1 00:00:00,000 --> 00:00:03,600 With this in place, there's only a 
couple of more things I want to talk about 2 00:00:03,600 --> 00:00:05,566 for this week with 3 00:00:05,566 --> 00:00:08,600 the integers modulo m and congruence classes and things like that. 4 00:00:08,600 --> 00:00:11,633 I want to talk about addition 
and multiplication tables. 5 00:00:11,633 --> 00:00:14,566 So here we have an addition 
table for the integers mod 4. 6 00:00:14,566 --> 00:00:18,166 So the integers mod 4 consist 
of 4 elements: 0, 1, 2, 3. 7 00:00:18,166 --> 00:00:19,333 8 00:00:19,333 --> 00:00:22,300 What is an addition table? It's just 
like your multiplication tables from 9 00:00:22,300 --> 00:00:24,133 grade school except with addition. 10 00:00:24,133 --> 00:00:26,866 It's going to give you a table for 
what happens when I add 0 and 0. 11 00:00:26,866 --> 00:00:29,166 Well I get the equivalence class of 0. 12 00:00:29,166 --> 00:00:32,533 If I add the equivalence class of 0 with 
1, I get the equivalence class of 1. 13 00:00:32,533 --> 00:00:36,000 If I add the equivalence class of 0 and 2, I get the equivalence class of 2, and so on. 14 00:00:36,000 --> 00:00:37,666 15 00:00:37,666 --> 00:00:39,500 Maybe something a little bit 
more interesting, let's look at the 16 00:00:39,500 --> 00:00:42,400 equivalence class of 2 
added to 0, that's to 2. 17 00:00:42,400 --> 00:00:44,100 2 added to 1 that's 3. 18 00:00:44,100 --> 00:00:46,900 The equivalence class of 2 added 
to the equivalence class of 2, 19 00:00:46,900 --> 00:00:48,600 that's the equivalence class of 4, 20 00:00:48,600 --> 00:00:52,000 but that's the same as the equivalence class of 0. So... 21 00:00:52,000 --> 00:00:54,600 I'm going to write 
down 0 instead of 4. 22 00:00:54,600 --> 00:00:57,566 Similarly, the equivalence class of 
2 plus the equivalence class of 3, 23 00:00:57,566 --> 00:01:00,233 that's the equivalence class of 5 24 00:01:00,233 --> 00:01:03,800 and that's going to be the 
equivalence class of 1. 25 00:01:03,800 --> 00:01:06,666 Similarly - I have a little typo here. 
So the equivalence class of 3 26 00:01:06,666 --> 00:01:08,566 plus the equivalence 
class of 3, that should be 27 00:01:08,566 --> 00:01:12,000 6 and when I reduce that 
I should get 2, not 1. 28 00:01:12,000 --> 00:01:14,200 29 00:01:14,200 --> 00:01:16,166 I'll correct that offline. 30 00:01:16,166 --> 00:01:17,266 31 00:01:17,266 --> 00:01:19,033 Okay, and now let's look 
at a multiplication table. 32 00:01:19,033 --> 00:01:21,566 So now if I'm multiplying 
two elements together, 33 00:01:21,566 --> 00:01:24,766 if I take 0 times 0 I get 0. The equivalence class of 0 34 00:01:24,766 --> 00:01:26,700 and the equivalence class 
of 1, that's going to give me 0. 35 00:01:26,700 --> 00:01:28,266 The equivalence class of 0 and 
the equivalence class of 3, 36 00:01:28,266 --> 00:01:30,700 it's going to give me 0. It's the same in all directions. 37 00:01:30,700 --> 00:01:34,200 If I take, let's say, 2, the equivalence class 
of 2 times the equivalence class of 1, 38 00:01:34,200 --> 00:01:36,333 that gives me 2. The 
equivalence class of 2 39 00:01:36,333 --> 00:01:38,766 times to the equivalence 
class of 2, well that's 4 40 00:01:38,766 --> 00:01:42,500 and the equivalence class of 4 is the same 
as the equivalence class of 0 modulo 4. 41 00:01:42,500 --> 00:01:44,866 So therefore I write 0 instead of 4. 42 00:01:44,866 --> 00:01:47,400 Similarly, the equivalence class of 2 
times the equivalence class of 3, 43 00:01:47,400 --> 00:01:49,233 it's going to give you the 
equivalence class of 6 44 00:01:49,233 --> 00:01:52,000 which is the same as the 
equivalence class of 2 mod 4. 45 00:01:52,000 --> 00:01:54,900 So hopefully you realize it's 
basically you just, you know, 46 00:01:54,900 --> 00:01:57,566 one way that I like to think 
about the integers mod 4, 47 00:01:57,566 --> 00:01:58,433 48 00:01:58,433 --> 00:02:00,133 or the integers 
mod m in general, 49 00:02:00,133 --> 00:02:03,200 is I like to think of m 
as somehow being 0. 50 00:02:03,200 --> 00:02:06,233 Right? So I can always add copies 
of m or subtract copies of m 51 00:02:06,233 --> 00:02:10,433 from my number and it's not going to change 
the equivalence class of the number, right? 52 00:02:10,433 --> 00:02:15,866 So 4 is the same as 0 which is the 
same as 8, and 16, and 12, and 400, 53 00:02:15,866 --> 00:02:17,433 they're all the same. Again why? 54 00:02:17,433 --> 00:02:20,233 Because they're all multiples of 4. 
All multiples of 4 are the same. 55 00:02:20,233 --> 00:02:21,833 56 00:02:21,833 --> 00:02:23,700 We'll see more about this 
in the upcoming week 57 00:02:23,700 --> 00:02:26,200 to get this idea down and 58 00:02:26,200 --> 00:02:28,033 help solidify it in your mind, 59 00:02:28,033 --> 00:02:31,333 but hopefully at least this 
gives you a starting point 60 00:02:31,333 --> 00:02:34,366 with the integers modulo 
m, what they mean, 61 00:02:34,366 --> 00:02:35,333 62 00:02:35,333 --> 00:02:39,533 and try to unravel what this 
notation is really doing. 63 00:02:39,533 --> 00:02:42,066 Thank you very much for listening. 
Hopefully this was informative. 64 00:02:42,066 --> 00:02:44,466 Hopefully you learned a little bit 
and will remember bits and pieces. 65 00:02:44,466 --> 00:02:48,000 Remembering everything, you 
know, the first time it's hard, right, 66 00:02:48,000 --> 00:02:51,200 but after a little bit of time with this topic, 67 00:02:51,200 --> 00:02:53,966 hopefully you'll help solidify it and 68 00:02:53,966 --> 00:02:56,466 keep it in mind. When you 
see it in future courses, 69 00:02:56,466 --> 00:03:00,133 you'll at least know oh I remember this 
issue of being well-defined, I remember 70 00:03:00,133 --> 00:03:03,166 that I had to show that oh if I do the 
same thing to two different objects 71 00:03:03,166 --> 00:03:05,733 that are actually the same 
I get the same result, 72 00:03:05,733 --> 00:03:09,033 something like this. I'm really hoping 
that this just gives you an overall view 73 00:03:09,033 --> 00:03:13,399 of what these topics are. So again thank 
you very much for listening and best of luck.