Continued Fraction Convergents Calculator
Enter the continued-fraction terms to list its convergents.
Result
How to use
- Enter your values in the fields above.
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- Use the Share button to copy a link to your result.
About this calculator
A continued fraction [a₀; a₁, a₂, a₃, …] represents a number as a nested sequence of integer terms, and its convergents are the successive fractions obtained by truncating that sequence at each step — each one the best possible rational approximation of the target value for its denominator size. Convergents are computed with the recurrence h_n = a_n·h_{n−1} + h_{n−2} and k_n = a_n·k_{n−1} + k_{n−2}, giving the convergent as h_n/k_n.
Mathematicians and number theorists use continued fraction convergents to find compact rational approximations of irrational constants like π or √2, and the same technique shows up in practical engineering problems such as choosing gear-tooth counts or musical tuning ratios that closely approximate an irrational target with small whole numbers.
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