1 00:00:04,840 --> 00:00:08,340 So let's now have a bit of a discussion about binary. 2 00:00:08,340 --> 00:00:10,810 What it is, and how you can actually use it. 3 00:00:10,810 --> 00:00:14,380 So binary is actually a number system just like decimal but 4 00:00:14,380 --> 00:00:19,560 whereas decimal is based on ten and uses the digits zero to nine binary 5 00:00:19,560 --> 00:00:23,870 is actually based on two and so therefore can only use the digits zero to one. 6 00:00:23,870 --> 00:00:27,580 So I'm just gonna show this little table on the screen. 7 00:00:27,580 --> 00:00:30,440 And in decimal, we're actually very familiar with 8 00:00:30,440 --> 00:00:34,690 the idea that the position of a digit represents its value. 9 00:00:34,690 --> 00:00:36,640 So you may remember doing hundreds. 10 00:00:36,640 --> 00:00:40,220 Tens and units in school to get used to that idea. 11 00:00:40,220 --> 00:00:43,590 So if we enter the digits into columns as you can see on the screen we 12 00:00:43,590 --> 00:00:46,430 can see that the column values are just powers of ten. 13 00:00:46,430 --> 00:00:49,430 So with the first column representing 10 to the power of 0, 14 00:00:49,430 --> 00:00:51,140 anything to the power of 0 is 1. 15 00:00:51,140 --> 00:00:53,290 In the second column is 10 to the power of 1. 16 00:00:53,290 --> 00:00:57,060 And anything to the power of 1 is itself and so on. 17 00:00:57,060 --> 00:00:59,750 So 10 squared, 10 cubed, etc. 18 00:00:59,750 --> 00:01:03,340 So, thus the 9 in the second column as you can see on the screen were 90, 19 00:01:03,340 --> 00:01:05,650 and the 9 in the third column is worth 900. 20 00:01:05,650 --> 00:01:09,960 So it's easy to see that the maximum number we can represent with four digits 21 00:01:09,960 --> 00:01:13,760 then is 9999, as you can see on the screen And 22 00:01:13,760 --> 00:01:17,040 if we need to store 10,000, then we have to go into a fifth column. 23 00:01:17,040 --> 00:01:20,160 So binary, as it turns out, works in exactly the same way. 24 00:01:20,160 --> 00:01:23,520 But because it's working in base two rather than base ten, 25 00:01:23,520 --> 00:01:26,530 the column headings are powers of two, they're not powers of ten. 26 00:01:29,780 --> 00:01:33,190 So as you can see in this screen, in binary each binary digit or 27 00:01:33,190 --> 00:01:35,090 bit is worth the power of 2, 28 00:01:35,090 --> 00:01:39,540 and it's corresponding to the column that it's in, just like in the decimal system. 29 00:01:39,540 --> 00:01:42,010 So 1 in the first column is still worth 1. 30 00:01:42,010 --> 00:01:44,700 In the next column it is worth 2, and so on. 31 00:01:44,700 --> 00:01:46,270 And as you can see from the slide, 32 00:01:46,270 --> 00:01:49,700 a byte where all the bits are 0 holds the value 0. 33 00:01:49,700 --> 00:01:53,250 If the first bit is one, then the value is one. 34 00:01:53,250 --> 00:01:56,920 So, as an example to represent decimal 47, 35 00:01:56,920 --> 00:02:02,560 we need 32 plus 8 which gives us 40 plus 4 which gives us a total of 44, and 36 00:02:02,560 --> 00:02:07,090 We need two, which gives us a total of 46, and then plus one, to give it 47. 37 00:02:07,090 --> 00:02:09,400 So in binary, the decimal value of 47 is 00101111, 38 00:02:09,400 --> 00:02:13,770 as you can see on the screen there. 39 00:02:13,770 --> 00:02:16,140 If every one of the first 8 bits is one. 40 00:02:16,140 --> 00:02:21,370 Now what we actually have is 128 plus 64 plus so on so forth. 41 00:02:21,370 --> 00:02:23,840 So it's easy in this case to look at the next column 42 00:02:23,840 --> 00:02:27,830 because we can actually have one less than it. 43 00:02:27,830 --> 00:02:28,680 Or there is 255. 44 00:02:28,680 --> 00:02:32,780 Now when we actually add 1 to 255 the number in the first column becomes 2. 45 00:02:32,780 --> 00:02:35,420 So we enter 0 and we actually carry 1. 46 00:02:37,470 --> 00:02:41,820 And on the next slide you can see that the number in the next column then becomes 2. 