TITLE: Why is modular arithmetic defined as a "similarity" and not an operation?
QUESTION [8 upvotes]: I was asking in the mathematics chatroom about the difference between $\equiv$ and $=$, and I fell into an interesting vein of discussion that I'd like to ask about. Modular arithmetic is apparently not defined as an operation, but a "similarity" (for lack of a better term), which is why $\equiv$ and not $=$ is used. However, I was also told that you can define an operation: 

like $a\mod m$ is the smallest positive number equivalent to $a\mod m$,
  - Akiva Weinberger
but then you'd write it like $a~{\rm mod}~m=b~{\rm mod}~m$ instead of $a\equiv b\mod m$. - Akiva Weinberger

I was also told that

The more useful point of view is that $a$ and $b$ fall into the same "category" (more formally, equivalence class), than that they have the same image under that operation. - Akiva Weinberger

but why is it more useful?
My one thought here is that it might not be useful because when if say that $a\mod m = b \mod m$, that doesn't necessarily mean that $a=b$; however, with other operations, such as addition, if you have $a+m=b+m$ you know that $a=b$ (I don't know the name of this "property", sorry). So maybe an operation must fulfill certain "properties" and modular arithmetic doesn't fulfill this one? I'm not sure though.

REPLY [7 votes]: Operational use of mod (remainder) is often more convenient in (mechanical) computational contexts, whereas the relational use (congruences)  yields more flexibility in theoretical contexts. 
The difference amounts to whether it is more convenient to work with general equivalence classes, or canonical / normal representatives ("reps") thereof. For example,  it would be quite cumbersome to state the laws of fraction arithmetic if one required all fractions to be in normal (reduced) form, i.e. in lowest terms. Instead, it proves more convenient to have the flexibility to work with arbitrary equivalent fractions. For example, this allows us to state the fraction addition rule in very simple form by first choosing convenient reps having a common denominator.
Analogously, in modular arithmetic the remainder $\,a\ {\rm mod}\ m\,$ may not be the most convenient choice of representative of the equivalence class $\,[a]_m  =\, a + m\,\Bbb Z.\,$ For example, the formula for casting out elevens exploits that  $\ {\rm mod}\ 11\!:\ 10\equiv -1\,\Rightarrow\,10^n\equiv (-1)^n\equiv \pm1 .\,$ which involves choosing a rep of least magnitude $\,\color{#c00}{\bf -1}\,$ vs. $\,\color{#0a0}{10}\in [10]_{11}\! = \{\ldots,\, -23,-12,\color{#c00}{\bf -1},\color{#0a0}{10},21,\,\ldots\}.\,$ Hence, analogous to fraction addition, we chose a rep $\,-1\,$ whose powers are easier to compute. Using least magnitude reps often simplifies other computations too, e.g. the Euclidean algorithm.
Thus the use of congruence classes (vs. canonical reps) provides much greater flexibility, which often yields great simplifications.