Repeating Decimal to Fraction Calculator
Example: For 2.1444…, enter 2
Example: For 2.1444…, enter 1
Example: For 2.1444…, enter 4
Result:
How to Convert a Repeating Decimal to a Fraction
A repeating decimal is a decimal representation of a number whose digits are periodic (repeating at regular intervals). Converting these into fractions is a fundamental algebraic skill that helps in simplifying complex mathematical expressions.
The Step-by-Step Conversion Formula
To convert a repeating decimal (like 0.1666…) manually, follow these steps:
- Let x equal the decimal: $x = 0.1666…$
- Identify the repeating part: Here, "6" repeats. It has a length of 1 digit.
- Multiply to move the repeating part: Multiply $x$ by $10^1$ (since one digit repeats) to get $10x = 1.666…$
- Subtract the original equation: $(10x = 1.666…) – (x = 0.1666…)$ gives $9x = 1.5$.
- Solve for x: $x = 1.5 / 9 = 15 / 90$.
- Simplify: $15 / 90$ simplifies to $1/6$.
Common Repeating Decimal Examples
| Decimal | Fraction | Simplified |
|---|---|---|
| 0.333… | 3/9 | 1/3 |
| 0.666… | 6/9 | 2/3 |
| 0.1212… | 12/99 | 4/33 |
| 0.142857… | 142857/999999 | 1/7 |
Frequently Asked Questions
What if only part of the decimal repeats?
This is called a mixed repeating decimal (e.g., 0.12333…). You multiply by powers of 10 to shift the non-repeating part to the left of the decimal, then solve as shown in our formula above.
Does every repeating decimal represent a rational number?
Yes. By definition, a rational number is any number that can be expressed as a ratio of two integers. Since all repeating decimals can be converted into fractions, they are all rational numbers.