Multiplication

Multiplication is an arithmetic operation in which two numbers are combined to produce a third number called the product. The standard symbol is × or ·; computers typically write it as *. The numbers being multiplied are called factors.

For natural numbers, multiplication corresponds to repeated addition: 4 × 3 means adding 4 three times (4 + 4 + 4 = 12). This interpretation does not work directly with fractions, negative numbers, or real numbers, so advanced mathematics defines multiplication more generally based on the properties of the operation.

Basic Properties

Multiplication of ordinary numbers has several properties: it is commutative, meaning a × b = b × a; it is associative, (a × b) × c = a × (b × c); and it is distributive over addition, a × (b + c) = a × b + a × c. The number 1 is the multiplicative identity, since a × 1 = a, while zero always produces zero.

Methods of Calculation

Schools typically teach multiplication tables and long multiplication, in which numbers are broken down by place value. Other methods exist, such as lattice multiplication and the ancient Egyptian method of doubling. Modern computers use more sophisticated algorithms, such as the Karatsuba method, which reduce the number of multiplications needed when working with very large numbers.

Extension to Advanced Mathematics

The concept of multiplication has been extended to objects that are not numbers. For example, matrices can be multiplied, but the result is not commutative: A × B is usually not equal to B × A. Multiplication also applies to complex numbers, vectors, and abstract groups in algebra. In general mathematics, any operation satisfying certain conditions may be recognized as a form of multiplication.