Fraction Repeating Cycle Length
Trace the exact base-ten remainder cycle of a fraction with reduced denominator at most 10000; terminating fractions have zero repeating cycle length.
Description
Trace the exact base-ten remainder cycle of a fraction with reduced denominator at most 10000; terminating fractions have zero repeating cycle length.
Long division eventually reaches zero or repeats a remainder. Tracking remainders identifies decimal preperiod and cycle lengths.
When to use this tool
Trace the exact base-ten remainder cycle of a fraction with reduced denominator at most 10000; terminating fractions have zero repeating cycle length.
- Prepare the supplied numerator, denominator fields for this operation.
- Compare the complete example with the representation used by your own input before carrying the result into another step.
How to use the tool and how it works
- Enter numerator, denominator. Use the exact text syntax, units and index conventions explained below.
- Run the tool. If input validation fails, correct the supplied fields before interpreting any result.
- Read result.cycleLength, result.nonrepeatingDigits. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.
Reduce the fraction and require denominator at most 10,000. Start with its nonnegative residue, record successive remainders while multiplying by ten modulo denominator, and stop at zero or the first repeated remainder. Return cycleLength and the preceding nonrepeatingDigits count.1
| Input field | Meaning and representation | Accepted input |
|---|---|---|
| numerator | Exact signed decimal integer text; denominator fields must be nonzero. | Optional + or - and 1 to 256 decimal digits |
| denominator | Exact signed decimal integer text; denominator fields must be nonzero. | Optional + or - and 1 to 256 decimal digits |
| Output field | Interpretation |
|---|---|
| result | Object containing the named outputs below. |
| result.cycleLength | The cycle length value under the stated formula or text convention. |
| result.nonrepeatingDigits | The nonrepeating digits value under the stated formula or text convention. |
{
"input": {
"numerator": "1",
"denominator": "6"
},
"output": {
"result": {
"cycleLength": 1,
"nonrepeatingDigits": 1
}
}
}Limitations and assumptions
- This returns lengths rather than digits. Terminating fractions have cycleLength zero; exact integers have both lengths zero.
- Numerator and denominator fields use exact decimal integer strings of at most 256 digits with an optional sign. Fractions are reduced with a positive denominator before the stated operation.
- All declared input fields are required and unexpected top-level fields are rejected. Numbers must be finite; lists are limited to 128 entries and ordinary text fields to 8,192 UTF-16 code units, with tighter limits where stated.
- JSON text inputs must parse within bounded nesting and size limits. Numeric outputs outside the safe finite range are rejected; large exact integer results use decimal strings. Indexes and half-open spans are zero-based unless a rank is explicitly described as one-based.
Alternative or Complementary methods
Use exact numerator/denominator operations when equality and boundary decisions matter. Convert to a decimal display only with an explicit rounding policy; a rounded display should not replace the exact result.
References
Similar or alternative tools
- Exact Fraction Addition
Exact rational addition using decimal integer text and coprime output; no IEEE-754 precision loss.
- Fraction Base Termination Check
Check exact rational termination in an explicit integer radix 2–36 by removing denominator factors shared with that radix.