Recurring Decimal Calculator

Calculator

Choose a mode. Exact fraction work uses integer arithmetic.

Recurring decimal Parentheses, brackets, overline text, and clear ellipsis patterns are accepted.

Decimal separator

Precision and display options

Displayed decimal places

Maximum cycle length

Rounding method

Keep trailing zeros Show scientific notation Live calculation after edits

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Example values

Recurring decimal	Exact fraction	Type	Cycle length
0.(3)	1/3	Pure recurring	1
0.(6)	2/3	Pure recurring	1
0.(09)	1/11	Pure recurring	2
0.1(6)	1/6	Mixed recurring	1
0.(142857)	1/7	Pure recurring	6
2.(45)	27/11	Recurring above one	2
0.12(3)	37/300	Mixed recurring	1
0.(9)	1	Equivalent terminating value	1

Formula used

Pure recurring decimal

When a block has n digits, place that block over a denominator containing n nines.

x = repeating block ÷ (10ⁿ − 1)

Example: 0.(27) = 27/99 = 3/11.

Mixed recurring decimal

Subtract the digits before repetition from all digits through one complete repeating block.

x = (all digits − non-repeating digits) ÷ [10^m(10ⁿ − 1)]

Here, m counts non-repeating digits. The value n counts repeating digits.

How to use this calculator

Choose the conversion, detection, verification, or comparison mode.
Enter the decimal with parentheses around repeating digits.
Adjust precision, notation, cycle length, and rounding options.
Select Calculate to create an exact result and explanation.
Copy, print, share, favorite, or download the result.

Understanding recurring decimals

What recurring decimals mean

A recurring decimal contains digits that repeat forever. The repeating section is called the repetend. Parentheses mark that section clearly. For example, 0.(27) means 0.272727 forever. Every recurring decimal represents a rational number. Therefore, it can be written as an exact fraction.

Pure and mixed forms

A pure recurring decimal starts repeating immediately after the point. The value 0.(6) is pure. A mixed recurring decimal begins with non-repeating digits. The value 0.1(6) is mixed. Both forms use subtraction to remove endless tails.

Why long division repeats

Long division creates a sequence of remainders. A denominator offers only finitely many possible remainders. When a remainder appears again, the following digits repeat. The cycle length measures digits between repeated remainders. A zero remainder means the decimal terminates.

Terminating denominators

A reduced fraction terminates only when its denominator contains factors two and five. Other prime factors create a recurring cycle. For example, eighths terminate because eight uses only factor two. Sevenths repeat because seven introduces another prime factor.

Why repeated nines equal one

The value 0.(9) equals one exactly. Algebra confirms this result. Let x equal 0.999 forever. Then ten x equals 9.999 forever. Subtracting x leaves nine x equal nine. Therefore, x equals one. No gap remains between them.

Exact arithmetic matters

Computer floating-point values can introduce tiny approximation errors. This calculator builds integers directly from entered digits. It reduces fractions using the Euclidean algorithm. Exact arithmetic protects long cycles and large values from common rounding mistakes.

Useful notation choices

Teachers use bars, dots, brackets, parentheses, or ellipses. Parentheses are easiest for plain text. An expression like 2.4(81) separates its fixed and recurring sections. Ellipses can become ambiguous when too few digits are supplied.

Checking a result

Convert the decimal into a fraction first. Then simplify both fractions. Equal reduced numerators and denominators confirm equivalence. Reverse conversion offers another check. Matching repeating blocks provide strong evidence. Exact fraction comparison gives the final conclusion.

Frequently asked questions

What is a recurring decimal?

It is a decimal whose digits repeat forever after some point. The repeating digits form a fixed cycle.

How do I enter repeating digits?

Place the repeating block inside parentheses. Enter 0.1(6) for 0.1666 forever.

Can the calculator handle negative values?

Yes. Place a minus sign before the decimal, numerator, or whole-number part.

Why does 0.(9) equal one?

No positive gap exists between them. Algebra and fraction conversion both produce exactly one.

What causes a decimal to terminate?

After reduction, its denominator must contain only factors of two and five.

What does cycle length mean?

It counts digits in the smallest repeating block. The block 142857 has length six.

Why was my ellipsis input rejected?

The supplied digits may not show two reliable repetitions. Parentheses remove that ambiguity.

Are large fractions supported?

Yes. The included string-based integer routines handle values beyond normal machine integer limits.