Definition:Multiplication

Notation
There are several variants on the notation for multiplication:


 * $n \times m$, which is usually used only when numbers are under consideration, e.g. $3 \times 5 = 15$;
 * $nm$, which is most common in algebra, but not with numbers unless parentheses are put round the numbers, e.g. $\left({3}\right)\left({4}\right) = 12$, for obvious reasons;
 * $n \cdot m$ or $n . m$, which have their uses in algebra, but has the danger of being confused with the decimal point.

Also see

 * Quaternion Multiplication

Commutativity of Multiplication
On all the above number sets, we have that multiplication is commutative:


 * Natural Number Multiplication is Commutative
 * Integer Multiplication is Commutative
 * Modulo Multiplication is Commutative
 * Rational Multiplication is Commutative
 * Real Multiplication is Commutative
 * Complex Multiplication is Commutative

Associativity of Multiplication
On all the above number sets, we have that multiplication is associative:


 * Natural Number Multiplication is Associative
 * Integer Multiplication is Associative
 * Modulo Multiplication is Associative
 * Rational Multiplication is Associative
 * Real Multiplication is Associative
 * Complex Multiplication is Associative

Historical Note
The symbol $\times$ was invented by, who was criticized by as it was in his view too similar to the letter $x$.