Portions of this entry contributed by Matt Insall (author's link)
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 
,
of integers modulo n. The addition in the ring
is determined from addition in
by computing the remainder, upon division by n, of the sum 
of two integers a and b. Similarly, for multiplication in the ring
,
For each positive integer n, the ring
has n elements, namely the equivalence classes of each of
the nonnegative integers less than n, under the equivalence relation R that is defined according to the rule
aRb iff n divides 
.
(under the
equivalence relation R) of a nonnegative integer a < n by a.
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 "11 + 1 = 0", or "7 + 8 = 3", or "
,
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.
Boolean Ring, Congruence, Modulus, Residue
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