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Now that my #VisualAlgebra class is in the books, I want to post a long "meta thread" of all 16 weekly threads, with daily summaries. Here's my entire class, including lectures, HW, & exams, in one convenient place.
And stayed tuned for some surprise announcements below!👇🧵
And stayed tuned for some surprise announcements below!👇🧵
We started #Week1 of #VisualAlgebra with a few quotes from "A Mathematician's Apology" on the beauty of mathematics, and then saw Cayley diagrams for the symmetries of the rectangle and triangle.
In #Week2 of #VisualAlgebra, we explored the Rubik's cube, more Cayley diagrams, group presentations, the impossibility of the word & halting problems, and we classified all frieze groups.
Finishing up our🧵👇 #VisualAlgebra class in #Week15 with divisibility and factorization. I'm a little short on visuals, but here are two really nice ones on what we'll be covering, made by @linguanumerate.
Henceforth, we'll assume that R is an integral domain.
1/8 Mon
Henceforth, we'll assume that R is an integral domain.
1/8 Mon
The integers have nice properties that we usually take for granted:
--multiplication commutes
--there aren't zero divisors
--every nonzero number can be factored into primes
--any 2 nonzero numbers have a unique gcd and lcm
--the Euclidean algorithm can compute these
2/8 M
--multiplication commutes
--there aren't zero divisors
--every nonzero number can be factored into primes
--any 2 nonzero numbers have a unique gcd and lcm
--the Euclidean algorithm can compute these
2/8 M
Some, but not all of these hold in general integral domains. This is what we'd like understand!
If b=ac, we say "a divides b", or "b is a multiple of a".
If a | b and b | a, they're "associates", written a~b.
HW: a~b iff they differ by a unit (i.e., a=bu).
3/8 M
If b=ac, we say "a divides b", or "b is a multiple of a".
If a | b and b | a, they're "associates", written a~b.
HW: a~b iff they differ by a unit (i.e., a=bu).
3/8 M
