multiplications

Multiplication is a binary operation that combines two elements to produce a product. In the common arithmetic sense, the operation on numbers uses the symbol × or · or is written by juxtaposition as ab. It can be understood as repeated addition for natural numbers, but extends beyond that interpretation in other contexts.

Key properties include closure, associativity, commutativity for many number systems, and distributivity of multiplication over addition. In the real or complex numbers, multiplication is commutative and associative, and a × (b + c) = a × b + a × c. Zero is an absorbing element: a × 0 = 0; 1 is the multiplicative identity.

An inverse exists for nonzero numbers in fields, called the multiplicative inverse or reciprocal, a⁻¹ where a × a⁻¹ = 1. For integers, not every nonzero element has an inverse within the integers.

Notational variants include a × b, a · b, or ab. In higher mathematics, multiplication generalizes to other objects, such as matrix multiplication, which is associative but not commutative in general, or multiplication in rings and fields, which must satisfy axioms defining a multiplicative structure.

Multiplication has wide applications in science and engineering, including scaling quantities, solving equations, and describing geometric area and volumes.