upper level math

 it's somewhat interesting how we get to certain places

 "xor is an addition in modulus space and can be generalized as such"

 "modern processors are able to easily handle 64-bit integers as an algebraic construct, but that construct does not completely correspond to actual integers"

 well

 my professor told me to imagine it like this

 you're working with ordinary numbers in the real plane down here on earth

 and soon you settle with a bunch of useful theorems and constructs

 but what about on mars, where we aren't dealing with real numbers anymore? what can we take from earth to an alien planet?

 it turns out, a lot

 let's take the modulus space from 1 - 2^63 to 2 ^ 63 as an example

 or take the space of continuous functions C[0, 1]

 it turns out that a functional distance metric can be defined for both of them

 so we take these results and "take them home" to earth using real-valued functions (typically by taking a sum or applying conditions to modify the numbers) into the real number plane

 but i think the most important thing that i learned is that math doesn't stop. you don't have to give up on something just because it's unfamiliar.