Addition

(mathematics) An operation by which two elements of a set are combined to yield a third; denoted +; usually reserved for the operation in an Abelian group or the group operation in a ring or vector space. The combining of complex quantities in which the individual real parts and the individual imaginary parts are separately added. The combining of vectors in a prescribed way; for example, by algebraically adding corresponding components of vectors or by forming the third side of the triangle whose other sides each represent a vector. Also known as composition.

One of the four fundamental operations of arithmetic and algebra. The term addition is applied to many kinds of objects other than numbers. For example, two vectors x, y are added to produce a third vector z obtained from them by the “parallelogram” law, and two sets A, B are added to form a third set C consisting of all the elements of A and of B. See also Calculus of vectors.

As an operation on pairs of real or complex numbers, addition is associative, (1), and commutative, Eq. (2); and multiplication is distributive over addition, Eq. (3). There are
1.

2.

3.
important mathematical structures in which an addition operation is defined that lacks one or more of these properties. See also Algebra; Division; Multiplication; Number theory; Subtraction.