multiplication

Multiplication is a binary operation that combines two numbers to produce a third number known as the product. In the case of natural numbers, it can be understood as repeated addition: a × b equals a added to itself b times. More generally, multiplication is a scaling operation that extends beyond integers to real numbers, complex numbers, and other algebraic structures. Common notations include a × b, a · b, or simply ab.

Key properties: commutativity (a × b = b × a); associativity ((a × b) × c = a × (b × c)); and distributivity over addition (a × (b + c) = a × b + a × c). There is a multiplicative identity, 1, since a × 1 = a for any a, and the zero property, a × 0 = 0. Multiplication is compatible with signs: multiplying by a negative number reverses the sign of the product.

Notational history: The times symbol ×, popularized in the 17th century by William Oughtred, is one traditional form; the centered dot (·) has also been widely used, and juxtaposition ab is common in algebraic contexts.

Geometric interpretation: For real numbers, multiplication corresponds to scaling: multiplying by a scales a by a factor of b. The product ab also represents the area of a rectangle with sides a and b, providing a geometric meaning to multiplication for positive numbers.

Extensions and applications: In higher mathematics, multiplication is defined in many contexts. Matrix multiplication combines rows and columns and is generally not commutative. Complex numbers multiply via polar form, multiplying moduli and adding arguments. Quaternions multiply in a noncommutative but associative way. Multiplication underpins computation, science, engineering, finance, and many areas of mathematics.