Recognizing Properties of Real Numbers
The real numbers behave in very regular ways. These behaviors are called the properties of the real numbers. Knowing these properties helps when evaluating formulas, working with equations, or performing algebra. Being familiar with these properties is helpful in all settings where numbers are used and manipulated.
The table below is a partial list of properties of real numbers.
| Property | Example | In Words |
|---|---|---|
| Distributive Property [latex]a \times (b + c) = a \times b + a \times c[/latex] |
[latex]5 \times (3 + 4) = 5 \times 3 + 5 \times 4[/latex] | Multiplication distributes across addition |
| Commutative Property of Addition [latex]a + b = b + a[/latex] |
[latex]3 + 7 = 7 + 3[/latex] | Numbers can be added in any order |
| Commutative Property of Multiplication [latex]a \times b = b \times a[/latex] |
[latex]10 \times 4 = 4 \times 10[/latex] | Numbers can be multiplied in any order |
| Associative Property of Addition [latex]a + (b + c) = (a + b) + c[/latex] |
[latex]4 + (3 + 8) = (4 + 3) + 8[/latex] | Doesn’t matter which pair of numbers is added first |
| Associative Property of Multiplication [latex]a \times (b \times c) = (a \times b) \times c[/latex] |
[latex]2 \times (5 \times 7) = (2 \times 5) \times 7[/latex] | Doesn’t matter which pair of numbers is multiplied first |
| Additive Identity Property [latex]a + 0 = a[/latex] |
[latex]17 + 0 = 17[/latex] | Any number plus [latex]0[/latex] is the number |
| Multiplicative Identity Property [latex]a \times 1 = a[/latex] |
[latex]21 \times 1 = 21[/latex] | Any number times one is the number |
| Additive Inverse Property [latex]a + (-a) = 0[/latex] |
[latex]14 + (-14) = 0[/latex] | Every number plus its negative is [latex]0[/latex] |
| Multiplicative Inverse Property [latex]a \times \frac{1}{a} = 1[/latex], provided [latex](a \neq 0)[/latex] |
[latex]3 \times \frac{1}{3} = 1[/latex] | Every non-zero number times its reciprocal is [latex]1[/latex] |
The names of the properties are suggestive. The commutative properties, for example, suggest commuting, or moving. Associative properties suggest which items are associated with others, or if order matters in the computation. The distributive property addresses how a number is distributed across parentheses.
In each of the following, identify which property of the real numbers is being applied.
- [latex]4+(8+13)=(4+8)+13[/latex]
- [latex]34×(\frac{1}{34})=1[/latex]
- [latex]14+27=27+14[/latex]
Using these properties to perform arithmetic quickly relies on spotting easy numbers to work with. Look for numbers that add to a multiple of [latex]10[/latex], or multiply to a multiple of [latex]10[/latex] or [latex]100[/latex].
Use properties of the real numbers to calculate the following:
- [latex]2×13×50[/latex]
- [latex]13+84+27[/latex]
- [latex]9×16×11[/latex]