Example 3 from Algebra and Trigonometry, 1.1 Real Numbers: Algebra Essentials

Differentiating Rational and Irrational Numbers

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

ⓐ $\sqrt{25}$
ⓑ $\frac{33}{9}$
ⓒ $\sqrt{11}$
ⓓ $\frac{17}{34}$
ⓔ $0.3033033303333 \dots$

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.

1
Solution
1. ⓐ $\sqrt{25} :$ This can be simplified as $\sqrt{25} = 5.$ Therefore, $\sqrt{25}$ is rational.
2. ⓑ $\frac{33}{9} :$ Because it is a fraction of integers, $\frac{33}{9}$ is a rational number. Next, simplify and divide.
  $\frac{33}{9} = \frac{\overset{11}{33}}{\underset{3}{9}} = \frac{11}{3} = 3. \bar{6}$
  So, $\frac{33}{9}$ is rational and a repeating decimal.
3. ⓒ $\sqrt{11} :$ This cannot be simplified any further. Therefore, $\sqrt{11}$ is an irrational number.
4. ⓓ $\frac{17}{34} :$ Because it is a fraction of integers, $\frac{17}{34}$ is a rational number. Simplify and divide.
  $\frac{17}{34} = \frac{\overset{1}{17}}{\underset{2}{34}} = \frac{1}{2} = 0.5$
  So, $\frac{17}{34}$ is rational and a terminating decimal.
5. ⓔ $0.3033033303333 \dots$ is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.

Your turn

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

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

ⓐ $\frac{7}{77}$
ⓑ $\sqrt{81}$
ⓒ $4.27027002700027 \dots$
ⓓ $\frac{91}{13}$
ⓔ $\sqrt{39}$

How did it go?

Your turn

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

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

ⓐ $\frac{7}{77}$
ⓑ $\sqrt{81}$
ⓓ $\frac{91}{13}$
ⓔ $\sqrt{39}$

Solution

Your turn

Solution

Your turn