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.
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...". • 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: :: \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". ==See also==