# Convenient way to write $x \bmod{n}$

I'm trying to figure out an easy way to write $x \bmod{n}$. For example, in this exercise, where I need to show that this is an homomorphism: $$f:\Bbb{Z}/12\Bbb{Z}\to\Bbb{Z}/4\Bbb{Z}:x\bmod{12}\mapsto x\bmod{4}$$

Writing or typing down all those mod's, makes it quite a mess. For example, I was thinking about something like this:

$$f(\overline{x+y}^{12})=f(x+y+12k)=\overline{x+y+12k}^4=\overline{x+y}^4+\overline{12k}^4=...$$

Just as an exmaple, my question is not about the math.
I'm quite sure that I'm not the first one with this problem, so how would you guys write this down ? Is there any "standard" for this ?

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well as a computer science student, i always use the % symbol for mod because that's what a lot of popular programming languages use. – Aden Dong Feb 22 '13 at 15:23
I usually use $\equiv$ rather than $=$, and only write $\mod n$ at the end. – Tara B Feb 22 '13 at 15:24
$x\bmod n$, x%n, $x+n\Bbb Z$, $[x]$ are all standard notations. The last one you can use to omit the modulus altogether, and can thus create ambiguity, but if you think the ambiguity is negligible then it might be worth it in my opinion. (So you would say $[x]\mapsto[x]$...) – anon Feb 22 '13 at 15:27
(Personally, I use $\equiv$ and $\bmod$ when in contexts that are not fully-fledged abstract algebra, whereas if the audience is familiar with ideals and quotient rings then I just use $=$ and often don't bother with $\bmod$.) – anon Feb 22 '13 at 15:29

Try $[x]_{12}$ and $[x]_4$ for the equivalences classes in the corresponding rings.

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Well, I guess I might as well put my comment as an answer.

I would usually use $\equiv$ rather than $=$, and only write mod $n$ at the end.

Though I agree with anon that there are situations in which it is unnecessary to write mod at all, if it's unlikely that there'd be any ambiguity.

I have also seen overlines used, as you suggest, but never with the superscript $n$, which I really don't recommend, as it looks too much like a power.

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I would never write anything attached to an overline; I've seen students do it, but it is horribly confusing with powers (indeed what you typed was the power of overlined formulae, not an index attached to the overline). As has been said you can write $x\equiv y\equiv z\pmod5$ (be sure to type \pmod5, then the spacing and parentheses are done for you), as long as there is just one modulus as in your second example. You can also just say: compute in $\Bbb Z/5\Bbb Z$, and then you can write equals signs.

When there are different moduli as in the first example, there is no really easy solution. I would write the first example as $$f: \Bbb Z/12\Bbb Z\to\Bbb Z/4\Bbb Z: x+12\Bbb Z\mapsto x+4\Bbb Z$$ although in the end even writing $\overline x\mapsto\overline x$ would be unambiguous (though possibly confusing). You could also say $f$ is the canonical projection $\Bbb Z/12\Bbb Z\to\Bbb Z/4\Bbb Z$.

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