Editing Modulo (mathematics)

{{Short description|Word with multiple distinct meanings}}
{{about|the general term in mathematics|the operation|Modulo|the mathematical system|Modular arithmetic}}
{{refimprove|date=December 2009}}

In mathematics, the term '''''modulo''''' ("with respect to a modulus of", the [[Latin]] [[ablative]] of ''[[wikt:modulus|modulus]]'' which itself means "a small measure") is often used to assert that two distinct mathematical objects can be regarded as equivalent—if their difference is accounted for by an additional factor. It was initially introduced into [[mathematics]] in the context of [[modular arithmetic]] by [[Carl Friedrich Gauss]] in 1801.<ref>{{Cite web|url=https://www.britannica.com/science/modular-arithmetic|title=Modular arithmetic|website=Encyclopedia Britannica|language=en|access-date=2019-11-21}}</ref> Since then, the term has gained many meanings—some exact and some imprecise (such as equating "modulo" with "except for").<ref>{{Cite web|url=http://catb.org/jargon/html/M/modulo.html|title=modulo|website=catb.org|access-date=2019-11-21}}</ref> For the most part, the term often occurs in statements of the form:

:''A'' is the same as ''B'' modulo ''C''

which is often equivalent to "''A'' is the same as ''B'' [[up to]] ''C''", and means
:''A'' and ''B'' are the same—except for differences accounted for or explained by ''C''.

==History==
''Modulo'' is a [[mathematical jargon]] that was introduced into [[mathematics]] in the book ''[[Disquisitiones Arithmeticae]]'' by [[Carl Friedrich Gauss]] in 1801.<ref>{{Cite journal|last=Bullynck|first=Maarten|date=2009-02-01|title=Modular arithmetic before C.F. Gauss: Systematizations and discussions on remainder problems in 18th-century Germany|journal=Historia Mathematica|volume=36|issue=1|pages=48–72|doi=10.1016/j.hm.2008.08.009|issn=0315-0860|doi-access=}}</ref>  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 share the same remainder when divided by ''n''. It is the [[Latin]] [[ablative]] of ''[[wikt:modulus|modulus]]'', which itself means "a small measure."<ref>{{Citation|title=modulo|url=https://www.thefreedictionary.com/modulo|work=The Free Dictionary|access-date=2019-11-21}}</ref> 

The term has gained many meanings over the years—some exact and some imprecise. The most general precise definition is simply in terms of an [[equivalence relation]] ''R'', where ''a'' is ''equivalent'' (or ''congruent)'' to ''b'' modulo ''R'' if ''aRb''.

==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''".

==See also==
{{Wiktionary|modulo}}
*[[Essentially unique]]
*[[List of mathematical jargon]]
*[[Up to]]

==References==
{{Reflist}}

==External links==
* [http://catb.org/jargon/html/M/modulo.html Modulo] in the [[Jargon File]]

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[[Category:Mathematical terminology]]