Two types of rational decimals
Every rational number has a decimal that either stops or repeats:
| Type | Example |
|---|---|
| Terminating decimal | 3/8 = 0.375 |
| Repeating decimal | 5/11 = 0.454545… |
The repeating block is called the period. In 0.454545…, the period is 45 (length 2).
Bar notation
Bar notation places a bar over the repeating digits:
- 0.333… = 0.3 with a bar over the 3
- 0.1666… = 0.16 with a bar over only the 6
- 0.142857142857… = a bar over 142857
Which fractions terminate?
Write the fraction in lowest terms and factor the denominator:
- Only 2s and/or 5s → terminating
- Any other prime factor → repeating
| Fraction | Denominator | Type |
|---|---|---|
| 7/20 | 2² × 5 | terminating (0.35) |
| 3/16 | 2⁴ | terminating (0.1875) |
| 2/3 | 3 | repeating (0.666…) |
| 5/12 | 2² × 3 | repeating (0.41666…) |
| 3/7 | 7 | repeating (0.428571…) |
Why? Our system is base 10 = 2 × 5, so only denominators made of 2s and 5s divide evenly into a power of 10.
Converting repeating decimals to fractions
One repeating digit: x = 0.777…
- 10x = 7.777…
- Subtract: 10x − x = 7
- 9x = 7 → x = 7/9
Two repeating digits: x = 0.363636…
- 100x = 36.3636…
- 99x = 36 → x = 36/99 = 4/11
Delayed repeat: x = 0.2555…
- 10x = 2.555… and 100x = 25.555…
- 90x = 23 → x = 23/90
Shortcut: a block of n repeating digits starting right after the point → put the block over n nines. 0.123123… = 123/999 = 41/333.
Is 0.999… = 1?
Yes! x = 0.999… → 10x = 9.999… → 9x = 9 → x = 1. Two decimals can name the same number.
Worked examples
Example 1: Is 9/40 terminating? 40 = 2³ × 5 → yes (0.225).
Example 2: Convert 0.444… to a fraction. 4/9.
Example 3: Convert 0.181818… to a fraction. 18/99 = 2/11.
Common mistakes
- Not simplifying the fraction before checking the denominator (3/6 has a 3 in its denominator, but it equals 1/2, which terminates).
- Putting the bar over too many or too few digits.
- Rounding a repeating decimal and treating it as exact.
Try it yourself
- Will 7/25 terminate?
- Will 1/6 terminate?
- Convert 0.555… to a fraction.
- Convert 0.272727… to a fraction.
- Convert 0.1333… to a fraction.
Answers: 1) yes (0.28) 2) no 3) 5/9 4) 3/11 5) 2/15
Quick recap
- Rational decimals terminate or repeat.
- Denominators with only 2s and 5s give terminating decimals.
- Use algebra (10x, 100x) to turn a repeating decimal into a fraction.