Modulo

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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.

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 a 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 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 if and only if 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.

Example

Using Gauss's definintion

13 is congruent 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.

Generally, to say:

A is the same as B modulo C

means, "more-or-less", as in:

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

Up to

The up to concept is often talked about this way, using modulo as a term alerting the hearer. The use of the term in modular arithmetic is a special case of that usage, and that is how this more general usage evolved. The operation of "modding out by C" is that of identifying with each other any two things that are the same modulo C.

Here are several ways in which modulo is used.

• "http and https are the same, modulo encryption." - means "the only difference between http and https is the addition of encryption".

• "These two characters are equal." "You mean, equal modulo case." - indicates that the first speaker's words are true only for a relaxed sense of equality. In computing, letter case is sometimes treated as significant, and sometimes not.

• "The two students performed equally well on the exam, modulo some minor computational mistakes." - means that the two students demonstrated an equal understanding of the material and its application, but at least one of them lost some points for minor computational mistakes which the other did not make.

• "This code is finished modulo testing" - means "this code is finished except for testing". Since testing is generally considered quite important, whereas in mathematics the use of modular arithmetic generally ignores the difference between modulo-equal numbers, use of a phrase like this might be deliberate understatement.

See also

• Up to