Continued Fraction Approximation Tool
Generate convergents from a continued fraction expansion to approximate irrationals or long decimals with manageable numerators and denominators.
Examples
- π with max denominator 500 ⇒ Result: 355/113 (≈ 3.141592920)
- √2 with max denominator 70 ⇒ Result: 99/70 (≈ 1.414285714)
FAQ
What happens when the denominator limit is reached?
The calculator scales the last stable convergent so the denominator stays within your limit, returning the closest achievable fraction.
Can I use integers?
Yes. Integers simply return themselves with denominator 1 because the expansion terminates immediately.
How accurate is the approximation?
The decimal shown is the convergent divided out to nine decimal places. For error bounds, compare the result with your original value manually.
Additional Information
- Outputs the convergent as numerator/denominator alongside its decimal form.
- If the denominator limit stops the expansion early, the tool adjusts the last convergent proportionally.
- Use a higher iteration cap to explore deeper convergents when the default 12 levels are insufficient.
- Continued fractions yield best-possible rational approximations for a given denominator bound.