WEBVTT 00:00.440 --> 00:01.200 Hello everyone. 00:01.240 --> 00:05.480 I'm Tiffany and in this lecture we are about to explore one of the most essential operations that a 00:05.480 --> 00:09.400 CPU performs, which is the addition of binary values. 00:09.400 --> 00:13.080 And specifically we want to add two n bit binary numbers. 00:13.080 --> 00:16.480 And to begin, let's quickly revisit how binary numbers are structured. 00:16.880 --> 00:17.160 Now. 00:17.160 --> 00:25.560 As you might recall from earlier lessons, the bits in a binary number are numbered from right to left, 00:26.400 --> 00:28.520 starting with the bit position zero. 00:28.840 --> 00:39.880 So this means the rightmost bit is the least significant bit, and as you move left, the bit becomes 00:39.920 --> 00:42.040 more significant. 00:42.880 --> 00:49.320 And our approach starts with understanding how to add two individual bits at the same bit position, 00:49.320 --> 00:53.240 which we will then expand into adding multi bit binary numbers. 00:53.280 --> 00:59.840 Now we will begin by examining a simple component called the half adder, and then build up to a more 00:59.840 --> 01:03.280 complete solution using a full adder. 01:04.200 --> 01:08.440 So now we will create our truth table. 01:09.480 --> 01:15.000 Now in digital circuits we can perform binary addition using various building blocks. 01:15.000 --> 01:18.560 And one of the most basic is the half adder which you will learn here. 01:18.680 --> 01:26.200 So a half adder adds two single bit binary values, typically taken from the same position in two different 01:26.200 --> 01:27.240 binary numbers. 01:27.440 --> 01:33.960 So here we have, let's say x I y I. 01:35.200 --> 01:42.200 We will also have carry I plus one and we will also have some. 01:43.160 --> 01:47.680 You have learned what does cavalry does from previous lectures. 01:48.320 --> 01:57.360 If you have forgot it I strongly suggest go watch that lecture and if you have forgotten it again I 01:57.400 --> 02:00.070 strongly suggest go to a doctor. 02:00.950 --> 02:01.710 And, uh. 02:01.750 --> 02:02.190 Yeah. 02:03.470 --> 02:10.190 So here we will build our blocks here in the half order zero zero. 02:10.710 --> 02:11.790 Again, carry is zero. 02:11.830 --> 02:13.510 The sum is going to be zero. 02:14.550 --> 02:19.390 So and one of the most basic here this this is one of the most basic errors here. 02:19.390 --> 02:27.030 The half adder adds two single byte binary values and which is taken from the their position in the 02:27.030 --> 02:27.870 binary numbers. 02:28.310 --> 02:31.270 And this sum sum I. 02:31.310 --> 02:31.830 Here. 02:32.950 --> 02:37.350 This is the result of the addition at this bit position. 02:37.350 --> 02:40.110 So this is result. 02:41.030 --> 02:49.070 And we have the carry I plus one which is the carry out bit that will be forwarded to the next higher 02:49.070 --> 02:52.230 position or higher bit position. 02:53.190 --> 02:57.310 Now here Zero one. 02:57.510 --> 02:58.390 What carry? 02:58.430 --> 03:01.750 We will have zero and the sum is going to be one again. 03:01.990 --> 03:03.310 One zero. 03:03.350 --> 03:04.790 We have learned that in previous lectures. 03:04.790 --> 03:08.670 If you remember, if you don't go to a doctor. 03:09.230 --> 03:12.710 So one one carry is going to be one of course. 03:12.710 --> 03:14.750 So it will move to one higher bit. 03:14.830 --> 03:16.910 The one higher bit is going to be one zero. 03:16.910 --> 03:18.830 And here this is the zero here. 03:18.830 --> 03:21.670 So this is the sum is going to be zero. 03:22.750 --> 03:23.430 And yeah. 03:23.830 --> 03:34.390 And from this truth table you can see that a sum is the result of an XOR operation between x I and x 03:35.390 --> 03:35.550 I. 03:35.990 --> 03:40.670 And the carry is the result of an operation between x I and y I. 03:40.950 --> 03:45.190 Let's see 000. 03:45.190 --> 03:46.430 Now here one one. 03:46.430 --> 03:47.550 And this is one again. 03:47.550 --> 03:49.990 So this is basically and operation. 03:49.990 --> 03:50.430 Right. 03:51.710 --> 04:00.390 And here we will also write our design, our logic table for this truth table. 04:00.430 --> 04:02.950 Here it is so hard to say. 04:02.990 --> 04:03.910 Truth table. 04:03.910 --> 04:04.710 I don't know why. 04:05.150 --> 04:05.430 So. 04:05.430 --> 04:05.830 Yeah. 04:06.150 --> 04:11.750 Uh, so we will have this again one. 04:12.750 --> 04:16.630 This is basically some will give us some here. 04:17.590 --> 04:19.750 It will connect to here again. 04:20.190 --> 04:21.830 We will have another one. 04:21.830 --> 04:23.350 It will also connect to here. 04:24.790 --> 04:28.470 And at the bottom we will have a carry. 04:29.430 --> 04:30.030 So what. 04:30.430 --> 04:39.230 This is basically some I and carry I plus one. 04:40.190 --> 04:44.070 This is x I and y I. 04:45.070 --> 04:52.110 So while uh this circuit is simple and functional it has key limitations. 04:52.110 --> 04:59.340 It only handles two input bits only x I and y I, so it does not take into account the carry bit from 04:59.380 --> 05:02.660 our previous lower bit position. 