Sprint 8 Converter + Math
Decimal to Fraction Converter
Approximate decimal as reduced fraction.
Fraction
10/1
How it works
Converting a decimal to a fraction requires finding the rational number that exactly represents it — which is always possible for terminating decimals and for repeating decimals, but requires different methods for each. The Decimal to Fraction Converter handles both cases and outputs the fraction reduced to lowest terms.
**Terminating decimals** For a terminating decimal, multiply by 10ⁿ (where n is the number of decimal places) to get an integer, then reduce: 0.75 = 75/100 = 3/4. 0.125 = 125/1000 = 1/8.
**Repeating decimals** For a repeating decimal, use the algebraic trick: let x = 0.333...; then 10x = 3.333...; subtract: 9x = 3; x = 3/9 = 1/3. For a mixed repeating decimal like 0.1666...: multiply to isolate the repeat: 10x = 1.666...; 100x = 16.666...; subtract: 90x = 15; x = 15/90 = 1/6.
**Simplification (GCD reduction)** The converter reduces all fractions to lowest terms by dividing numerator and denominator by their greatest common divisor (GCD), computed using the Euclidean algorithm. 0.6 = 6/10 → GCD(6,10) = 2 → 3/5.
**Practical uses** Financial calculations: interest rates (3.375% = 27/800), currency exchange rates, and loan terms are often more precisely expressed as fractions. Audio engineering: sample rate ratios (44100:48000 = 147:160 after reduction) are expressed as fractions for resampling calculations.
**Limitation: irrational numbers** Irrational numbers like π (3.14159...) and √2 (1.41421...) cannot be expressed exactly as fractions. Any decimal approximation of an irrational number will convert to an approximating fraction, not an exact representation.
Privacy: all calculation runs in the browser. No data is transmitted.
Frequently Asked Questions
- 0.1 = 1/10 as a fraction. However, computers using binary floating-point (IEEE 754 double precision) cannot represent 1/10 exactly — the binary expansion of 0.1 is 0.0001100110011... (repeating). The stored value is approximately 0.1000000000000000055511151231257827021181583404541015625. This is why 0.1 + 0.2 ≠ 0.3 exactly in most programming languages. Financial applications use decimal arithmetic (not binary float) to avoid this issue.
- 1/7 = 0.142857142857... (6-digit repeating cycle). The algebraic method: let x = 0.142857142857...; multiply by 10⁶ = 1,000,000: 1,000,000x = 142857.142857...; subtract: 999,999x = 142857; x = 142857/999,999 = 1/7. In general, a n-digit repeating block after the decimal point converts to (repeating block)/(10ⁿ − 1).
- A fraction is in lowest terms (simplest form) when the numerator and denominator share no common factor greater than 1 — their GCD is 1. To reduce: divide both by their GCD. 24/36 → GCD(24,36) = 12 → 2/3. You can also reduce stepwise: 24/36 ÷ 2 = 12/18 ÷ 2 = 6/9 ÷ 3 = 2/3. The Euclidean algorithm finds the GCD fastest: GCD(36,24) = GCD(24,12) = GCD(12,0) = 12.
- Terminating and repeating decimals can always be converted to exact fractions (they represent rational numbers). Non-terminating non-repeating decimals represent irrational numbers (like π, √2, e) and cannot be expressed as fractions. Any decimal approximation of an irrational number (e.g., 3.14159) will convert to a rational approximation (314159/100000), not the exact value. The converter uses the continued fraction algorithm to find the best rational approximation for any decimal input.