Example 1.114 from Elementary Algebra, 1.8 The Real Numbers

The Real Numbers: example 1.114

Given the numbers $−7, \frac{14}{5}, 8, \sqrt{5}, 5.9, − \sqrt{64},$ list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

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Solution
1. ⓐRemember, the whole numbers are 0, 1, 2, 3, … and 8 is the only whole number given.
2. ⓑ The integers are the whole numbers, their opposites, and 0. So the whole number 8 is an integer, and $−7$ is the opposite of a whole number so it is an integer, too. Also, notice that 64 is the square of 8 so $− \sqrt{64} = −8 .$ So the integers are $−7, 8, − \sqrt{64} .$
3. ⓒ Since all integers are rational, then $−7, 8, − \sqrt{64}$ are rational. Rational numbers also include fractions and decimals that repeat or stop, so $\frac{14}{5} and 5.9$ are rational. So the list of rational numbers is $−7, \frac{14}{5}, 8, 5.9, - \sqrt{64} .$
4. ⓓ Remember that 5 is not a perfect square, so $\sqrt{5}$ is irrational.
5. ⓔ All the numbers listed are real numbers.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

For the given numbers, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers: $−3, − \sqrt{2}, 0. \overset{–}{3}, \frac{9}{5}, 4, \sqrt{49} .$

How did it go?

Solution

Your turn

Solution

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