47 00:02:41,820 --> 00:02:45,930 So again we enter 0 and carry the 1 and so on. 48 00:02:45,930 --> 00:02:50,030 Once we reach the eighth bit what we do is set that to 0. 49 00:02:50,030 --> 00:02:52,540 And we have to move into another part. 50 00:02:52,540 --> 00:02:57,400 Now unlike a decimal where we don't have the concept of restricting the number of 51 00:02:57,400 --> 00:03:01,070 digits because we're not really trying to store decimal numbers in a computer or 52 00:03:01,070 --> 00:03:06,230 such, binary doesn't have that Or binary has a problem or has a restriction. 53 00:03:06,230 --> 00:03:07,240 Because binary, 54 00:03:07,240 --> 00:03:12,630 we actually have the concept of eight bits making a byte, by a eighth, hence byte. 55 00:03:12,630 --> 00:03:16,780 And because we're actually storing binary numbers in physical computer hardware, 56 00:03:16,780 --> 00:03:21,080 a decision had to be made early on for example As to how many transistors 57 00:03:21,080 --> 00:03:24,320 at the hyper level would be used in each memory location. 58 00:03:24,320 --> 00:03:26,570 So the number they eventually choose was 8. 59 00:03:26,570 --> 00:03:29,130 And as a result, when dealing with binary numbers, 60 00:03:29,130 --> 00:03:31,560 we have to consider a digit moving into the next byte. 61 00:03:31,560 --> 00:03:35,950 But it's nothing really more than starting a new column like we did in decimal 62 00:03:35,950 --> 00:03:37,030 when we added 1 to 9999. 63 00:03:37,030 --> 00:03:41,980 It also turns out it's actually very easy to display numbers in binary in Python. 64 00:03:41,980 --> 00:03:45,800 So I'm just gonna switch over to my Intellijay, IDE, and 65 00:03:45,800 --> 00:03:48,080 we're just gonna type a bit of code in. 66 00:03:48,080 --> 00:03:51,530 As I mentioned it's very easy to display numbers in binary in Python. 67 00:03:51,530 --> 00:03:54,430 Now we covered formatting strings and using replacement fields, 68 00:03:54,430 --> 00:03:58,530 the one thing we haven't mentioned is that you can add a number base to the format To 69 00:03:58,530 --> 00:04:02,580 display a number in binary hexadecimal or octal which we'll look at shortly. 70 00:04:02,580 --> 00:04:06,510 So, the following program I'm about to type in will display the numbers from 0 71 00:04:06,510 --> 00:04:07,240 to 16 in binary. 72 00:04:07,240 --> 00:04:12,060 So we can type for i, in range, 17, and then 73 00:04:15,280 --> 00:04:20,380 print Is a replacement field zero column 74 00:04:20,380 --> 00:04:25,270 greater than two in 75 00:04:28,640 --> 00:04:33,890 is, and then we can top zero on the replacement field greater than O eight B. 76 00:04:33,890 --> 00:04:36,870 In closing off the field, top format I. 77 00:04:36,870 --> 00:04:39,960 And, I made an arrow there at the start of this line. 78 00:04:39,960 --> 00:04:42,140 However I can fix that or close off the replacement field. 79 00:04:42,140 --> 00:04:46,320 So what we're doing is we're showing the decimal equivalent 80 00:04:46,320 --> 00:04:50,390 of the field with the 2, and we're right align, right aligning to the value of i, 81 00:04:50,390 --> 00:04:51,660 then we're specifying a of 8. 82 00:04:51,660 --> 00:04:57,410 Over here the, the binary right align and adding beta with the value of i in binary. 83 00:04:57,410 --> 00:05:03,450 So if we actually run that You can see we're getting the right result. 84 00:05:03,450 --> 00:05:08,110 Zero in binary is 0000 one in binary is and so on and so forth, as per the slides. 85 00:05:08,110 --> 00:05:11,520 So looking at the pattern of the binary numbers from 0 to 16 is a good 86 00:05:11,520 --> 00:05:14,650 way to understand how binary numbers work. 87 00:05:14,650 --> 00:05:17,950 What I'm talking about is we can see the one moving across here. 88 00:05:17,950 --> 00:05:23,020 So we've got the value zero which is gonna be all zeroes, one has got the bit 89 00:05:23,020 --> 00:05:26,640 right most bit set and notice when we move to two the next bit gets set and 90 00:05:26,640 --> 00:05:31,980 then three the two is set, four we've got the next bit set and so on and so forth. 