Convert terminating and recurring repeating decimals into simplified irreducible fractions and mixed numbers. Features step-by-step algebraic proofs, Greatest Common Divisor reductions, and visual number lines.
Converting a decimal to a fraction transforms floating-point numbers into exact rational proportions:
Use the standard mode for terminating decimals like 0.75, or switch to the repeating tab for recurring values like 0.333... or 0.166...
Follow the powers-of-10 base scaling for terminating values, or observe the simultaneous equation subtraction proof for repeating decimals.
Read the simplified proper or improper fraction, mixed number form, greatest common divisor reduction, and reciprocal.
Converting recurring decimals like 0.142857142857... into exact 1/7 requires simultaneous linear equation subtraction that students often misplace.
Digital calipers output 0.4375 inches; converting to 7/16 inches allows machinists to select standard imperial drill bits and stock metal gauges.
In financial interest modeling and physics formulas, using exact fractions (like 1/3) avoids cumulative IEEE-754 floating-point rounding errors.
Transforming numbers like 3.875 into 3 7/8 simplifies kitchen recipe measuring spoons and woodworking cut lists.
Separates non-repeating and repeating digits to solve complex mixed recurring decimals like 0.1666... (1/6) or 1.4545... (1 5/11).
Shows the full algebraic equation system (e.g. 100x - x = 99x) step by step for academic and homework verification.
Computes the Greatest Common Divisor to reduce large unreduced fractions like 625/1000 down to simplest form (5/8).
Properly preserves signs across whole integer parts and numerator fractional parts for negative decimals.
For a terminating decimal with \(k\) decimal digits after the point:
For a repeating decimal with \(k\) repeating digits under the vinculum bar:
Count 3 decimal places, scale over 1000, and simplify by the greatest common divisor (125):
Let x = 0.666... Multiply by 10 to shift one recurring period, then subtract the equations:
| Decimal Value | Unreduced Fraction | Simplified Lowest Terms | Common Application |
|---|---|---|---|
| 0.125 | 125/1000 | 1/8 | 1/8 inch drill bit / 1/8 cup (2 tbsp) |
| 0.25 | 25/100 | 1/4 | Quarter dollar / 1/4 pound portion |
| 0.333... | 3/9 | 1/3 | One third cup / recurring proportion |
| 0.625 | 625/1000 | 5/8 | 5/8 inch socket wrench / hardware |
| 0.875 | 875/1000 | 7/8 | 7/8 inch plumbing & pipe fitting |
To quickly convert any terminating decimal, write the decimal digits as the numerator and place 1 followed by as many zeroes as there are decimal digits in the denominator. Then reduce using the greatest common divisor.
Digital calipers measure in decimal thousandths of an inch (e.g. 0.4375). Using an exact decimal to fraction conversion matches your readout directly to standardized fractional drill sets (7/16 in.) without guesswork.
Write the decimal digits without the decimal point as the numerator. In the denominator, write a 1 followed by as many zeros as there are decimal places. For example, 0.625 has three decimal places, so it becomes 625/1000. Finally, divide top and bottom by their GCD (125) to get 5/8.
Use an algebraic system. Let x equal the repeating decimal (e.g., x = 0.4545...). Because two digits repeat, multiply by 100: 100x = 45.4545... Subtract the first equation from the second: 100x - x = 45.4545... - 0.4545..., giving 99x = 45. Divide both sides by 99: x = 45/99, which reduces by 9 to 5/11.
Let x = 0.999... Multiplying by 10 gives 10x = 9.999... Subtracting the original equation gives 10x - x = 9.999... - 0.999..., resulting in 9x = 9. Dividing both sides by 9 yields x = 1. In real analysis, both 0.999... and 1.0 represent the exact same point on the real number line.
Only rational numbers (decimals that terminate or repeat with a fixed recurring cycle) can be written as exact fractions of two integers. Irrational numbers like pi (3.14159...) or the square root of 2 (1.41421...) have non-terminating, non-repeating decimal expansions that cannot be expressed as exact integer ratios.
Convert the absolute positive value into a fraction following standard methods, then reapply the negative sign to the resulting numerator or place it directly in front of the fraction bar. For example, -2.4 becomes -2 4/10 = -2 2/5 = -12/5.
