Decimal to Fraction Calculator
Convert a decimal into a fraction in simplest form, with the mixed number form shown too.
Last updated · View algorithm, constants and default inputs (JSON)
About this calculator
This converter approximates a decimal with a reduced fraction using continued fractions. The denominator is capped at 10,000,000 and a numeric tolerance is used, so some results are approximations rather than exact identities.
Worked example
calculated with the default values below
Decimal 0.375 Fraction 3/8 · Numerator 3 · Denominator 8 · Whole part 0 · Remainder numerator 3
General background references
Disclaimer
These calculators give general mathematical results. They suit study, planning and estimation — verify anything that feeds a graded submission, a published figure or a financial commitment.
Implementation source
This record describes the implementation used by this page. It is not independent validation. External references, when available, support only the stated formula or definition; estimates and custom rules retain their own assumptions.
Formula and variables
a_j = ⌊b_j⌋; b_{j+1} = 1 ÷ (b_j − a_j); h_j = a_j h_{j−1} + h_{j−2}
- x
- Input decimal (b₀)
- b_j
- Value at step j (b₀ = x)
- a_j
- Continued-fraction term
- h_j
- Numerator convergent
- k_j
- Denominator convergent
Start b₀ = x with h₋₁=1, h₀=0, k₋₁=0, k₀=1 (same recurrence for k). Stop when k_j > 10⁷, remainder < 10⁻¹², or after 60 terms. Fraction = h_j/k_j; mixed form splits whole and remainder.
Frequently asked questions
Why does the decimal-to-fraction converter use continued fractions?
The continued-fraction method finds a reduced fraction within a numeric tolerance, with a denominator cap of 10,000,000. Values such as 0.375 resolve exactly to 3/8, but a finite decimal input cannot reveal an intended repeating pattern; other results may be approximations.