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 ([latex]\mathbb{W}[/latex]), integers ([latex]\mathbb{Z}[/latex]), decimals ([latex]\mathbb{D}[/latex]) and rational numbers ([latex]\mathbb{Q}[/latex]). Therefore, it retains the same properties.
Properties
The multiplication of real numbers is an operation wholly defined in [latex]\mathbb{R}[/latex] that has the following properties :- commutative property : for every a and b, [latex]a × b = b × a[/latex];
- associative property : for every a, b, c, [latex]a × (b × c) = (a × b) × c[/latex];
- distributive property : for every a, b, and c, [latex](a + b) × c = (a × c) + (b × c)[/latex];
- identity property of multiplication : for every a, [latex]a × 1 = 1 × a = a[/latex];
- 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
- [latex]\dfrac{3}{7}[/latex] × [latex]\dfrac{2}{5}[/latex] = [latex]\dfrac{6}{35}[/latex]
- [latex]\sqrt{2} × \sqrt{3} = \sqrt{6}[/latex]
- 6 × 2π = 12π
- 9 × [latex]\sqrt{2}[/latex] = 9[latex]\sqrt{2}[/latex]