Addition (mathematics)

Addition is the basic arithmetic operation of combining quantities to form a sum. It applies to numbers, vectors, matrices and abstract algebraic structures and has key properties like commutativity and associativity.

Overview

Addition is a fundamental mathematical operation that combines two or more elements of the same type to produce a single element called their sum. In elementary arithmetic it combines quantities such as whole numbers, fractions, or decimals; the symbol + denotes the operation. The objects being added are commonly called addends or summands.

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Basic properties

For most familiar number systems, addition satisfies several basic laws: it is commutative (a + b = b + a), associative ((a + b) + c = a + (b + c)), and has an identity element (0, since a + 0 = a). In arithmetic with integers and real numbers each element has an additive inverse (−a) so that a + (−a) = 0. Addition distributes over multiplication in the usual number systems.

Extensions and examples

Beyond simple numbers, addition can be defined on vectors and matrices by combining corresponding components: for example, (1,2) + (3,4) = (4,6). Vector addition is used throughout physics and engineering to combine forces, velocities and other quantities; see vectors. Matrices are added entrywise when they share the same dimensions; see matrices. Many programming languages implement + as a primitive operator for numeric types and often overload it for strings and other structures.

History and notation

Practical addition dates back to early counting methods using tokens, tallies or pebbles. The modern plus sign (+) and the word "plus" appear in European arithmetic texts from the late Middle Ages and early Renaissance; the symbol became standardized as arithmetic and algebraic notation developed. Summation of many terms is often written using the capital sigma Σ and is central to series and calculus.

Uses, variations and notable distinctions

Repeated addition underlies multiplication: multiplying n by m is adding n to itself m times.
In modular arithmetic addition wraps around a fixed modulus (clock arithmetic).
Abstract algebra studies binary operations called additions on groups and rings; such operations may lack commutativity in other contexts.
Addition is distinct from concatenation (joining symbols) and from set union; the operations behave differently and apply in different structures.