Repeating Decimal to Fraction Calculator

Rational Number Theory & Algebraic Decimal Elimination

Every repeating decimal represents a rational number that can be expressed as a quotient of two integers (a) / (b):

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• The Algebraic Elimination Algorithm**:

Let x be the repeating decimal with m non-repeating digits and k repeating digits:

10^m+kx - 10^m x = Integer Difference

(10^k - 1)10^m x = Integer x = Integer(10^k - 1)10^m

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• Infinite Geometric Series Form**:

The repeating part is an infinite geometric series with common ratio r = 10^-k:

S = Σ_n=1^∞ a r^n-1 = (a) / (1 - r) = R / 10^m+k1 - 10^-k = (R) / (10^m (10^k - 1))

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• Euler's Identity for 0.9 = 1**:

10x - x = 9x = 9 x = 1

How to Use This Calculator

Enter your Integer Whole Number Part (e.g. 0 or 1) (e.g. 0), Non-Repeating Decimal Part (before bar, e.g. 1 for 0.1666...) (e.g. 1 (leave blank if none)), Repeating Decimal Part / Repetend (under the bar, e.g. 6) (e.g. 6 for 0.1666...), and Rational Fraction Benchmark — choose from Custom Repeating Decimal, 0.16̄ (0.1666... → 1/6), 0.3̄ (0.3333... → 1/3). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

x = intPart + nonRepNum / 10^m + repNum / (10^m * (10^k - 1))

Worked Example

A civil engineer converting a measured decimal of 1.83 inches obtains the exact fractional socket size of 1\ (5) / (6) in (or (11) / (6)), eliminating rounding errors in CNC machining.

Understanding Your Result

Percentages are relative values — context determines what is "good" or "bad." A 50% discount is significant; a 50% failure rate is alarming. Percentage change and percentage points are different: moving from 4% to 5% is a 1 percentage point increase, but a 25% relative increase. When comparing percentages across different base values, always check whether absolute differences are meaningful — a 10% improvement on a large base creates more absolute value than 10% on a small base.