Multiplication - Properties

Properties

For the natural numbers, integers, fractions, and real and complex numbers, multiplication has certain properties:

Commutative property

The order in which two numbers are multiplied does not matter:

.

Associative property

Expressions solely involving multiplication are invariant with respect to order of operations:

Distributive property

Holds with respect to multiplication over addition. This identity is of prime importance in simplifying algebraic expressions:

Identity element

The multiplicative identity is 1; anything multiplied by one is itself. This is known as the identity property:

Zero element

Any number multiplied by zero is zero. This is known as the zero property of multiplication:

Zero is sometimes not included amongst the natural numbers.

There are a number of further properties of multiplication not satisfied by all types of numbers.

Negation

Negative one times any number is equal to the opposite of that number.

Negative one times negative one is positive one.

The natural numbers do not include negative numbers.

Inverse element

Every number x, except zero, has a multiplicative inverse, such that .

The natural numbers and integers do not have inverses.

Order preservation

Multiplication by a positive number preserves order: if a > 0, then if b > c then ab > ac. Multiplication by a negative number reverses order: if a < 0 and b > c then ab < ac.

The complex numbers do not have an order predicate.

Other mathematical systems that include a multiplication operation may not have all these properties. For example, multiplication is not, in general, commutative for matrices and quaternions.