In mathematics, the term modulo is often used to assert that two distinct mathematical objects can be regarded as equivalent—if their difference is accounted for by an additional factor. It was initially introduced into mathematics in the context of modular arithmetic by Carl Friedrich Gauss in 1801. Since then, the term has gained many meanings—some exact and some imprecise. For the most part, the term often occurs in statements of the form: which means
History
Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by CarlFriedrich Gauss in 1801. Given the integers a, b and n, the expression a ≡ b means that a − b is an integer multiple of n, or equivalently, a and b both share the same remainder when divided byn. It is the Latin ablative of ', which itself means "a small measure." The term has gained many meanings over the years—some exact and some imprecise. The most general precise definition is simply in terms of an equivalence relationR, where a is equivalent to bmodulo' R if aRb. More informally, the term is found in statements of the form: which means
Usage
Original use
Gauss originally intended to use "modulo" as follows: given the integers a, b and n, the expression a ≡ b means that a − b is an integer multiple of n, or equivalently, a and b both leave the same remainder when divided by n. For example: means that
In computing, it is typically the modulo operation: given two numbers, a and n, amodulon is the remainder of the numerical division of a by n, under certain constraints.
In category theory as applied to functional programming, "operating modulo" is special jargon which refers to mapping a functor to a category by highlighting or defining remainders.
Structures
The term "modulo" can be used differently—when referring to different mathematical structures. For example:
* Used as a verb, the act of factoring out a normal subgroup from a group is often called "modding out the..." or "we now mod out the...".
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 subset, then add a finite piece to it, and get the second subset as a result.
In general, modding out is a somewhat informal term that means declaring things equivalent that otherwise would be considered distinct. For example, suppose the sequence 1 4 2 8 5 7 is to be regarded as the same as the sequence 7 1 4 2 8 5, because each is a cyclicly-shifted version of the other: In that case, the phrase "modding out by cyclic shifts" can also be used.
