Operation under which each ordered pair (

*a*,*b*) of real numbers is made to correspond to the number (*a*×*b*) called the product of the numbers*a*and*b*.The multiplication of real numbers is a generalization of multiplication on the set of whole numbers (\(\mathbb{W}\)), integers (\(\mathbb{Z}\)), decimals (\(\mathbb{D}\)) and rational numbers (\(\mathbb{Q}\)). Therefore, it retains the same properties.

### Properties

The multiplication of real numbers is an operation wholly defined in \(\mathbb{R}\) that has the following properties :

- commutative property : for every
*a*and*b*, \(a × b = b × a\); - associative property : for every
*a*,*b*,*c*, \(a × (b × c) = (a × b) × c\); - distributive property : for every
*a*,*b*, and*c*, \((a + b) × c = (a × c) + (b × c)\); - identity property of multiplication : for every
*a*, \(a × 1 = 1 × a = a\); - symmetric element (inverse) : for every a not equal to zero, there exists a real number written as
*a*^{−1}such that*a*×*a*^{−1}= 1; - multiplication property of zero element : for every
*a*,*a*× 0 = 0 ×*a*= 0; - total order : for every
*a*> 0 and every*b*and*c*, if*b*<*c*then*ab*<*ac*.

### Examples

- 12 × 15 = 180
- 2.5 × 3.7 = 9.25
- \(\dfrac{3}{7}\) × \(\dfrac{2}{5}\) = \(\dfrac{6}{35}\)
- \(\sqrt{2} × \sqrt{3} = \sqrt{6}\)
- 6 × 2π = 12π
- 9 × \(\sqrt{2}\) = 9\(\sqrt{2}\)