Understanding the conversion
Common terminating decimals correspond to simple fractions worth memorizing. The table below lists frequently used equivalents.
| Decimal | Fraction | Percentage |
|---|---|---|
| 0.5 | 1/2 | 50% |
| 0.25 | 1/4 | 25% |
| 0.75 | 3/4 | 75% |
| 0.2 | 1/5 | 20% |
| 0.125 | 1/8 | 12.5% |
| 0.625 | 5/8 | 62.5% |
| 0.0625 | 1/16 | 6.25% |
- A decimal terminates exactly when its fraction in lowest terms has a denominator whose only prime factors are 2 and 5. That is why 1/3, 1/6 and 1/7 produce repeating decimals.
- Repeating decimals cannot be entered exactly: 0.333333 converts to 333333/1000000, not to 1/3. Use the algebraic method (let x = 0.333..., then 10x - x = 3, so x = 1/3) for repeating decimals.
- This calculator accepts up to nine decimal places; beyond that, floating-point representation limits exact conversion.
What is decimal to fraction conversion?
A terminating decimal is a number whose digits end after a finite number of decimal places, such as 0.625 or 0.32. Every terminating decimal can be written exactly as a fraction whose denominator is a power of ten: the digits after the decimal point form the numerator, and the denominator is 10 raised to the number of decimal places. 0.625 has three decimal places, so it equals 625/1000.
The resulting fraction is then reduced to lowest terms by dividing numerator and denominator by their greatest common divisor (GCD). For 625/1000, the GCD is 125, giving 5/8. A fraction of integers is called a rational number, and every terminating decimal is rational.
Repeating decimals such as 0.333... (which equals 1/3 exactly) also represent rational numbers, but they require a different algebraic method and cannot be entered exactly as a finite decimal. Entering 0.333333 converts that exact six-digit decimal (333333/1000000), not 1/3. This calculator handles terminating decimals with up to nine decimal places.
How to use this decimal to fraction calculator
- Enter the decimal value you want to convert, for example 0.625 or 2.5. Negative decimals are supported.
- Read the simplified fraction. The sign is carried on the numerator for negative inputs.
- Read the percentage equivalent, which is the decimal multiplied by 100.
- To verify by hand, write the digits after the decimal point over the matching power of ten and divide both by their greatest common divisor.
The place-value conversion method
Count the digits after the decimal point (call the count k). Multiply the decimal by 10^k to get an integer numerator, and use 10^k as the denominator. Then divide both by their greatest common divisor to reach lowest terms.
Worked example: 0.625. It has k = 3 decimal places, so 0.625 = 625/1000. The GCD of 625 and 1000 is 125. Dividing both: 625/125 = 5 and 1000/125 = 8, so 0.625 = 5/8. As a percentage, 0.625 x 100 = 62.5%.
Second example: 0.32 = 32/100. The GCD of 32 and 100 is 4, so 0.32 = 8/25 = 32%.
Decimals greater than 1 convert the same way: 2.5 = 25/10 = 5/2, which can also be written as the mixed number 2 1/2.
Common mistakes
- Expecting 0.33 or 0.333333 to convert to 1/3 — only the infinite repeating decimal 0.333... equals 1/3; any finite entry converts to a different, exact fraction.
- Using the wrong power of ten: 0.625 is 625/1000 (three places), not 625/100.
- Forgetting to simplify: 625/1000 and 5/8 are equal, but lowest terms is the conventional answer.
- Dropping the sign on negative decimals — the negative sign belongs to the numerator of the resulting fraction.
자주 묻는 질문
How do I convert a decimal to a fraction?
Write the digits after the decimal point as the numerator over 10 raised to the number of decimal places, then simplify by the greatest common divisor. For example, 0.625 = 625/1000; the GCD of 625 and 1000 is 125, so 0.625 = 5/8.
What is 0.625 as a fraction?
0.625 equals 5/8. The conversion: 0.625 has three decimal places, so it equals 625/1000. Dividing numerator and denominator by their greatest common divisor, 125, gives 5/8. As a percentage, 0.625 is 62.5%.
Can every decimal be written as a fraction?
Every terminating decimal and every repeating decimal can be written exactly as a fraction of integers — these are the rational numbers. Irrational numbers such as pi (3.14159...) and the square root of 2 have decimal expansions that neither terminate nor repeat, and they cannot be written exactly as any fraction of integers.
How do I convert a repeating decimal like 0.333... to a fraction?
Use algebra rather than place value. Let x = 0.333...; multiplying by 10 gives 10x = 3.333...; subtracting the first equation from the second gives 9x = 3, so x = 3/9 = 1/3. A finite entry such as 0.333333 is a different number and converts to 333333/1000000 instead.
How do I convert a decimal greater than 1?
The same place-value method applies. For example, 2.5 = 25/10 = 5/2 after dividing by the GCD of 5. The improper fraction 5/2 can also be written as the mixed number 2 1/2; both represent the same value.
참고 자료
- Weisstein, Eric W. "Decimal Expansion." MathWorld — A Wolfram Web Resource. mathworld.wolfram.com.
- Weisstein, Eric W. "Rational Number." MathWorld — A Wolfram Web Resource. mathworld.wolfram.com.
- Niven I. Numbers: Rational and Irrational. Mathematical Association of America, 1961.