WEBVTT 00:00.480 --> 00:05.160 Before we dive into the technical side, let's address something upfront here. 00:05.200 --> 00:12.880 I know that most people, myself included, don't really enjoy math, especially when it gets abstract 00:12.880 --> 00:14.720 or overly technical. 00:15.040 --> 00:21.400 However, the math we will use in today's lecture is not boring math. 00:21.440 --> 00:29.760 To be honest, it's practical, visual, and absolutely essential for understanding how computers work 00:29.960 --> 00:32.000 at a lowest level. 00:32.040 --> 00:36.120 Now, you do not need to be a math expert to follow along. 00:36.280 --> 00:41.840 What you will need is a bit of patience and willingness to look at the numbers in a new way. 00:42.040 --> 00:49.360 Once you start seeing how simple operations like addition subtraction work inside the computer, it 00:49.360 --> 00:52.160 becomes very logical and surprisingly interesting. 00:52.200 --> 00:58.880 Now this lecture will set the foundation for everything we will do with the low level computing, which 00:58.880 --> 01:05.200 is the number one requirement of the low level computer is math. 01:05.600 --> 01:11.950 Right now, especially in system programming, uh, digital forensics and embedded systems, you will 01:11.950 --> 01:15.710 always need the low level computing math. 01:15.750 --> 01:20.990 Now, in the digital world, everything must fit into a fixed number of bits. 01:21.150 --> 01:21.350 Now. 01:21.350 --> 01:21.790 That's right. 01:21.830 --> 01:29.230 No matter how big or a smaller number is, it must be squeezed into a limited space. 01:29.350 --> 01:34.350 Now, previously, as you remember, we discussed how data types are stored in a memory. 01:34.750 --> 01:46.510 Um, a variable declared as integer, for example, might get 32 bits here, 32 time bits, which is 01:46.510 --> 01:50.830 either 1 or 0 right now. 01:51.270 --> 02:02.990 Um, and the character only gets eight bits eight times bit, which is it is either 1 or 0. 02:03.350 --> 02:09.590 The limit is absolute, so there's no overflow zone or extra margin. 02:09.830 --> 02:13.660 Now this limitation creates complications for arithmetic operations. 02:13.700 --> 02:19.940 Now, if you do a calculation that goes beyond the space available here, something has to give. 02:19.980 --> 02:20.380 Right. 02:20.420 --> 02:22.780 And what gives this accuracy? 02:22.820 --> 02:29.300 Now you will either lose data or wrap around to a different result entirely. 02:29.580 --> 02:34.460 And in our daily lives we do math on paper or in our heads. 02:34.500 --> 02:40.100 We assume that we can keep writing more digits and so on and so forth. 02:40.100 --> 02:43.500 But in computing computing, we can't. 02:43.740 --> 02:48.500 Now the CPU has to make decisions based on a set of number of bits. 02:49.140 --> 02:58.100 And to manage these constraints, the CPU include single bit bit condition flags in the architecture. 02:58.140 --> 03:05.860 Now these are part of registers called the status register or flags register or flags. 03:06.020 --> 03:14.380 And the two flags we will talk about here is carry flag and overflow flag. 03:14.540 --> 03:15.580 Carry flag. 03:15.620 --> 03:19.210 Now this is used in unsigned arithmetic. 03:19.370 --> 03:24.410 Now it tells us whether a carry out of the most significant bit has occurred. 03:24.570 --> 03:31.770 Now, in simple terms, it means the result was too big to fit and in the overflow flag. 03:32.410 --> 03:34.730 This is used in a signed arithmetic. 03:34.930 --> 03:42.330 It tells us whether the sign bit is incorrect due to the operation, indicating that an overflow has 03:42.330 --> 03:45.610 occurred in a signed context. 03:45.610 --> 03:51.770 So this is an unsigned and this is signed arithmetic. 03:52.010 --> 03:55.530 Now we will go deeper into these flags in upcoming lectures. 03:55.530 --> 04:01.610 But for now just remember they tell us when something went wrong in a binary. 04:02.050 --> 04:03.130 Now welcome back. 04:03.890 --> 04:07.170 In this lecture we are going to talk about how computers do arithmetic. 04:07.490 --> 04:12.850 Now we will start with what's familiar which is a decimal numbers. 04:13.010 --> 04:18.610 Now before we jump into binary it is extremely helpful to understand how we manually perform addition 04:18.930 --> 04:22.210 and subtraction with decimal numbers. 04:22.450 --> 04:26.760 Now, because the logic is the same just with different digits and base values. 04:27.200 --> 04:35.440 And since this is a visual visual lecture, I will go with you with a step by step drawings as we go. 04:35.600 --> 04:38.920 Now in the decimal addition you already know. 04:39.080 --> 04:40.720 Let's start with a simple example. 04:41.000 --> 04:47.480 Now let's say we are adding 67 and 79 addition. 04:47.680 --> 04:56.360 Now we begin by adding the digits in the ones place right seven and nine which is 16. 04:56.680 --> 04:58.720 So that's more than ten. 04:58.720 --> 05:06.560 So we write down six and carry one to the tens place here. 05:06.840 --> 05:13.720 And now we have six from seven nine and plus one carry. 05:14.000 --> 05:18.640 What we have here is seven plus seven which is 14. 05:18.680 --> 05:24.840 Since we don't have any numbers here or here we will just end the subtraction. 05:24.840 --> 05:26.640 And that's our summary. 05:26.840 --> 05:32.430 And yeah that's basically the summary of this decimal addition. 05:32.710 --> 05:36.710 Now let's break down the algorithm that makes this work right now. 05:36.830 --> 05:40.630 Algorithm for for decimal addition. 05:40.870 --> 05:45.390 Now let's suppose x or let's use a different color. 05:45.390 --> 05:52.030 Here x and y are two n digit decimal numbers. 05:52.030 --> 05:55.750 Now we will add them digit by digit from right to left. 05:56.270 --> 06:00.070 Now let's call x I. 06:00.550 --> 06:10.470 This is the digit from the number x at position I and y I the digit from the number y at position I, 06:10.910 --> 06:14.630 and we have the carry and sum. 06:14.910 --> 06:16.990 And why does this work here? 06:17.270 --> 06:19.430 Because we are using positional notation. 06:19.630 --> 06:25.950 Every digit to the left here is worth ten times more. 06:26.310 --> 06:27.630 Here this 10th. 06:27.630 --> 06:29.470 And this is 100th. 06:29.670 --> 06:31.670 So and this goes on right. 06:31.910 --> 06:36.670 And that is how the addition algorithm works.