Fraction to Decimal Calculator
Convert any fraction, mixed number, or negative fraction to a decimal instantly. Detects repeating decimals with period length, shows step-by-step long division, and converts repeating decimals back to exact fractions.
๐ Enter Fraction
Enter positive or negative numerator, denominator must be non-zero.
Limit: ยฑ1 trillion to prevent display issues.
Press Enter to switch tabs.
๐Step-by-Step CalculationClick to expand โผClick to collapse โฒ
๐How it WorksClick to expand โผClick to collapse โฒ
Definition
Convert any fraction to a decimal instantly. A fraction's decimal terminates only if the denominator's prime factors are 2 and/or 5; otherwise it repeats.
๐ Remainder-Cycle Method
During long division, we track remainders. When a remainder repeats, the digits begin repeating. This method detects the exact repeating block (repetend) and its period length.
โก Euclidean GCD Algorithm
We use the Euclidean algorithm to find the greatest common divisor (GCD) of the numerator and denominator, then divide both by it to simplify the fraction to lowest terms.
๐ Input Types Supported
Simple Fractions
a/b format with positive/negative numerators
Mixed Numbers
a b/c format with whole number parts
Decimals
0.75, 0.(3), 1.2(34) formats
๐ Conversion Methods
Fraction โ Decimal
Long division with remainder tracking. Detects terminating vs repeating patterns automatically.
Decimal โ Fraction (Reverse)
Algebraic manipulation: let x = 0.(3), then 10x = 3.(3), subtract to get 9x = 3, so x = 3/9 = 1/3.
โจ Key Features
Instant
No button needed
Repeating
Period detection
Steps
Show work
Copy
All formats
๐ Decimal Notation
Terminating: 3/8 = 0.375 (finite decimal places)
Repeating: 1/3 = 0.(3) or 0.ฬ3 (parenthesis or overline notation)
Mixed repeating: 1/6 = 0.1(6) (non-repeating + repeating parts)
๐ Related Calculators
โFrequently Asked Questions
How do I know if a fraction's decimal terminates or repeats?โผ
A fraction in lowest terms terminates if and only if the denominator has only 2 and/or 5 as prime factors. Otherwise, it repeats. For example, 1/8 = 0.125 (terminates because 8 = 2ยณ), but 1/3 = 0.(3) (repeats because 3 is prime).
How do I convert a repeating decimal to a fraction?โผ
Use algebraic manipulation: let x = 0.(3), multiply by 10^n where n is the repeat length to get 10x = 3.(3), then subtract: 10x - x = 3, so 9x = 3, and x = 3/9 = 1/3 reduced.
What is a repetend and how do you find its length?โผ
The repetend is the repeating block of digits. Its length (period) is found using the remainder-cycle method during long division. When a remainder repeats, the digits begin repeating. For 1/7, the period is 6: 0.(142857).
How do I convert mixed numbers to decimals?โผ
Convert the mixed number to an improper fraction first. For example, 3 1/4 = (3ร4 + 1)/4 = 13/4 = 3.25. Our calculator handles this automatically.
Why do different calculators round away the repeating part?โผ
Most calculators only show a fixed number of decimal places. Our calculator detects true repeating decimals and shows them in both bar notation 0.ฬ3 and parenthesis notation 0.(3), along with the exact period length.