by 1MileCrash
Tags: arithmetic, lunatic, modular, rantings
 Mentor P: 18,058 Lunatic Rantings about Modular Arithmetic. Modulo 0 and modulo 1 are indeed two opposite concepts. But I think the right way of seeing this is with the notion of ideals. An ideal of $\mathbb{Z}$ is a nonempty subset $I\subseteq \mathbb{Z}$ such that 1) $0\in I$. 2) If $x,y\in I$, then $x+y\in I$. 3) If $x\in I$ and $n\in \mathbb{Z}$, then $nx\in I$. As can easily be verified, both $\{0\}$ and $\mathbb{Z}$ are ideals. Furthermore, for any $n\in \mathbb{Z}$, we have that $n\mathbb{Z}=\{nx~\vert~x\in \mathbb{Z}\}$ is an ideal. Now, with an ideal I, we can associate a modular arithmetic. Indeed, we say that x=y (mod I) if and only if $x-y\in I$. Modulo $\{0\}$ is then just equality. Modulo $\mathbb{Z}$ is something that is always true. And we can see that this is indeed two opposite situations: $\{0\}$ is the smallest possible ideal, while $\mathbb{Z}$ is the largest. Considering ideals in $\mathbb{Z}$ is not terribly exciting. Indeed, one can prove that the only ideals are the ones of the form $n\mathbb{Z}$. So all the modular arithmetics are just the ones you already know. But ideals are very useful in structures other than $\mathbb{Z}$, where modular arithmetic is maybe more restrictive!!