Modulus

The word modulus has several different meanings in mathematics with respect to complex numbers, congruences, elliptic integrals, quadratic invariants, sets, etc.

The modulus of a congruence  is the number . It is the "base" with respect to which a congruence is computed (i.e.,  gives the number of multiples of  that are "thrown out"). For example, when computing the time of day using a 12-hour clock obtained by adding four hours to 9:00, the answer, 1:00, is obtained by taking  (i.e., adding the hours with modulus 12).

In many computer languages (such as FORTRAN or the Wolfram Language), the common residue of  (mod ) is written mod(b,m) (FORTRAN) or Mod[b,m]