In mathematics, the term modulo (“with respect to a modulus of”, the Latin ablative of modulus which itself means “a small measure”) 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 (such as equating “modulo” with “except for”). For the most part, the term often occurs in statements of the form :

A is the same as B modulo C

which means

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

1 History
2 Usage
- 2.1 Original use
- 2.2 Computing
- 2.3 Structures
- 2.4 Modding out
3 See also
4 References
5 External links

1 History

Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. Given the integers a , b , n a, b, n a,b,n, the expression “ a ≡ b ( m o d n ) a ≡ b\ (mod\ n) a≡b (mod n)”, pronounced “ a a a is congruent to b b b modulo n n n”, means that a − b a - b a−b is an integer multiple of n n n, or equivalently, a , b a, b a,b both share the same remainder when divided by n n n. it is the Latin ablative of modulus, which itself means “a small measure”.

The term has gained many meaning over the years–some exact and some imprecise. The most general precise definition is simply in terms of an equivalence relation R R R, where a a a is equivalent (or congruent) to b b b modulo R R R if a R b aRb aRb. More informally, the term is found in statements of the form:

A is the same as B modulo C

which means

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

2 Usage

2.1 Original use

Gauss originally intended to use “modulo” as follows: given the integers a , b , n a, b, n a,b,n, the expression a ≡ b ( m o d n ) a ≡ b\ (mod\ n) a≡b (mod n) (pronounced " a a a is congruent to b b b modulo n n n) means that a − b a - b a−b is an integer multiple of n n n, or equivalently, a , b a, b a,b leave the same remainder when divided by n n n. For example:

13 is congruent to 63 modulo 10

means that

13 - 63 is a multiple of 10 (equiv., 13 and 63 differ by a multiple of 10).

2.2 Computing

In computing and computer science, the term can be used in several ways:

In computing, it is typically the modulo operation: given two numbers (either integer or real), a , n a, n a,n, a a a modulo n n n is the remainder of the numerical division of a a a by n n 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.

2.3 Structures

2.4 Modding out

3 See also

4 References

5 External links

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Modulo (mathematics)

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