repeating decimal to fraction calculator

What this repeating decimal to fraction calculator does

This tool converts a repeating decimal (also called a recurring decimal) into an exact simplified fraction. Instead of rounding, it preserves the precise value using integer math. That means your answer is mathematically exact, not an approximation.

You can use this calculator for:

• Pure repeating decimals like 0.(3) or 2.(81)
• Mixed decimals like 1.2(45) where part of the decimal does not repeat
• Terminating decimals like 12.75 (it still converts and simplifies)

How to enter repeating decimals correctly

Use parentheses around the repeating block

The calculator uses a common notation: the digits in parentheses repeat forever.

• 0.(3) = 0.3333...
• 0.1(6) = 0.16666...
• 3.41(27) = 3.41272727...

If there is no repeating part, just type a regular decimal number and the tool will return a simplified fraction.

The math behind the conversion

Suppose you have a number in the form:

I.A(B)

Where:

• I = integer part
• A = non-repeating decimal digits (length n)
• B = repeating block (length r)

The exact fraction is built using powers of 10 and the repeating factor (10^r - 1). The calculator computes the numerator and denominator, then reduces the result using the greatest common divisor (GCD).

Because the calculation is done with integer arithmetic, it avoids floating-point precision errors that can happen in standard decimal calculations.

Worked examples

Example 1: 0.(3)

The value is one-third, so the fraction is 1/3.

Example 2: 1.2(45)

This has integer part 1, non-repeating part 2, and repeating part 45. After applying the repeating-decimal formula and simplifying, you get an exact rational fraction.

Example 3: 12.75

This is a terminating decimal. The calculator converts it to 1275/100 and simplifies to 51/4.

Why converting repeating decimals to fractions matters

• Algebra: exact values make equations cleaner and easier to solve
• Standardized tests: many questions expect fraction form
• Engineering and science: exact rational forms reduce rounding error in symbolic work
• Teaching and learning: seeing decimal-to-fraction structure builds number sense

Quick tips

• Use parentheses for repeating digits, not commas.
• If your decimal is negative, include a minus sign (for example, -0.(6)).
• Check both the simplified fraction and decimal preview to verify your input.

Frequently asked questions

Can every repeating decimal be written as a fraction?

Yes. Every repeating decimal is a rational number and can be represented exactly as a fraction.

What about non-repeating, non-terminating decimals?

Those are irrational numbers (like π or √2), and they cannot be written as exact fractions of integers.

Does the calculator simplify automatically?

Yes. Results are always reduced to lowest terms.