About

Converting a recurring decimal to a fraction is a fundamental skill in algebra that bridges the gap between estimated values and exact quantities. In scientific computing, engineering, and pure mathematics, keeping numbers in their fractional form (`1/3` instead of `0.333...`) ensures that precision is never lost during subsequent calculations. Rounding errors can compound rapidly in complex algorithms, leading to significant deviations in the final result.

This tool is designed for students, educators, and professionals who need to determine the exact rational number representation of a decimal. It doesn't just give the answer; it demonstrates the algebraic logic used to derive the fraction, detecting patterns in the input string up to 100 decimal places.

Formulas

To convert a recurring decimal to a fraction, we use algebra to eliminate the repeating part. If a number x has a repeating cycle of length n, we multiply by 10ⁿ.

x = 0.d₁d₂...

Subtracting the original equation removes the infinite decimal tail:

10ⁿx − x10ⁿ − 1 = Result

Reference Data

Decimal Type	Example Input	Fraction Result	Mathematical Classification
Terminating	0.5	1/2	Rational Number
Simple Recurring	0.333...	1/3	Rational (1 digit cycle)
Complex Recurring	0.1666...	1/6	Rational (Mixed)
Long Cycle	0.142857...	1/7	Rational (6 digit cycle)
Percentage	0.75	3/4	Rational
Integer	1.0	1/1	Integer
Small Recurring	0.0909...	1/11	Rational (2 digit cycle)
Mixed Integer	2.555...	23/9	Improper Fraction

Frequently Asked Questions

The algorithm analyzes the input string from right to left, looking for repeating substrings. If you input '0.333333', it detects '3' as the cycle. For '0.121212', it detects '12'. The more repetitions you provide, the more accurate the detection.

Algebraically, if x = 0.999..., then 10x = 9.999.... Subtracting x from 10x gives 9x = 9, which simplifies to x = 1. This is a standard property of real numbers.

Yes. If no repeating pattern is detected (e.g., '0.25'), it treats it as a terminating decimal and converts it directly to a fraction (25/100 -> 1/4).

The tool parses input strings directly to avoid floating-point errors standard in programming. It can handle inputs with up to 100 decimal places to find long cycles (like 1/7 or 1/17).