Example 7.4 from Prealgebra, 7.1 Rational and Irrational Numbers

Rational and Irrational Numbers: example 7.4

Determine whether each of the numbers in the following list is a ⓐ whole number, ⓑ integer, ⓒ rational number, ⓓ irrational number, and ⓔ real number.

−7, \frac{14}{5}, 8, \sqrt{5}, 5.9, - \sqrt{64}

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.

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

We'll summarize the results in a table.

Number Whole Integer Rational Irrational Real

$−7$ $✓$ $✓$ $✓$

$\frac{14}{5}$ $✓$ $✓$

$8$ $✓$ $✓$ $✓$ $✓$

$\sqrt{5}$ $✓$ $✓$

$5.9$ $✓$ $✓$

$- \sqrt{64}$ $✓$ $✓$ $✓$

Number	Whole	Integer	Rational	Irrational	Real
$−7$		$✓$	$✓$		$✓$
$\frac{14}{5}$			$✓$		$✓$
$8$	$✓$	$✓$	$✓$		$✓$
$\sqrt{5}$				$✓$	$✓$
$5.9$			$✓$		$✓$
$- \sqrt{64}$		$✓$	$✓$		$✓$

Your turn

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

Determine whether each number is a ⓐ whole number, ⓑ integer, ⓒ rational number, ⓓ irrational number, and ⓔ real number: $−3, - \sqrt{2}, 0 . \overset{–}{3}, \frac{9}{5}, 4, \sqrt{49} .$

How did it go?

Solution

Your turn

Solution

Your turn