91 00:05:31,980 --> 00:05:34,700 You can also extend the range to print out high binary numbers so 92 00:05:34,700 --> 00:05:38,810 experiment with this and get a feel for how the binary numbers increase as you add 93 00:05:38,810 --> 00:05:41,810 one to it and that will really help you understand. 94 00:05:41,810 --> 00:05:44,360 The conversions that are actually happening when 95 00:05:44,360 --> 00:05:46,660 you're converting from decimal to binary. 96 00:05:46,660 --> 00:05:49,160 Now subtraction actually works the same as it does in decimal, so 97 00:05:49,160 --> 00:05:51,000 I'm going to move back to the next slide. 98 00:05:53,140 --> 00:05:56,450 So as I mentioned subtraction also works Works the same as it does in decimal. 99 00:05:56,450 --> 00:05:59,960 If there's not a number big enough in the current column 100 00:05:59,960 --> 00:06:01,860 then you can actually borrow from the next column. 101 00:06:01,860 --> 00:06:04,430 Now you'll be glad to know that we're not going to look at multiplication and 102 00:06:04,430 --> 00:06:05,540 division here. 103 00:06:05,540 --> 00:06:07,280 If you were to perform multiplication or 104 00:06:07,280 --> 00:06:12,380 division long hand the process is identical to what you would do in decimal, 105 00:06:12,380 --> 00:06:15,100 although you'd actually tend to have many more digits to deal with in 106 00:06:15,100 --> 00:06:19,310 binary just because we're dealing with a base-2 number system. 107 00:06:22,290 --> 00:06:25,120 So this next slide shows seven, which in binary is one, 108 00:06:25,120 --> 00:06:26,430 one, one as you can see on the screen there. 109 00:06:26,430 --> 00:06:30,790 And it shows up being multiplied by three or one, one in binary. 110 00:06:30,790 --> 00:06:33,270 And it shows you that long hand if you want to experiment with it, 111 00:06:33,270 --> 00:06:37,100 but you really don't have to perform arithmetic like this on binary numbers 112 00:06:37,100 --> 00:06:38,410 That's what we've got computers for. 113 00:06:38,410 --> 00:06:41,070 It's just good to have a sort of basic understanding of this 114 00:06:41,070 --> 00:06:44,200 without necessarily needing to know or to know it in detail. 115 00:06:44,200 --> 00:06:46,930 Now there's a few more operations that can be performed on 116 00:06:46,930 --> 00:06:51,440 binary numbers that are beyond the basic arithmetic that we're used to and 117 00:06:51,440 --> 00:06:53,080 understanding those can actually be useful. 118 00:06:53,080 --> 00:06:58,590 So I'm just gonna move to the next slide Now 119 00:06:58,590 --> 00:07:02,730 this slide shows what happens when you shift a binary number left. 120 00:07:02,730 --> 00:07:06,000 So just like a decimal when you want to multiply by ten, 121 00:07:06,000 --> 00:07:08,350 you insert a zero at the end of the Of the number. 122 00:07:08,350 --> 00:07:12,090 In binary the same thing, multiplies by two instead of ten. 123 00:07:12,090 --> 00:07:17,230 So shifting right divides by two just like it divides by ten in decimal. 124 00:07:17,230 --> 00:07:19,860 And as it turns out, you can have a binary point in binary as well. 125 00:07:19,860 --> 00:07:24,500 So 11 or 1011, as you can see on the screen there, 126 00:07:24,500 --> 00:07:28,060 Shift right would become 101.1 which is 5.5 in decimal. 127 00:07:28,060 --> 00:07:34,200 Now the final three operations we want to look at briefly are going to be or, 128 00:07:34,200 --> 00:07:36,930 and, and xall, which is short for exclusive all. 129 00:07:36,930 --> 00:07:38,220 The first thing or. 130 00:07:38,220 --> 00:07:42,210 So the binary or operation checks each bid and its two upper hands. 131 00:07:42,210 --> 00:07:45,410 And for each bid it actually sets the bid in the result to one 132 00:07:45,410 --> 00:07:48,290 if either of the upper hands had a one in that position. 133 00:07:48,290 --> 00:07:53,470 And probably the next lab will help you understand that. 134 00:07:53,470 --> 00:08:00,090 So this slide actually shows 4 or 8 giving 12 and then 12 or 11 giving 15. 