to be 0 as well."},{"Start":"01:19.790 ","End":"01:25.580","Text":"We\u0027re looking for the first occurrence of a 0 and a 1 and the first place we find it."},{"Start":"01:25.580 ","End":"01:30.350","Text":"In this number, implied binary point would be here."},{"Start":"01:30.350 ","End":"01:33.260","Text":"But where we find the first 0,"},{"Start":"01:33.260 ","End":"01:35.830","Text":"1, which is where we\u0027d want it to be."},{"Start":"01:35.830 ","End":"01:39.375","Text":"What we\u0027d like to do is to shift the mantissa."},{"Start":"01:39.375 ","End":"01:46.675","Text":"Shift the mantissa so that the binary point is 2 places over from the beginning."},{"Start":"01:46.675 ","End":"01:49.795","Text":"That would give us a mantissa that would look like this."},{"Start":"01:49.795 ","End":"01:51.540","Text":"It\u0027s the same number,"},{"Start":"01:51.540 ","End":"01:54.870","Text":"but 2 of the leading 0s have been removed."},{"Start":"01:54.870 ","End":"01:57.530","Text":"I can do that, that\u0027s absolutely fine,"},{"Start":"01:57.530 ","End":"02:05.105","Text":"but I\u0027ll now need to adjust my exponent by the amount of places I shifted by which was 2."},{"Start":"02:05.105 ","End":"02:12.570","Text":"I\u0027m going to need to take 2 of my exponent and so I\u0027m now looking to make 3,"},{"Start":"02:12.570 ","End":"02:16.210","Text":"and that would be 0011."},{"Start":"02:16.210 ","End":"02:22.790","Text":"I now have my adjusted mantissa and my adjusted exponent,"},{"Start":"02:22.790 ","End":"02:27.658","Text":"and I can write those down in the final answer,"},{"Start":"02:27.658 ","End":"02:34.190","Text":"0101010111."},{"Start":"02:34.190 ","End":"02:35.900","Text":"However, looking at this,"},{"Start":"02:35.900 ","End":"02:38.050","Text":"I\u0027ve only got 2,"},{"Start":"02:38.050 ","End":"02:40.850","Text":"4, 6, 8, 10 bits."},{"Start":"02:40.850 ","End":"02:44.300","Text":"The answer is to be in a 16-bit encoding,"},{"Start":"02:44.300 ","End":"02:47.840","Text":"so I need to pad it by having 2 bits at the end before I add"},{"Start":"02:47.840 ","End":"02:51.440","Text":"my 4 bits worth of exponent and so I\u0027ve done that."},{"Start":"02:51.440 ","End":"02:55.700","Text":"There\u0027s my 2 zeros and then I can add the exponent onto the end."},{"Start":"02:55.700 ","End":"02:58.420","Text":"Now I have completed Part a,"},{"Start":"02:58.420 ","End":"03:04.445","Text":"that is the normalized version of the original number that we were given."},{"Start":"03:04.445 ","End":"03:09.395","Text":"If you were to work this out by converting this to binary,"},{"Start":"03:09.395 ","End":"03:11.785","Text":"multiplying it by 2^5,"},{"Start":"03:11.785 ","End":"03:17.630","Text":"you\u0027d find it\u0027s exactly the same number as this mantissa value multiplied by"},{"Start":"03:17.630 ","End":"03:21.920","Text":"2^3 and so very straightforward calculation"},{"Start":"03:21.920 ","End":"03:25.355","Text":"to do that now that we\u0027ve done some exercises on it previously."},{"Start":"03:25.355 ","End":"03:27.920","Text":"Part B, let\u0027s have a look at Part B."},{"Start":"03:27.920 ","End":"03:29.930","Text":"It\u0027s going to be exactly the same thing again."