English

Etymology

From Latin modulo, ablative of modulus.

Pronunciation

/ˈmɒdjʊləʊ/

Preposition

modulo

1. Given a specified modulus of.

   21 and 84 are congruent to each other modulo 9, since both numbers leave the same remainder, 3, when divided by 9.

   Thus 21 modulo 9 is 3 - because when 21 is divided by 9, the remainder is 3.

2. In the context of "mathematics|colloquial": Except for differences accounted for by.

   "A is the same as B modulo C" means A is the same as B except for differences accounted for by C.

3. In the context of "extended use|colloquial": With due allowance for (a specified exception or particular detail).

   The "tail recursion modulo cons" technique allows functions that would be tail recursive but for a cons call to be transformed into a tail recursive form.

   All mammals, modulo the monotremes, give birth to live young.

Translations

• Icelandic: mátað við

Noun

1. the operation or function that returns the remainder of one number divided by another

See also

• congruent
• remainder

Italian

Noun

1. form
2. unit, module
3. In the context of "mathematics|physics|lang=it": modulus
4. In the context of "astronomy|lang=it": module

Related terms

• modulare

Verb

modulo

1. Form of First-person singular present tense, modulare

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.

• (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).