NOTE 1 By extension, a scalar is also an element of a set for which an addition and a commutative multiplication are defined, each with a neutral element, such that any element has an opposite and any element other than the neutral element for addition has an inverse.

NOTE 2 Sets of scalars, including the extension of Note 1, are usually called fields in mathematics. The set of real numbers and the set of complex numbers are infinite fields. An example of a finite field is the set of two elements 0 and 1 subject to Boolean algebra (where 1 + 1 = 0).