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 Integer s ''a'', ''b'' and ''n'', the expression ''a'' ≡ ''b'' ( ''n'') (pronounced "''a'' is congruent to ''b'' '''modulo''' ''n''") means that ''a'' and ''b'' have the same Remainder when divided by ''n'', or equivalently, that ''a'' − ''b'' is a multiple of ''n''. For more details, see Modular Arithmetic .
- In Computing , given two integers, ''a'' and ''n'', ''a'' ''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 an Ideal if the difference between them is in the ideal.
- Two subsets of an infinite set are precisely if their Symmetric Difference is finite, that is, you can take 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'' ''R'' if ''aRb''.
- In the mathematical community, the word 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) .
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