With this in place, there's only a couple of more things I want to talk about
for this week with
the integers modulo m and congruence classes and things like that.
I want to talk about addition and multiplication tables.
So here we have an addition table for the integers mod 4.
So the integers mod 4 consist of 4 elements: 0, 1, 2, 3.
What is an addition table? It's just like your multiplication tables from
grade school except with addition.
It's going to give you a table for what happens when I add 0 and 0.
Well I get the equivalence class of 0.
If I add the equivalence class of 0 with 1, I get the equivalence class of 1.
If I add the equivalence class of 0 and 2, I get the equivalence class of 2, and so on.
Maybe something a little bit more interesting, let's look at the
equivalence class of 2 added to 0, that's to 2.
2 added to 1 that's 3.
The equivalence class of 2 added to the equivalence class of 2,
that's the equivalence class of 4,
but that's the same as the equivalence class of 0. So...
I'm going to write down 0 instead of 4.
Similarly, the equivalence class of 2 plus the equivalence class of 3,
that's the equivalence class of 5
and that's going to be the equivalence class of 1.
Similarly - I have a little typo here. So the equivalence class of 3
plus the equivalence class of 3, that should be
6 and when I reduce that I should get 2, not 1.
I'll correct that offline.
Okay, and now let's look at a multiplication table.
So now if I'm multiplying two elements together,
if I take 0 times 0 I get 0. The equivalence class of 0
and the equivalence class of 1, that's going to give me 0.
The equivalence class of 0 and the equivalence class of 3,
it's going to give me 0. It's the same in all directions.
If I take, let's say, 2, the equivalence class of 2 times the equivalence class of 1,
that gives me 2. The equivalence class of 2
times to the equivalence class of 2, well that's 4
and the equivalence class of 4 is the same as the equivalence class of 0 modulo 4.
So therefore I write 0 instead of 4.
Similarly, the equivalence class of 2 times the equivalence class of 3,
it's going to give you the equivalence class of 6
which is the same as the equivalence class of 2 mod 4.
So hopefully you realize it's basically you just, you know,
one way that I like to think about the integers mod 4,
or the integers mod m in general,
is I like to think of m as somehow being 0.
Right? So I can always add copies of m or subtract copies of m
from my number and it's not going to change the equivalence class of the number, right?
So 4 is the same as 0 which is the same as 8, and 16, and 12, and 400,
they're all the same. Again why?
Because they're all multiples of 4. All multiples of 4 are the same.
We'll see more about this in the upcoming week
to get this idea down and
help solidify it in your mind,
but hopefully at least this gives you a starting point
with the integers modulo m, what they mean,
and try to unravel what this notation is really doing.
Thank you very much for listening. Hopefully this was informative.
Hopefully you learned a little bit and will remember bits and pieces.
Remembering everything, you know, the first time it's hard, right,
but after a little bit of time with this topic,
hopefully you'll help solidify it and
keep it in mind. When you see it in future courses,
you'll at least know oh I remember this issue of being well-defined, I remember
that I had to show that oh if I do the same thing to two different objects
that are actually the same I get the same result,
something like this. I'm really hoping that this just gives you an overall view
of what these topics are. So again thank you very much for listening and best of luck.