# Notation for modulo: congruence relation vs operator

If a and b are congruent modulo a number c, we might write $a \equiv b \pmod c$. When writing programs, it's often useful to compute the remainder after division, and in pseudocode we might write a = b mod c, where mod is understood to be a a binary operator which assings a = b - c * floor(b / c).

When writing an academic paper, should any distinction be drawn between these two uses? Specifically, when describing how to compute a set of parameters, we are currently specifying many using imperative (how-to-compute, e.g., modulus-operator-like) syntax (and a few with the declarative, modular-congruence-like syntax). For instance, would this:

a = b + c / d
e = f (mod g)


Be better rendered as this:

a = b + c / d
e = f mod g


In formal writing? I tend to believe the latter is more appropriate, but this might be a folk belief.

EDIT: Also note that an "=" is currently being used. If we stick with the "mod" inside parentheses, should we be using "$\equiv$"?

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There are two related concepts: the relation, and the operator. The operator is very common in Computer Science.

The relation notation corresponds to the binary relation on integers. $a\equiv b\pmod{n}$ (or $a=b\pmod{n}$; the former is more common, but there is nothing to stop you from using the latter) if and only if $n|b-a$.

In computer science, the modulo operator is common: specifically, $a\bmod n$ means "the remainder when dividing $a$ by $n$". In other words, $a\bmod n$ is the smallest positive integer $r$ such that $a\equiv r\pmod{n}$.

Formally, the two notations refer to two different things: one is a relation, one is an operator. You should certainly keep the distinction clear in a paper, in my humble opinion.

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A humble opinion that I get to enforce from time to time as a referee, of course... (-; –  Arturo Magidin Jan 23 '12 at 16:59
That was my understanding as well, but it is reassuring to hear it from somebody with a reputation of almost 100k in the Math SE. So I should go with the second of the two examples in my post? –  Patrick87 Jan 23 '12 at 17:00
@Patrick87: I would certainly prefer the second; however, it is never amiss to make explicit what may be confusing to some readers in a paper, so a quick parenthetical note the first time you use the notation that says something like "where $x\bmod y$ represents the nonnegative remainder when dividing $x$ by $y$" would not be amiss and would probably be welcomed by most readers. –  Arturo Magidin Jan 23 '12 at 17:01
OK, awesome. One last question. Would it be best just to write $e = f - g \cdot \left \lfloor f / g \right \rfloor$, to eliminate any confusion? Or would that be considered pedantic? –  Patrick87 Jan 23 '12 at 17:06
@Patrick87: Personally, I think it's easier reading to write "$e = f\bmod g$, where $x\bmod y$ is the nonnegative remainder when dividing $x$ by $y$" (and omitting the explanation after the first time), than writing "$e = f-g\lfloor f/g\rfloor$", which is valid for non-integer values and which may take longer to process, even if it says the same thing. –  Arturo Magidin Jan 23 '12 at 17:16