|Definition:|| operation performed on a set, assigning a unique element of the set to any ordered pair of elements a and b of the set, with the following properties: |
- associativity: , where c is also an element of the set,
- if an addition is performed on the set, distributivity: and
Note 1 to entry: Multiplication is defined for natural numbers and extended to other classes of numbers and to mathematical entities such as polynomials and matrices. Multiplication is also defined for quantities and units, even if they are not of the same kind, so that addition cannot be defined.
Note 2 to entry: Multiplication is not necessarily commutative, for example in the case of matrices.
Note 3 to entry: Each element in a multiplication of two or more elements is called a factor. The term "factor" is also used for a quotient of two quantities of the same kind (see IEV 112-01-04). In the multiplication of two elements, the first is called "multiplier" and the second "multiplicator".
Note 4 to entry: The multiplication of entities a and b is expressed by the words "a multiplied by b" or "a times b" and denoted by a ⋅ b, a × b, or ab. The symbol ∏ is used to denote successive multiplications, for example a2 ⋅ a3 ⋅ a4 ⋅ a5 ⋅ a6 ⋅ a7 is denoted by .