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In mathematics, the term * modulo* ("with respect to a modulus of", the Latin ablative of

*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.

^{[1]}It was initially introduced into mathematics in the context of modular arithmetic by Carl Friedrich Gauss in 1801.

^{[2]}Since then, the term has gained many meanings—some exact and some imprecise (such as equating "modulo" with "except for").

^{[3]}For the most part, the term often occurs in statements of the form:

*A*is the same as*B*modulo*C*

which 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.^{[4]} Given the integers *a*, *b* and *n*, the expression "*a* ≡ *b* (mod *n*)", pronounced "*a* is congruent to *b* modulo *n*", means that *a* − *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 *modulus*, which itself means "a small measure."^{[5]}

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*.^{[1]} More informally, the term is found in statements of the form:

*A*is the same as*B*modulo*C*

which means

*A*and*B*are the same—except for differences accounted for or explained by*C*.

## Usage

### Original use

Gauss originally intended to use "modulo" as follows: given the integers *a*, *b* and *n*, the expression *a* ≡ *b* (mod *n*) (pronounced "*a* is congruent to *b* modulo *n*") means that *a* − *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 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.
^{[6]}

### Structures

The term "modulo" can be used differently—when referring to different mathematical structures. For example:

- Two members
*a*and*b*of a group are congruent modulo a normal subgroup, if and only if*ab*^{−1}is a member of the normal subgroup (see quotient group and isomorphism theorem for more). - Two members of a ring or an algebra are congruent modulo an ideal, if the difference between them is in the ideal.
- Used as a verb, the act of 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...".

- Used as a verb, the act of factoring out a normal subgroup (or an ideal) from a group (or ring) is often called "
- 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 as being one space modulo another; thus, for example, that a cohomology is the space of 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 1 4 2 8 5 7 is to be regarded as the same as the sequence 7 1 4 2 8 5, because each is a cyclicly-shifted version of the other:

In that case, the phrase *"modding out by cyclic shifts*" can also be used.

## See also

Look up in Wiktionary, the free dictionary.modulo |

## References

- ^
^{a}^{b}"The Definitive Glossary of Higher Mathematical Jargon — Modulo".*Math Vault*. 2019-08-01. Retrieved 2019-11-21. **^**"Modular arithmetic".*Encyclopedia Britannica*. Retrieved 2019-11-21.**^**"modulo".*catb.org*. Retrieved 2019-11-21.**^**Bullynck, Maarten (2009-02-01). "Modular arithmetic before C.F. Gauss: Systematizations and discussions on remainder problems in 18th-century Germany".*Historia Mathematica*.**36**(1): 48–72. doi:10.1016/j.hm.2008.08.009. ISSN 0315-0860.**^**"modulo",*The Free Dictionary*, retrieved 2019-11-21**^**Barr; Wells (1996).*Category Theory for Computing Science*. London: Prentice Hall. p. 22. ISBN 0-13-323809-1.

## External links

- Modulo in the Jargon File