multiplying

Multiplying, in arithmetic, is the operation that combines two numbers to produce their product. It can be understood as repeated addition: for a positive integer n, a × n equals a added to itself n times. The standard binary operation is denoted by a × b or a · b, with the multiplication of real numbers forming the real number product.

Properties: The product of two real numbers is closed under real numbers. Multiplication is associative: (a × b) × c = a × (b × c). It is commutative for real numbers: a × b = b × a. The identity element is 1 since a × 1 = a. The zero property: a × 0 = 0 for any a. Sign rules: (−a) × b = −(a × b) and (−a) × (−b) = a × b.

Fractions and decimals follow the usual rules: a × (m/n) = (a × m)/n; with decimals multiply as integers after removing decimal points. Multiplication distributes over addition: a × (b + c) = a × b + a × c. It is used to scale quantities and to compute areas and volumes as products of side lengths.

Beyond basic arithmetic, multiplication is defined for larger systems such as complex numbers, matrices, and vectors, and is a fundamental operation in algebra. While its exact rules depend on the structure, the same ideas—combining objects to produce a product and obeying associativity and distributivity—often apply.