Modulo (mathematics)

The word modulo (Latin, with respect to a modulus of ___) is the Latin ablative of modulus which itself means "a small measure." It was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. Ever since, however, "modulo" has gained many meanings, some exact and some imprecise. Generally, to say:

A is the same as B modulo C

means

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

Usage

• (This usage is from Gauss's book.) 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 more details, see modular arithmetic.
• In computing, given two numbers (either integer or real), a and n, a modulo n is the remainder after numerical division of a by n, under certain constraints. See modulo operation.
• 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.
• 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 as result the second subset.
• 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.
• The most general precise definition is simply in terms of an equivalence relation R. We say that a is equivalent or congruent to b modulo R if aRb.
• Generally, 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:

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

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

• "Operating modulo" is special jargon in Category theory as applied to functional programming, in the sense of mapping a functor to a category by highlighting or defining remainders.

Example

Using Gauss's definition

13 is congruent to 63 modulo 10

to mean

13 and 63 differ by a multiple of 10

However, the word modulo has acquired several related definitions with time, many of which have become integrated into popular mathematical jargon.

See also

• Up to