05:02.660 --> 05:10.940 You remember if it was one and one we will as a carry we will get one, but the sum is going to be zero, 05:11.380 --> 05:17.460 so it does not take into account the carry bit from a previous which is lower bit position. 05:17.460 --> 05:21.700 And this is where full adder comes into play. 05:22.740 --> 05:25.180 So a full adder what is the full adder? 05:25.620 --> 05:34.820 A full adder extends the concept of the half adder by introducing a third input, the carry bit from 05:34.820 --> 05:39.060 the previous stage and a full adder. 05:41.260 --> 05:46.780 Is a full adder is a digital circuit that extends the functionality of the half adder, so it takes 05:46.780 --> 05:53.780 into account not only the two bits being added to the circuit in a current position, but also carry 05:53.820 --> 05:55.620 from the previous edition. 05:55.620 --> 05:56.860 So to clarify. 05:56.900 --> 06:00.260 Suppose you are adding two multi-bit binary numbers. 06:01.980 --> 06:11.660 So when you add the bits in position zero, a carry out might be produced like this one, right? 06:11.860 --> 06:13.700 We have the carry out one here. 06:14.380 --> 06:18.620 So this carry must then be included in the addition at the position first. 06:18.940 --> 06:21.060 In this case the position first is this one. 06:21.860 --> 06:23.300 And so on so forth. 06:23.660 --> 06:25.780 It will go on like goes on like that. 06:25.820 --> 06:30.540 So basically the full adder makes this possible. 06:31.260 --> 06:34.100 So the full adder has three inputs. 06:34.100 --> 06:38.780 Let's actually instead of x and y because I hate saying x and y. 06:39.220 --> 06:44.100 Let's just say a and b I. 06:45.260 --> 06:50.260 We will have a carry at the right. 06:50.300 --> 06:54.930 The left carry I on the right. 06:54.930 --> 07:03.530 We will have carry I plus one and some I. 07:04.450 --> 07:04.650 Yeah. 07:04.690 --> 07:05.050 Basically. 07:05.090 --> 07:05.570 Basically. 07:05.570 --> 07:06.210 That's it. 07:07.130 --> 07:10.130 Now let's draw the line. 07:11.050 --> 07:14.450 1234. 07:15.610 --> 07:17.730 So it basically has two outputs. 07:18.290 --> 07:26.090 The sum I which is the result of adding carry I I and b I. 07:27.210 --> 07:38.130 And we also have the carry I plus one which is the carry out bit which will be used in the next higher 07:38.170 --> 07:39.530 order addition. 07:40.370 --> 07:43.010 Now let's actually calculate that right. 07:43.130 --> 07:49.450 So if we have a and I b I we will here have again zero. 07:49.490 --> 07:52.050 Sum is zero and the carry is zero. 07:53.130 --> 07:58.210 If you add zero one it will have zero. 07:58.530 --> 08:00.490 Sum is going to be one. 08:01.410 --> 08:02.730 Because of this one here. 08:02.730 --> 08:05.490 And then the carry I we will have zero again. 08:06.210 --> 08:10.730 If we add one and zero the result is will be pretty same. 08:11.810 --> 08:17.690 And if we add one and one we will have carry one. 08:18.050 --> 08:20.010 We will have some one. 08:20.490 --> 08:23.890 And yeah we have carry plus r. 08:23.930 --> 08:26.330 So we will add one bit to the left. 08:27.290 --> 08:31.810 And yeah in the carry we will have uh zero. 08:32.850 --> 08:40.130 And if in the carry which is the second, if in the carry if we have one and in X we have zero zero 08:40.530 --> 08:41.570 which is the addition. 08:41.570 --> 08:46.370 After this again in the carry we will have one. 08:46.530 --> 08:51.010 So what we will have here is what 011. 08:52.050 --> 08:52.890 So some some. 08:52.930 --> 08:53.290 Yeah. 08:53.330 --> 08:53.810 Sorry. 08:54.370 --> 08:55.530 This should be zero. 08:55.770 --> 08:57.250 And here we will have one. 08:58.170 --> 09:09.810 And if we add like zero one and we have the carry my one here again here what we will have, it's one. 09:10.730 --> 09:16.690 And again here we will basically have the one. 09:16.730 --> 09:18.450 We will have zero of the sum. 09:19.450 --> 09:24.050 And here if we have we have 10001. 09:24.090 --> 09:29.850 Now we one zero carry is going to be one because of this. 09:30.570 --> 09:39.530 And here we will again have 00110 and 1111. 09:40.210 --> 09:47.010 Again here in the carry will have one in the carry I we will have one. 09:47.480 --> 09:50.240 and in the sum we will have one. 09:50.520 --> 09:54.680 So this table shows all possible input combinations. 09:54.680 --> 10:02.200 And as you can see, when 1 or 3 of the inputs are one, the sum is one. 10:02.920 --> 10:10.440 When exactly two of the inputs are one a carry here is generated here. 10:11.640 --> 10:12.120 So here. 10:13.400 --> 10:16.360 Now from this we derive a logical equation. 10:17.000 --> 10:25.480 So sum equation is basically sum I equals carry I xor. 10:26.400 --> 10:28.160 You will learn XOR later. 10:29.120 --> 10:34.920 X a and xor b I. 10:35.960 --> 10:42.280 So this equation here basically shows the sum of sum is one. 10:42.280 --> 10:46.520 If an odd number of the three inputs are one.