},{"Start":"03:29.930 ","End":"03:39.170","Text":"Let\u0027s put a line here just to indicate what our exponent is and it\u0027s 0100,"},{"Start":"03:39.170 ","End":"03:46.009","Text":"which I can say straightaway is going to be 4 and now let\u0027s look at the mantissa."},{"Start":"03:46.009 ","End":"03:48.710","Text":"The mantissa has an implied binary point there."},{"Start":"03:48.710 ","End":"03:50.805","Text":"Of course, we don\u0027t like that,"},{"Start":"03:50.805 ","End":"03:53.055","Text":"that\u0027s not normalized 11."},{"Start":"03:53.055 ","End":"03:57.320","Text":"We\u0027re looking for the first occurrence where there is not 2 1s next to each other."},{"Start":"03:57.320 ","End":"04:03.020","Text":"The first place where we find it is here where we find a 10"},{"Start":"04:03.020 ","End":"04:09.065","Text":"and so we want to shift 1 to 3 places exponent."},{"Start":"04:09.065 ","End":"04:12.835","Text":"We want to take 3 off 4."},{"Start":"04:12.835 ","End":"04:21.090","Text":"Obviously that\u0027s going to give us 1 so our exponent is going to be that now."},{"Start":"04:21.090 ","End":"04:22.900","Text":"Got the exponent,"},{"Start":"04:22.900 ","End":"04:26.135","Text":"I\u0027ll just write out the mantissa again."},{"Start":"04:26.135 ","End":"04:27.950","Text":"This time shifted over,"},{"Start":"04:27.950 ","End":"04:33.335","Text":"so it\u0027s 101011 and 0, 0, 0."},{"Start":"04:33.335 ","End":"04:37.493","Text":"I\u0027ve taken it from this part of the number above."},{"Start":"04:37.493 ","End":"04:39.905","Text":"Great. I\u0027m pretty much done."},{"Start":"04:39.905 ","End":"04:43.335","Text":"What I need to do as obviously pad as well."},{"Start":"04:43.335 ","End":"04:45.233","Text":"Let\u0027s write the initial,"},{"Start":"04:45.233 ","End":"04:48.749","Text":"pad the number down and work out how much padding we need afterwards."},{"Start":"04:48.749 ","End":"04:53.670","Text":"It\u0027s 10101100."},{"Start":"04:53.670 ","End":"04:57.960","Text":"I need to add another 2 bits there, 2,"},{"Start":"04:57.960 ","End":"04:59.730","Text":"4, 6, 8,"},{"Start":"04:59.730 ","End":"05:02.280","Text":"10, another 4 zeros on the end,"},{"Start":"05:02.280 ","End":"05:09.470","Text":"and I\u0027ve got 12 bits now and then I add my exponent at the end and now I\u0027ve got 12 bits,"},{"Start":"05:09.470 ","End":"05:11.285","Text":"16 bits in total."},{"Start":"05:11.285 ","End":"05:12.995","Text":"Just check that again, 2,"},{"Start":"05:12.995 ","End":"05:14.795","Text":"4, 6, 8,"},{"Start":"05:14.795 ","End":"05:18.480","Text":"10, 12, 14, 16."},{"Start":"05:18.480 ","End":"05:20.970","Text":"I\u0027m done with Part b."},{"Start":"05:20.970 ","End":"05:24.105","Text":"The final one, Part c. Once again,"},{"Start":"05:24.105 ","End":"05:25.770","Text":"let\u0027s partition off our exponent."},{"Start":"05:25.770 ","End":"05:29.209","Text":"You might notice now that this is a negative exponent."},{"Start":"05:29.209 ","End":"05:32.925","Text":"If I write that down 1110,"},{"Start":"05:32.925 ","End":"05:36.450","Text":"the column headings for that would be 1, 2, 4,"},{"Start":"05:36.450 ","End":"05:43.050","Text":"minus 8 as the previous and what that would give us is minus 8 plus 6,"},{"Start":"05:43.050 ","End":"05:44.790","Text":"which is minus 2."