Calculator Inputs
Enter a fraction and a base. The tool detects whether the expansion terminates or repeats, then identifies the preperiod and the recurring cycle.
Example Data Table
| Fraction | Base | Expansion | Preperiod Length | Period Length | Status |
|---|---|---|---|---|---|
| 1/2 | 10 | 0.5 | 1 | 0 | Terminating |
| 1/3 | 10 | 0.(3) | 0 | 1 | Repeating |
| 7/12 | 10 | 0.58(3) | 2 | 1 | Repeating |
| 5/28 | 10 | 0.17(857142) | 2 | 6 | Repeating |
| 1/8 | 2 | 0.001 | 3 | 0 | Terminating |
Formula Used
Step recurrence: rk+1 = (b × rk) mod d
Digit extraction: ak = floor((b × rk) / d)
Cycle rule: when a remainder repeats, the decimal block between those two appearances is the period.
First reduce the fraction to lowest terms. Then divide in the chosen base using repeated remainders. If the remainder reaches zero, the expansion terminates. If the same remainder appears again, digits from its first appearance to its repeat form the recurring cycle.
The base-factor test offers a quick check. A reduced denominator terminates only when all its prime factors also divide the chosen base. In base 10, only powers of 2 and 5 terminate.
How to Use This Calculator
- Enter the numerator and denominator for your fraction.
- Select the base you want to analyze, from 2 through 16.
- Set the maximum number of digits to inspect for long cycles.
- Press Find Period to generate the result summary above the form.
- Review the expansion, preperiod length, period length, and remainder steps.
- Use the CSV or PDF buttons to save the output.
FAQs
1. What does period mean here?
The period is the repeating block of digits in a fraction’s base expansion. For 1/3 in base 10, the period is 3 because that single digit repeats forever.
2. What is the preperiod?
The preperiod is the non-repeating part that appears before the repeating block starts. In 7/12 = 0.58(3), the digits 5 and 8 form the preperiod.
3. Why do some fractions terminate?
A reduced fraction terminates when the denominator’s prime factors are all included in the selected base. In base 10, denominators using only 2s and 5s terminate.
4. Why can the base change the answer?
Different bases change both the extracted digits and the factor test. A fraction may terminate in one base and repeat in another because the denominator interacts differently with the base.
5. Does simplifying the fraction matter?
Yes. Simplifying removes common factors first, which reveals the true denominator structure. Period length and termination behavior should always be checked on the reduced fraction.
6. What does max digits do?
It sets the upper limit for the long-division search. Most fractions finish much sooner, but larger limits help when the repeating cycle is long.
7. Can this handle bases above 10?
Yes. The calculator supports bases 2 through 16. For bases above 10, digits greater than 9 are displayed with letters, such as A through F.
8. What does the chart show?
The chart compares preperiod length, period length, and generated digits. When a repeating block exists, it also shows the frequency of each digit inside that block.