Modulo

Some of the pages that link to this one should link to modular arithmetic. Please help fix those. This article treats more general use of this term by mathematicians than its use in modular arithmetic.

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The original use of modulo in mathematics: modular arithmetic

The word modulo was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. That book introduced a number of new ideas, among them modular arithmetic.

Far more general use in modern mathematics

But the word modulo is now used much more generally in mathematics.

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.

That is, 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.

Examples of precise definitions

• 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.

A long list of examples and the technical details need to be added here. The phrase "to mod out" should be explained.