# Multiplication of Real Numbers

## Multiplication of Real Numbers

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$$\sqrt{2}$$