Addition

Addition is a fundamental mathematical operation that combines two numbers to produce their sum. It is denoted by the plus sign (+) and written as a + b. The sum represents the total amount obtained by bringing together quantities a and b. The concept extends to many mathematical structures that support an addition-like operation, such as vectors and matrices.

Key properties include closure within a number system, commutativity (a + b = b + a), and associativity ((a + b) + c = a + (b + c)). The additive identity is 0, since a + 0 = a for any number a. Every number has an additive inverse −a, satisfying a + (−a) = 0; subtraction can be viewed as addition of an additive inverse. In multiplication, addition is distributive over multiplication (a × (b + c) = ab + ac), but not the other way around.

In richer contexts, addition is defined component-wise for vectors and matrices, and modular addition is addition modulo n. In analysis, addition is extended to infinite processes via summation: for a sequence (a_n), the finite sums s_N = ∑_{n=1}^N a_n and the infinite series ∑ a_n. Addition is also foundational to counting, measurement, and computation across disciplines.

Historically, addition has ancient roots in counting systems used by early civilizations. The symbol + was developed in Europe in the late medieval to early Renaissance period and became standard in algebra and arithmetic. Today, addition is taught from elementary arithmetic through advanced mathematics and underlies many algorithms and numerical methods.