How this calculation works
The repeating block forms a geometric series. Multiply by a power of ten to align its cycle and subtract the repeated tail. Add the whole part, then reduce the resulting integer ratio.
Denominator = 10^prefixLength × (10^repeatLength−1)A worked example
0.1(6) means 0.1666… . Its non-repeating prefix is 1 and repeating block is 6, so it is exactly 1/6.
Good to know
A parenthesized block denotes endless repetition. This is not an approximation of a measured decimal. Prefix and repeat blocks each have at most nine digits; a repeating block of zeros represents a terminating value.
Reference: OpenStax: Rational numbers and repeating decimals (opens in a new tab) ↗
Mark exactly which digits repeat
The notation 0.1(6) means 0.1666…: the 1 occurs once, and the 6 repeats indefinitely. It is not the finite decimal 0.16 or the repeated block 16. Separating the whole part, non-repeating prefix and repeating block removes that ambiguity.
The repeat length one contributes 10¹−1=9, and the one-digit prefix contributes another factor 10. The resulting fraction reduces to 1/6. No finite decimal approximation is used in the arithmetic.
The continued fraction calculator can inspect the same recovered numerator and denominator by a separate Euclidean process.
A repeating tail is a geometric series
For 0.(3), the tail is 3/10+3/100+3/1000+…, which sums to 1/3. With a prefix, shift the repeating tail before forming the same exact ratio. The output reduces common factors before display.
The continued fraction calculator can take this result as a numerator and denominator and show successive approximants. The fraction calculator handles ordinary fraction operations once you have the exact rational.
A stated pattern is an assumption
The form trusts that your stated digit block truly repeats forever. A few observed decimal digits cannot establish that a measured quantity or an irrational constant has that pattern.
A repeating block of zeros is allowed and represents a terminating decimal. A block of nines can yield a value equal to the next terminating decimal; for example 0.(9) equals 1. These are mathematical identities, not rounding artifacts.
A few questions, answered
What does 0.(9) become?
It reduces exactly to 1/1, because the infinite repeating decimal equals one.
May the non-repeating part be empty?
Yes. Leave that field blank when the cycle begins immediately after the decimal point.
Does the tool accept an arbitrary pasted decimal?
No. State its whole part, optional finite prefix and repeating block explicitly.