Continued Fraction and Convergent Approximations
Expand a real number into a regular continued fraction and list every fraction approximation (convergent p/q) until a target accuracy or maximum denominator is met. Expressions such as pi or the golden ratio are supported.
Input
Example values (tap to fill in)
Stopping condition
Operators: + - * / ^ (power). Functions: sin cos tan exp log (natural log) ln log10 sqrt cbrt abs. Constants: pi (π), e. You may also enter plain integers or decimals.
Result
Best fraction approximation of pi ≒ 3.141592654
355 / 113
= 3.14159292 (Reached target accuracy)
Absolute error
2.668e-7
Denominator q
113
Terms (number of coefficients)
4
Continued fraction notation
List of fraction approximations (convergents)
| Term k | Coefficient a_k | Fraction p/q | Value | Absolute error |
|---|---|---|---|---|
| 0 | 3 | 3 / 1 | 3 | 0.141592654 |
| 1 | 7 | 22 / 7 | 3.14285714 | 0.001264489 |
| 2 | 15 | 333 / 106 | 3.14150943 | 8.322e-5 |
| 3 | 1 | 355 / 113 | 3.14159292 | 2.668e-7 |
How it works
- Enter a real value and the tool repeatedly extracts the integer part and the reciprocal to find each coefficient of the regular continued fraction, computing the fraction approximations p/q one by one.
- Choose the stopping condition as either a target accuracy (absolute error) or a maximum denominator. With accuracy, it stops once the error falls to that value or below; with a denominator limit, it shows the best fraction that does not exceed the given upper bound.
- The value can also be an expression. Use pi for the circle constant, e for the base of the natural logarithm, and forms such as sqrt(2) or (1+sqrt(5))/2 are evaluated directly. Plain integers and decimals are fine too.
- The result table lists, for each term, the coefficient a_k, the fraction approximation p/q, its value, and the absolute error from the target. Lower rows have smaller errors, and the row of the chosen best fraction is highlighted.
- For pi you can watch the accuracy jump as the denominator grows slightly, through 22/7, 333/106, and 355/113. For the golden ratio the numerator and denominator are consecutive Fibonacci numbers.
- If you enter a rational number, the expansion terminates when it divides evenly, and that fraction is shown as the exact value.
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Continued Fraction and Convergent Approximations