},{"Start":"05:44.790 ","End":"05:48.040","Text":"We start off with minus 2 as our exponents."},{"Start":"05:48.040 ","End":"05:52.535","Text":"We\u0027re making a smaller number by this exponent."},{"Start":"05:52.535 ","End":"05:56.975","Text":"Let\u0027s now move on to what the mantissa would be."},{"Start":"05:56.975 ","End":"06:00.080","Text":"Once again, we\u0027re looking to take"},{"Start":"06:00.080 ","End":"06:04.625","Text":"the implied binary point and move it along to somewhere more acceptable."},{"Start":"06:04.625 ","End":"06:09.285","Text":"We\u0027re looking for the first occurrence of 10,"},{"Start":"06:09.285 ","End":"06:12.255","Text":"and we find it over here."},{"Start":"06:12.255 ","End":"06:16.170","Text":"We need to adjust by 1, 2 places."},{"Start":"06:16.170 ","End":"06:20.310","Text":"We started with minus 2 and we need to"},{"Start":"06:20.310 ","End":"06:24.560","Text":"take 2 off that and that will obviously give us minus 4,"},{"Start":"06:24.560 ","End":"06:31.365","Text":"so minus 4 is going to be minus 8 plus 4 and that\u0027s it."},{"Start":"06:31.365 ","End":"06:35.285","Text":"There is our experiment for this adjusted number."},{"Start":"06:35.285 ","End":"06:39.530","Text":"Let\u0027s just write down the mantissa underneath in red as we\u0027ve done for previous ones"},{"Start":"06:39.530 ","End":"06:47.115","Text":"1011100000, 1 missing."},{"Start":"06:47.115 ","End":"06:49.560","Text":"There we go. I\u0027ve got 1,"},{"Start":"06:49.560 ","End":"06:51.270","Text":"2, 3, 4, 5,"},{"Start":"06:51.270 ","End":"06:52.980","Text":"6, 7, 8, 9,"},{"Start":"06:52.980 ","End":"06:56.390","Text":"10, digits for my mantissa,"},{"Start":"06:56.390 ","End":"07:05.115","Text":"I need 12 so I\u0027m going to pad it with another 2 on the end there, so 101110,"},{"Start":"07:05.115 ","End":"07:15.660","Text":"2, 3, 4 and then 2 more and then I add my exponent at the end 1100."},{"Start":"07:15.660 ","End":"07:17.490","Text":"Just check that I\u0027ve got that right."},{"Start":"07:17.490 ","End":"07:19.695","Text":"2, 4, 6,"},{"Start":"07:19.695 ","End":"07:21.255","Text":"8, 10,"},{"Start":"07:21.255 ","End":"07:22.500","Text":"12, 14,"},{"Start":"07:22.500 ","End":"07:28.685","Text":"16 and I\u0027m done and I\u0027ve normalized all 3 numbers now."},{"Start":"07:28.685 ","End":"07:30.200","Text":"As I said earlier,"},{"Start":"07:30.200 ","End":"07:33.800","Text":"you could work out what the original number"},{"Start":"07:33.800 ","End":"07:38.119","Text":"was in the method that we\u0027ve used before by converting"},{"Start":"07:38.119 ","End":"07:42.530","Text":"this to its bits point binary equivalent and then multiplying"},{"Start":"07:42.530 ","End":"07:47.160","Text":"it by 2 to the exponent and then do the same thing for our new number."},{"Start":"07:47.160 ","End":"07:48.770","Text":"You should find they work out to be"},{"Start":"07:48.770 ","End":"07:51.955","Text":"exactly the same decimal value or you\u0027ve done something wrong."},{"Start":"07:51.955 ","End":"07:56.640","Text":"That\u0027s us done then with normalized numbers. Thank you very much."}],"ID":26328}],"Thumbnail":null,"ID":237671}]

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1.2

11