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How to tell whether a decimal is rational

The rule: terminating or repeating means rational

A number is rational if it can be written as a fraction of two integers. Its decimal expansion does one of two things: it stops, or a fixed block repeats forever. A decimal that continues forever without a repeating block is irrational. A bar marks the repeating part: $0.2 \overline{3} = 0.2333 \dots$ . An ellipsis can also show repetition when the pattern is visible, as in $0.919191 \dots$ . If no repeating pattern is stated or visible, the worksheet treats the continuing decimal as nonrepeating.

Problem

Classify each number as rational or irrational and explain why. Example 1. $0.19$ Example 2. $0. \overline{8}$ Example 3. $0.111 \dots$ Example 4. $0.919191 \dots$ Example 5. $0.888$ Example 6. $0.2 \overline{3}$ How would the answer change for a decimal that continues forever with no repeating pattern?

Conversion examples. Write $3 \frac{1}{5}$ , $2 \frac{5}{8}$ , and $1 \frac{5}{12}$ as decimals, and say whether each terminates or repeats.

Answer

0.19

0. \overline{8}

0.111 \dots

0.919191 \dots

0.888

and

0.2 \overline{3}

are rational. A stated nonterminating, nonrepeating decimal is irrational. Also,

3 \frac{1}{5} = 3.2

and

2 \frac{5}{8} = 2.625

terminate;

1 \frac{5}{12} = 1.41 \overline{6}

repeats. All three are rational.

Checked by converting representative terminating and repeating cases to fractions.

Step-by-step solution

1. Terminating decimals

Both $0.19$ and $0.888$ stop after finitely many places. Put the digits over the matching power of ten:

0.19 = \frac{19}{100}, 0.888 = \frac{888}{1000} = \frac{111}{125}

So both are rational.

2. A bar means repetition

The bar in $0. \overline{8}$ means $0.888 \dots$ , and the bar in $0.2 \overline{3}$ means $0.2333 \dots$ . Repeating decimals are rational. For example, if $x = 0. \overline{8}$ , then

10 x - x = 8.888 \dots - 0.888 \dots = 8

9 x = 8, x = \frac{8}{9}

For $y = 0.2 \overline{3}$ , subtract the nonrepeating first digit before using the same idea:

y = \frac{1}{5} + \frac{1}{30} = \frac{7}{30}

3. A visible repeating block

The dots in $0.111 \dots$ continue the digit $1$ , so the number equals $1/9$ . In $0.919191 \dots$ , the block $91$ repeats. If $z = 0.919191 \dots$ , then

100 z - z = 91

99 z = 91, z = \frac{91}{99}

Both are rational.

4. No terminating point and no repeating block

A decimal that continues forever and has no repeating pattern cannot be written as a ratio of integers. It is irrational. The key is not the dots alone - it is whether the continuation repeats.

5. Mixed numbers can terminate or repeat

Separate the whole part from the fractional part: $3 \frac{1}{5} = 3 + 0.2 = 3.2, 2 \frac{5}{8} = 2 + 0.625 = 2.625.$ These decimals terminate. For the third example, $1 \frac{5}{12} = 1 + 0.41666 \dots = 1.41 \overline{6} .$ This decimal repeats. It remains rational because the original number is a fraction: $1 \frac{5}{12} = 17/12$ . A repeating decimal should not be replaced with a finite truncation such as 1.416.

Quick check

Stops? Rational. Repeat bar or an explicit repeating block? Rational. Continues forever with no repetition? Irrational.

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