135 00:08:00,090 --> 00:08:02,980 So that's how OR actually works. 136 00:08:02,980 --> 00:08:08,290 The next one, the AND operation, This sets each bit of the result to one, 137 00:08:08,290 --> 00:08:11,570 if the corresponding bit in both upper hands is one. 138 00:08:11,570 --> 00:08:12,900 So let's have a look at that in the next slide. 139 00:08:15,790 --> 00:08:20,720 So as you can see here, 12 and eight is eight because there's only one position 140 00:08:20,720 --> 00:08:25,070 that's got both a one, or that has a one in both 12 and eight as you can see there. 141 00:08:25,070 --> 00:08:29,870 So eight and seven is zero, as no positions contain a one in both numbers. 142 00:08:29,870 --> 00:08:34,740 And just moving on to the final slide, this is Exclusive OR, or XOR. 143 00:08:34,740 --> 00:08:35,789 And with this one, 144 00:08:35,789 --> 00:08:39,669 XOR sets the corresponding bit of the result to one if either but not 145 00:08:39,669 --> 00:08:43,929 both of the upper ANDs has a one in that position as you can see in this slide. 146 00:08:43,929 --> 00:08:46,600 Now, XOR is useful because if you repeat the operation, 147 00:08:46,600 --> 00:08:47,670 you get the original number back. 148 00:08:47,670 --> 00:08:49,020 Now we'll move on to the next slide. 149 00:08:52,100 --> 00:08:56,410 So in this one you can actually see 12 XOR eight is four or equals four and 150 00:08:56,410 --> 00:08:58,970 four XOR eight is equal to 12. 151 00:08:58,970 --> 00:09:03,190 Now this has got useful applications in cryptography, for example. 152 00:09:03,190 --> 00:09:07,030 And was also used to draw the cursor on early monochrome screens. 153 00:09:07,030 --> 00:09:11,690 By XORing each row of a character's eight bit representation on the screen The image 154 00:09:11,690 --> 00:09:14,850 would actually be inverted to represent the cursor being on that character. 155 00:09:14,850 --> 00:09:17,430 So XOR again restores the character. 156 00:09:17,430 --> 00:09:18,760 Now moving on to the last slide. 157 00:09:21,390 --> 00:09:25,010 This actually shows you a much smaller four by five character 158 00:09:25,010 --> 00:09:26,830 XOR in each row with one one one one. 159 00:09:26,830 --> 00:09:28,550 And you can sort of see what happened there. 160 00:09:28,550 --> 00:09:31,060 Now that's really all you need to know about binary at this stage, 161 00:09:31,060 --> 00:09:34,450 we're not really going to perform binary arithmetic manually 162 00:09:34,450 --> 00:09:36,470 Because we've got computers to do that. 163 00:09:36,470 --> 00:09:39,610 But, that said understanding how numbers are stored 164 00:09:39,610 --> 00:09:42,870 inside the computer is actually very useful especially when we have 165 00:09:42,870 --> 00:09:45,970 to consider how many bytes we have to allocate for storing our data. 166 00:09:45,970 --> 00:09:48,010 Now it also explains why, for 167 00:09:48,010 --> 00:09:52,240 example ASCII cannot be used to represent all the characters that are used in 168 00:09:52,240 --> 00:09:57,330 the various languages around the world as ASCII characters must fit into one byte. 169 00:09:57,330 --> 00:10:00,950 Now Unicode resolves this by using two or four bytes per character and 170 00:10:00,950 --> 00:10:02,840 allowing many more characters to be represented. 171 00:10:02,840 --> 00:10:06,420 Now if some of that or all of that didn't make a great deal of sense, don't worry. 172 00:10:06,420 --> 00:10:09,110 It's useful to understand what's going on inside the computer 173 00:10:09,110 --> 00:10:11,940 But is not actually essential and the remainder of this course 174 00:10:11,940 --> 00:10:15,340 does not require you to actually understand the binary number system. 175 00:10:15,340 --> 00:10:17,170 So that's it, I'm going to finish the video here. 176 00:10:17,170 --> 00:10:21,100 In the next video we're going to start looking at hexidecimal and octal. 177 00:10:21,100 --> 00:10:22,460 So see you in the next video.