Modular arithmetic is the arithmetic of congruences, sometimes known informally as "clock arithmetic." In modular arithmetic, numbers "wrap around" upon reaching a given fixed quantity, which is known as the modulus (which would be 12 in the case of hours on a clock, or 60 in the case of minutes or seconds on a clock).
Formally, modular arithmetic is the arithmetic of any nontrivial homomorphic image of the ring of integers. For any such homomorphic image 
of 
, there is an integer 
such that 
is isomorphic to the ring 
of integers modulo 
. The addition in the ring 
is determined from addition in 
by computing the remainder, upon division by 
, of the sum 
of two integers 
and 
. Similarly, for multiplication in the ring 
, one multiplies two integers 
and 
, and computes the remainder upon division of 
by 
.
For each positive integer 
, the ring 
has 
elements, namely the equivalence classes of each of the nonnegative
integers less than 
,
under the equivalence relation 
that is defined according to the rule 
iff 
divides 
. It is natural and common to denote the equivalence class
![[a]](/images/equations/ModularArithmetic/Inline26.svg)
(under the equivalence relation 
) of a nonnegative integer 
by 
.
For example, in arithmetic modulo 12 (for which the associated ring is 
), the allowable numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8,
9, 10, and 11. This arithmetic is sometimes referred to as "clock arithmetic"
because the additive structure here is the same as that used to determine times for
a twelve-hour clock, except that 0 is often replaced, on a clock, by 12. Example
calculations in arithmetic modulo 12 include statements like "
", or "
", or "
," although the equal sign 
is commonly replaced with the congruence sign 
in such statements to indicate that modular arithmetic is
being used. More explicitly still, a notation such as
 |
is frequently used.
Arithmetic modulo 2 is sometimes referred to as "Boolean arithmetic", because the ring 
is the canonical example of a Boolean ring.
