Convert any decimal number into a simplified fraction.
Enter any decimal number (positive or negative, terminating or repeating pattern).
Simplified fraction
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Enter a decimal to see the fraction and step-by-step working.
The Decimal to Fraction Converter transforms any decimal number into its equivalent simplified fraction. Enter a terminating decimal like 0.75 or a repeating decimal like 0.333..., and the calculator returns the simplified fraction along with step-by-step working showing how the conversion was performed.
Fraction = Decimal × 10^n ÷ 10^n, then simplify
Where n is the number of decimal places. The denominator becomes 10^n, then you simplify by dividing both numerator and denominator by their GCD.
0.75: Two decimal places → 75/100. GCD(75, 100) = 25. Simplified: 3/4.
0.625: Three decimal places → 625/1000. GCD(625, 1000) = 125. Simplified: 5/8.
2.4: One decimal place → 24/10. GCD(24, 10) = 2. Simplified: 12/5 or 2 2/5.
0.333... (repeating): Let x = 0.333... Multiply by 10: 10x = 3.333... Subtract: 10x − x = 3 → 9x = 3 → x = 1/3.
Repeating decimals require an algebraic trick. For 0.818181... (repeating "81"):
The pattern: if n digits repeat, multiply by 10^n, subtract, and solve. This always works for purely repeating decimals. For decimals with a non-repeating prefix (like 0.1666...), the algebra is slightly more involved but follows the same principle.
Not every decimal can be expressed as a fraction. Irrational numbers — π (3.14159...), √2 (1.41421...), e (2.71828...) — have decimal expansions that never terminate and never repeat, so no integer denominator can capture them exactly. The calculator will return an approximation for these (e.g., π ≈ 22/7 or 355/113), but the conversion is never exact.
This distinction between rational (fraction-convertible) and irrational (non-convertible) decimals is fundamental in mathematics. The set of rational numbers is countable (you can list them), while the set of irrational numbers is uncountable — meaning "most" real numbers cannot be expressed as fractions. Fortunately, the decimals you encounter in everyday life (measurements, money, percentages) are almost always rational.
For terminating decimals: write the decimal as a fraction with denominator 10, 100, 1000, etc., then simplify. Example: 0.75 = 75/100 = 3/4.
Use algebra. Let x = 0.333... Then 10x = 3.333..., so 10x − x = 3, giving 9x = 3, so x = 1/3.
0.625 = 625/1000. GCD(625, 1000) = 125. So 625/1000 = 5/8. Verify: 5 ÷ 8 = 0.625.
No. Irrational numbers like π (3.14159...) and √2 (1.41421...) cannot be expressed as fractions of integers. Their decimal expansions never terminate or repeat.
Convert the absolute value to a fraction, then reapply the sign. Example: −0.5 = −1/2.
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