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\(\sqrt{2}\)