## Math Basics## More About Numbers |

## How does modular arithmetic work? |

In modular arithmetic, numbers “wrap around” when they reach a fixed quantity. This is also called the *modulus*—thus the name modular arithmetic—with the standard way of writing the form as “mod 12” or “mod 2.”

In this case, if the two numbers *b* (also called the base) and *c* (also called the remainder) are subtracted (*b* - *c*), and their difference is a number integrally divisible by *m*, or (*b* - *c*)/*m*, then *b* and *c* are said to be congruent modulo m. Mathematically, *“b* is congruent to *c* (modulo *m)”* is written as follows, with the symbol for congruence (≡):

*b ≡ c* (mod *m*)

But if *b* - *c* is not integrally divisible by *m,* then it is said, *“b* is not congruent to *c* (modulo *m*),” or

*b* ≢ *c* (mod m)

More formally, modular arithmetic includes any “non-trivial homomorphic image of the ring of integers.” We can interpret this interesting definition using a clock. The modulus would be the number 12 on the clock (arithmetic modulo 12), with an associated ring labeled C_{12} and the allowable numbers being 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. Another example is arithmetic modulo 2, with an associated ring of C_{2}, or allowable numbers of 1 and 2.