Modulo (mathematics)

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. It was initially introduced into mathematics in the context of modular arithmetic by Carl Friedrich Gauss in 1801.^[1] Since then, the term has gained many meanings--some exact and some imprecise (such as equating "modulo" with "except for").^[2] 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.

Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801.^[3] 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."^[4]

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 (or congruence) relation R, where a is equivalent (or congruent) to b modulo R if aRb.

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).

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.^[5]

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...".
• 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.

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:

${\displaystyle {\begin{array}{ccccccccccccc}&1&&4&&2&&8&&5&&7\\\searrow &&\searrow &&\searrow &&\searrow &&\searrow &&\searrow &&\searrow \\&7&&1&&4&&2&&8&&5\end{array}}}$

In that case, one is "modding out by cyclic shifts".

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