Converting Fractions to Decimals
Every fraction can be expressed as a decimal by dividing the numerator by the denominator. The result is either a terminating decimal (ends) or a recurring decimal (repeats indefinitely).
The Method
fraction a/b → decimal = a ÷ bCommon Fractions Reference
- 1/2 = 0.5 | 1/4 = 0.25 | 3/4 = 0.75
- 1/3 = 0.333… | 2/3 = 0.666…
- 1/5 = 0.2 | 2/5 = 0.4 | 3/5 = 0.6
- 1/8 = 0.125 | 3/8 = 0.375 | 5/8 = 0.625
- 1/6 = 0.1666… | 5/6 = 0.8333…
- 1/7 = 0.142857142857… (6-digit repeating)
Terminating vs Recurring
A fraction a/b (in lowest terms) produces a terminating decimal only when the denominator b has no prime factors other than 2 and 5. All other denominators produce recurring decimals.
When to Use Each Form
- Fractions: Exact arithmetic, ratios, cooking, algebra
- Decimals: Money, measurements, scientific calculations, calculators
Convert fractions instantly: Free Fraction to Decimal Converter
Common Fractions Reference Table
- 1/2 = 0.5
- 1/3 = 0.333... (repeating)
- 1/4 = 0.25
- 1/5 = 0.2
- 1/6 = 0.1666... (repeating)
- 1/7 = 0.142857... (repeating, period 6)
- 1/8 = 0.125
- 2/3 = 0.666... (repeating)
- 3/4 = 0.75
- 5/8 = 0.625
Terminating vs Repeating Decimals
A fraction produces a terminating decimal if and only if the denominator (in lowest terms) has no prime factors other than 2 and 5. For example, 3/8 terminates because 8 = 2³; 3/4 terminates because 4 = 2². But 1/3 repeats because 3 is a prime factor of the denominator other than 2 or 5. 1/7 repeats with a 6-digit cycle because 7 is prime. Understanding this helps predict whether a fraction will give a clean decimal before calculating.
Frequently Asked Questions
How do I convert a repeating decimal back to a fraction?
For a single repeating digit like 0.333...: let x = 0.333..., then 10x = 3.333..., so 10x - x = 3, giving 9x = 3, x = 1/3. For a two-digit repeat like 0.090909...: let x = 0.0909..., then 100x = 9.0909..., 100x - x = 9, 99x = 9, x = 9/99 = 1/11.
Why does 1/7 produce such a long repeating decimal?
1/7 = 0.142857142857... with a period of 6 digits. The period length of a repeating decimal 1/p (where p is prime and p ≠ 2, 5) equals the smallest integer k such that 10^k ≡ 1 (mod p). For p=7, this is k=6. The six-digit block 142857 is a cyclic number: its multiples (×2 = 285714, ×3 = 428571...) are rotations of the same digits.
Is 0.999... really equal to 1?
Yes, mathematically 0.999... = 1 exactly, not approximately. Proof: let x = 0.999..., then 10x = 9.999..., so 10x - x = 9, giving 9x = 9, x = 1. This surprises many people but is a rigorous consequence of how infinite series work in real number arithmetic.