Editing Modulo (mathematics) (section)

==Usage==
===Original use===
{{main|modular arithmetic}}
Gauss originally intended to use "modulo" as follows: given the [[integer]]s ''a'', ''b'' and ''n'', the expression ''a'' ≡ ''b'' (mod ''n'') (pronounced "''a'' is congruent to ''b'' modulo ''n''") means that ''a''&nbsp;−&nbsp;''b'' is an integer multiple of ''n'', or equivalently, ''a'' and ''b'' both leave the same remainder when divided by ''n''. For example:
: 13 is congruent to 63 modulo 10

means that
: 13 − 63 is a multiple of 10 (equiv., 13 and 63 differ by a multiple of 10).

===Computing===
In [[computing]] and [[computer science]], the term can be used in several ways:
* In [[computing]], it is typically the [[modulo]] operation: given two numbers (either integer or real), ''a'' and ''n'', ''a'' modulo ''n'' is the [[remainder]] of the numerical [[Division (mathematics)|division]] of ''a'' by ''n'', under certain constraints.
* In [[category theory]] as applied to functional programming, "operating modulo" is special jargon which refers to mapping a functor to a category by highlighting or defining remainders.<ref>{{cite book |page=22 |title=Category Theory for Computing Science |last=Barr |last2=Wells |location=London |publisher=Prentice Hall |year=1996 |isbn=0-13-323809-1 }}</ref>

===Structures===
The term "modulo" can be used differently—when referring to different mathematical structures. For example:
* Two members ''a'' and ''b'' of a [[group (mathematics)|group]] are congruent modulo a [[normal subgroup]], [[if and only if]] ''ab''<sup>−1</sup> is a member of the normal subgroup (see [[quotient group]] and [[isomorphism theorem]] for more).
* Two members of a [[ring (mathematics)|ring]] or an algebra are congruent modulo an [[ideal (ring theory)|ideal]], if the difference between them is in the ideal.
** Used as a verb, the act of [[Quotient group|factoring]] out a normal subgroup (or an ideal) from a group (or ring) is often called "''modding out'' the..." or "we now ''mod out'' the...".
* Two subsets of an infinite set are '''equal modulo finite sets''' precisely if their [[symmetric difference]] is finite, that is, you can remove a finite piece from the first subset, then add a finite piece to it, and get the second subset as a result.
* A [[short exact sequence]] of maps leads to the definition of a [[Quotient space (topology)|quotient space]] as being one space modulo another; thus, for example, that a [[cohomology]] is the space of [[differential form|closed forms]] modulo exact forms.

===Modding out===
In general, '''''modding out''''' is a somewhat informal term that means declaring things equivalent that otherwise would be considered distinct. For example, suppose the sequence&nbsp;1&nbsp;4&nbsp;2&nbsp;8&nbsp;5&nbsp;7 is to be regarded as the same as the sequence&nbsp;7&nbsp;1&nbsp;4&nbsp;2&nbsp;8&nbsp;5, because each is a cyclicly-shifted version of the other:
:: <math>
\begin{array}{ccccccccccccc}
& 1 & & 4 & & 2 & & 8 & & 5 & & 7 \\
\searrow & & \searrow & & \searrow & & \searrow & & \searrow & & \searrow & & \searrow \\
& 7 & & 1 & & 4 & & 2 & & 8 & & 5
\end{array}
</math>
In that case, one is ''"modding out by cyclic shifts''".