The word modulo
(Latin, with respect to a modulus of ___) is the Latin ablative
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.
- (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 a multiple of 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 of a ring or an algebra are congruent modulo an ideal if the difference between them is in the ideal.
- Two members a and b of a group are congruent modulo a normal subgroup iff ab−1 is a member of the normal subgroup. See quotient group and isomorphism theorem.
- 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 infinite set, then add a finite piece to it, and get as result the second infinite set.
- 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.
- In the mathematical community, the word modulo is also used informally, in many imprecise ways. Generally, to say "A is the same as B modulo C" means, more-or-less, "A and B are the same except for differences accounted for or explained by C". See modulo (jargon).