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So we're doing modular arithmetic in my proof class. I have a weird cycle when learning something new in pure math, I think "wow, this is just exceedingly indepth version of something learned by gradeschool children." Then I find something (on my own, not in the text book, just thinking about it. It's usually a case the book may consider too trivial to be worth mentioning, but those are ALWAYS the things I find the most interesting because it is where things become beautiful to me) that is majorly cool about the topic, realize how incredible it is that these "gradeschool children" don't really know the depth of what there doing, and then I become insanely obsessed.

Modular arithmetic was a shining example. My class is small, taught by a young, carefree professor who is very bright, and we generally are very open and allowed to talk whenever, its always about math because everyone in the class loves math. When he first gave the definition of congruence mod n, I remarked "So we're doing 4th grade clock problems where the clock has n hours." "YES!" he responded.

Reading up in the text book, they talked about the "uninteresting" case of mod 1. Any two numbers are congruent mod 1 because there difference is of course a multiple of 1. I thought that was interesting, not uninteresting. They went on to say that generally, there are n equivalence classes for congruence mod n. So in mod 1's case, that one equivalence class is the set of all, because they are all congruent. Cool!

So I began to wonder about the "opposite extreme" because there always seems to be one. I realized the following:

The only multiple of 0 is 0.

Then the difference between two numbers can only be a multiple of 0 if it is in fact 0.

The only way that the difference between two numbers is 0 is if they are equal.

So then I realized, that congruence mod 0 is the

But it gets cooler. The equivalence class thing. Congruence mod 1 has 1, the set of all. Congruence mod 0, since numbers are only congruent to themselves, there are

So, congruence mod 0 and congruence mod 1 are exactly opposite as far as I can see. Which means something to me, because I see a lot of reasons to think of 0 and 1 as opposites rather than 0 and "infinity" or "a lot" being opposites.

Yeah, I'm a bit bonkers when it comes to these things, whatever. Hope you enjoyed the rant.

Modular arithmetic was a shining example. My class is small, taught by a young, carefree professor who is very bright, and we generally are very open and allowed to talk whenever, its always about math because everyone in the class loves math. When he first gave the definition of congruence mod n, I remarked "So we're doing 4th grade clock problems where the clock has n hours." "YES!" he responded.

Reading up in the text book, they talked about the "uninteresting" case of mod 1. Any two numbers are congruent mod 1 because there difference is of course a multiple of 1. I thought that was interesting, not uninteresting. They went on to say that generally, there are n equivalence classes for congruence mod n. So in mod 1's case, that one equivalence class is the set of all, because they are all congruent. Cool!

So I began to wonder about the "opposite extreme" because there always seems to be one. I realized the following:

The only multiple of 0 is 0.

Then the difference between two numbers can only be a multiple of 0 if it is in fact 0.

The only way that the difference between two numbers is 0 is if they are equal.

So then I realized, that congruence mod 0 is the

*same*as equality. I found that extremely badass. Do I mean to say that normal arithmetic is modular arithmetic with the modulus as 0? Yes, I do, that's how I view it now. Maybe that's wrong.But it gets cooler. The equivalence class thing. Congruence mod 1 has 1, the set of all. Congruence mod 0, since numbers are only congruent to themselves, there are

*infinite*equivalence classes, namely, singleton sets of all.So, congruence mod 0 and congruence mod 1 are exactly opposite as far as I can see. Which means something to me, because I see a lot of reasons to think of 0 and 1 as opposites rather than 0 and "infinity" or "a lot" being opposites.

Yeah, I'm a bit bonkers when it comes to these things, whatever. Hope you enjoyed the rant.

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