Modulus

The modulus of a vector r, usually written |r|, or sometimes simply r, is defined to be the length of the vector r. If r is written in terms of three orthonormal vectors (ie: mutually perpendicular vectors of unit length), i, j and k such that:

r = xi+yj+zk,

the modulus of r is the square root of x²+y²+z².

It should be noted that the modulus of a vector is a scalar.

Modulus is also a synonym for:

• the absolute value of a complex number, defined to be the distance between the origin and the complex number, thought of as a point in the plane.
• the absolute value of a real number
• the number by which two numbers are said to be congruent in modular arithmetic. A related, incorrect use of 'modulus' is for the remainder after integer division (see below).

In materials science modulus (or elastic modulus) is a measure of the stiffness of a material. See also Young's modulus

In computer programming, the modulus operator gives the remainder of integer division, ie modular arithmetic. Mathematicians tend to call this modulo, for example:

13 = 3 modulo 10

13 = 3 mod 10

This is because 10 is here the modulus of the congruence, and modulo (a Latin ablative) stands for 'to the modulus'.

C programmers will write:

13 % 10 = 3.