Modulus

For the term relating to complex numbers, see Complex number.

Aside from its unrelated use in complex numbers, the term modulus refers to the remainder, that is, the remainder of integer division. For example, in 11 / 3 the remainder is 2. This could be stated as <math>mod(11, 3) = 2\,</math>, though mathematicians much more commonly express it as "11 is congruent to 2 modulo 3" and write it symbolically as "<math>11 \equiv 2\mod 3</math>".

The notion of congruence is actually more general than the remainder function. We could also say "2 is congruent to 11 modulo 3", or "<math>2 \equiv 11\mod 3</math>".

"<math>x \equiv y\mod k</math>" means that <math>x - y\,</math> is an integer multiple of <math>k\,</math>.

The remainder operation is also important in computer programming, and the term "modulus" is used to refer to this operation. In C-like languages it is denoted with a percent sign:

x = 11 % 3; // x is now 2

In complex numbers, the modulus is the analogue of the absolute value, given by <math>|a+bi| = \sqrt{a^2 + b^2}</math>.