Modular Arithmetic: Under the Hood

Modular arithmetic visually! For aspiring mathematicians already familiar with modular arithmetic, this video describes how to formalize the concept mathematically: to define the integers modulo n, to define the operations of addition and multiplication, and check that these are well-defined.
This video is appropriate for an introduction to proof course, for undergraduate mathematics majors, or for the mathematically inclined, especially those interested in cryptography or number theory.
This is part of a series on modular arithmetic: • Modular Arithmetic Vis...
For associated materials:
proofofconcept...

[@Yahya-gb8zn]

I love you so mucchhh. This makes Visually tremendously more sense…

[@jacobrshields]

Thank you for this video! The bit at the end about proving there are no integer solutions to a polynomial equation using modulo arithmetic was very cool to see.

[@jonahunderhill]

Took me a bit to get a proof of multiplication being well defined, but here goes:
We have n|a-a' and n|b-b'. We want n|ab-a'b'.
Since n|b-b', we get n|a(b-b') since multiplying an arbitrary integer won't change whether it's divisible
Since n|a-a', we get n|(a-a')b' for the same reason
The sum of 2 things with a common factor will have that same common factor, so n|a(b-b') + (a-a')b'
n|ab-ab'+ab'-a'b', distributing the multiplication
n|ab-a'b'

[@dontreadmyprofilepicture1024]

thank u

[@user-wr4yl7tx3w]

thank you for the awesome content

[@IntegralMoon]

The proof at the end - which you built up to over the last two videos, was done a little bit hastily for my liking. Instead of using symbols, perhaps it could be useful to justify every step with an example. For me, the sep where you said n | a - A and n | b - B gives n | (a-A) + (b-B) was something I had to stop and ponder for some time before I could properly convince myself that n should divide the sum. But overall, I'm very excited to see what you come up with in the future :)

[@ProofofConceptMath]

Thanks! That's very helpful feedback. I think you're right, it gets a bit denser right there. I wish there were a way to edit and improve UA-cam videos when I get feedback.

[@IntegralMoon]

@@ProofofConceptMath There's probably no need :) Just keep making awesome videos and you will improve naturally

[@marlonbrando_0]

This series is the best explanation of modular arithmetic. Thank you for making it.