WEBVTT 00:00.400 --> 00:01.240 Welcome back. 00:01.440 --> 00:07.560 And in this lecture we are going to take a deeper look at subtraction, focusing on entirely on how 00:07.600 --> 00:11.400 borrowing works step by step in decimal numbers. 00:11.440 --> 00:17.000 Now this is important not only because subtraction is fundamental, but also because everything we do 00:17.040 --> 00:22.320 here will directly apply to how subtraction is performed in binary. 00:22.680 --> 00:28.040 Now, computers subtracting binary, but they follow the exact same logical process. 00:28.280 --> 00:30.040 And we are about to see here. 00:30.040 --> 00:36.320 And as always, since we are doing this in a visual lecture, I will go through what to draw on your 00:36.360 --> 00:39.200 notebook to practice on your own. 00:39.880 --> 00:48.960 Now let's reuse our some example numbers that we used previously 67 subtraction 79. 00:49.560 --> 00:56.480 Now let's go digit by digit starting from the right most column from here. 00:57.160 --> 00:57.960 Now this. 00:57.960 --> 01:06.620 As you remember this is ones tens and hundreds and this goes on like that. 01:07.060 --> 01:07.700 Hundreds. 01:07.740 --> 01:08.340 Thousands. 01:08.380 --> 01:09.300 Ten thousands. 01:09.620 --> 01:10.100 Hundreds. 01:10.100 --> 01:10.740 Thousands. 01:11.180 --> 01:14.340 But now we will work with just ones and tens. 01:14.540 --> 01:15.940 Now you have seven. 01:16.220 --> 01:20.260 Subtract nine, which is not possible. 01:20.500 --> 01:24.180 Seven is smaller than nine. 01:24.860 --> 01:25.860 What to do now? 01:26.620 --> 01:36.180 We borrow one one from six which is next digit to the left in the tens place. 01:36.740 --> 01:43.700 When we borrow, that one becomes a ten to add our digit. 01:44.300 --> 01:53.420 So seven becomes 17 minus nine, which becomes eight. 01:54.020 --> 02:01.020 After the borrow, the six we had here in the tens place is now five. 02:01.580 --> 02:07.500 But now we are doing five minus seven again. 02:08.180 --> 02:12.410 It's not possible Since five is less than seven. 02:13.050 --> 02:15.250 So once again we need to borrow. 02:15.730 --> 02:20.050 But this time we don't have a digit to the left to borrow from. 02:20.650 --> 02:22.450 Now in this two digit number. 02:22.930 --> 02:30.810 So we imagine borrowing from zero here or from an implied digit which gives us ten. 02:31.570 --> 02:37.930 And what we will see here 15 minus seven equals eight. 02:38.490 --> 02:42.170 So eight here and eight here. 02:43.050 --> 02:51.770 Now after all the borrowing and the subtraction, we arrive at the digit 88. 02:52.370 --> 02:56.330 But the result needs to be examined carefully. 02:56.610 --> 03:02.770 The value 67 -79 equals eight. 03:02.890 --> 03:06.090 88 is not correct. 03:06.650 --> 03:12.170 The value 8 to 8 here is not the actual result of 67 -79. 03:12.650 --> 03:22.350 It's a number that came out of treating 67 as if it were, as if it were bigger than 79, and forcing 03:22.350 --> 03:25.510 the subtraction to work by borrowing. 03:26.070 --> 03:34.470 In doing so, we completely ignored the most important part of subtraction which number is actually 03:34.670 --> 03:35.270 larger. 03:35.910 --> 03:38.230 But what we do here. 03:38.750 --> 03:42.950 Now we imagine a zero here, one here. 03:43.510 --> 03:47.190 So if we make the six seven. 03:47.790 --> 03:48.390 So what? 03:48.430 --> 03:53.870 500 -88, we get the result. 03:54.630 --> 03:59.310 But that adds another complexity to our arithmetic here. 03:59.710 --> 04:02.310 But it is also a solution. 04:02.910 --> 04:06.870 In this case 12 is the solution. 04:07.390 --> 04:09.710 But we want to get that 12 easily. 04:09.710 --> 04:10.870 There's an easier way. 04:11.630 --> 04:16.110 In the real math we will just change these places here. 04:16.110 --> 04:21.010 So 79 -67. 04:21.650 --> 04:22.330 What now? 04:23.050 --> 04:23.490 Two. 04:23.890 --> 04:24.370 One. 04:25.730 --> 04:26.970 This is positive number. 04:27.650 --> 04:33.490 So the correct result, because we are subtracting a larger number from a smaller number. 04:34.010 --> 04:40.170 So how did we end up with 88 instead of 12. 