Chapter 1.1, Problem 1E

In Exercises 1–4, make the given changes in the indicated examples of this section, and then answer the given questions.

In the first line of Example 1, change the 5 to ‒7 and the ‒19 to 12. What other changes must then be made in the first paragraph?

EXAMPLE 1 Identifying rational numbers and irrational numbers

The numbers 5 and –19 are integers. They are also rational numbers because they can be written as $\frac{5}{1}$ and $\frac{-19}{1}$, respectively. Normally, we do not write the 1’s in the denominators.

The numbers $\frac{5}{8}$ and $\frac{-11}{3}$ are rational numbers because the numerator and the denominator of each are integers.

The numbers $\sqrt{2}$ and π are irrational numbers. It is not possible to find two integers, one divided by the other, to represent either of these numbers. In decimal form, irrational numbers are nonterminating, nonrepeating decimals. It can be shown that square roots (and other roots) that cannot be expressed exactly in decimal form are irrational. Also, $\frac{22}{7}$ is sometimes used as an approximation for π, but it is not equal exactly to π. We must remember that $\frac{22}{7}$ is rational and π is irrational.

The decimal number 1.5 is rational since it can be written as $\frac{3}{2}$. Any such terminating decimal is rational. The number 0.6666 . . . , where the 6’s continue on indefinitely, is rational because we may write it as $\frac{2}{3}$. In fact, any repeating decimal (in decimal form, a specific sequence of digits is repeated indefinitely) is rational. The decimal number 0.6732732732 . . . is a repeating decimal where the sequence of digits 732 is repeated indefinitely (0.6732732732 . . . = $\frac{1121}{1665}$).