04:40.170 --> 04:42.730 Which is actually we will just change it to minus. 04:42.730 --> 04:49.570 I will say that we perform the mechanical borrowing without checking which number was bigger. 04:50.170 --> 04:54.650 So yes, the difference in the magnitude is 12. 04:55.170 --> 05:03.050 But our manual borrowing when starting with 67 incorrectly produced 88 here, which as I said, if you 05:03.330 --> 05:11.250 just you remember once tenths and hundredths, if you end here on the left side of six, we have hundreds, 05:11.250 --> 05:11.730 right. 05:11.850 --> 05:21.650 So if you borrow one from the hundreds and 100 minus our result here equals 12 -12. 05:22.250 --> 05:26.480 Again this is just this just adds another complexity to our subtraction. 05:26.480 --> 05:31.080 So we will not use this model in our lecture. 05:31.080 --> 05:32.680 But here is an easier model. 05:32.680 --> 05:40.560 Just change the places with the higher value if you are subtracting higher from less. 05:40.840 --> 05:45.440 And there we are 12 and add just minus here. 05:45.800 --> 05:46.400 That's it. 05:47.120 --> 05:48.640 Then we will write. 05:48.640 --> 05:53.880 The correct final result is -12 here. 05:54.400 --> 05:58.280 Now always compare your numbers before subtracting. 05:58.280 --> 06:03.800 So if the top number is smaller your result will be negative and the borrowing without considering that 06:03.840 --> 06:07.040 will give you a misleading answer. 06:07.200 --> 06:18.120 Which, uh needs another another layer layer of arithmetic here, which is not needed. 06:18.760 --> 06:19.200 Right. 06:19.880 --> 06:27.160 Let's do another example here 60 or 33 -43. 06:27.880 --> 06:30.780 So 33 is less than 43. 06:30.780 --> 06:36.060 So we will change it to three -33 and ten. 06:36.740 --> 06:39.340 And we will just add minus here. 06:40.100 --> 06:41.980 And here we are. 06:42.660 --> 06:44.980 Now this illustrates an important point. 06:45.500 --> 06:48.260 The digits alone don't tell the whole story. 06:48.260 --> 06:52.300 You must always look at the original problem to understand whether your result should be positive or 06:52.300 --> 06:52.900 negative. 06:53.060 --> 07:03.060 In programming and computing, signed and unsigned number representations help enforce this distinction 07:03.100 --> 07:08.060 automatically using flags and bit encoding, which we will explore later. 07:08.700 --> 07:11.460 And this result makes sense. 07:12.060 --> 07:20.380 If you have $67 and someone asks you to give the 79, you are going to be 12 short. 07:20.940 --> 07:24.340 That's what -12 represents in this context. 07:24.340 --> 07:32.700 In programming, the system needs to know this too, which is why things like sign bits and two's complement 07:32.700 --> 07:33.260 exist. 07:33.260 --> 07:36.370 But we will cover this later in this next. 07:36.850 --> 07:42.370 Now, this example emphasizes how subtraction by borrowing focuses on the mechanical part of the operation 07:42.530 --> 07:43.690 while interpreting. 07:43.690 --> 07:49.130 The final result also requires considering which number is bigger, even though we got the negative 07:49.130 --> 07:51.450 result which is expected. 07:51.610 --> 07:55.610 67 is less than 79. 07:56.130 --> 08:03.370 What we really care about here is step by step borrowing process in subtraction. 08:04.010 --> 08:09.610 Borrowing seems like just a classroom trick when you are learning subtraction in school, but in competing, 08:09.610 --> 08:13.410 the logic is at the heart of subtraction. 08:13.410 --> 08:14.650 Circuits work. 08:14.970 --> 08:21.250 When we get into binary subtraction, we will apply this exact same algorithm, which is two digits 08:21.290 --> 08:22.530 zero and one. 08:23.170 --> 08:28.290 And once you understand borrowing, you will also understand how subtraction creates flags in CPU, 08:28.450 --> 08:29.610 just like addition does. 08:29.770 --> 08:35.770 In this next lecture, we will move on to the binary subtraction and you will see how beautifully this 08:35.770 --> 08:36.930 logic carries over. 08:36.970 --> 08:39.170 Let's continue on typhoon and I'm waiting.