Decimal to Continued Fraction Calculator
Enter a decimal number to expand it as a continued fraction.
Result
How to use
- Enter your values in the fields above.
- Press Calculate to see your result instantly.
- Use the Share button to copy a link to your result.
About this calculator
A continued fraction writes a number in the form [a0; a1, a2, a3, …], built by repeatedly taking the integer part of a value and then inverting and continuing with what's left over, a process that, for a rational decimal, eventually terminates, and for an irrational number like π continues indefinitely, with each additional term giving a better rational approximation of the original value.
This representation has real uses beyond pure curiosity: continued fractions produce the best possible rational approximations of a number for a given denominator size, which is exactly how the ratio 355/113 was found as a remarkably accurate approximation of π, and the same technique underlies historical calendar reforms (finding simple fractions close to the length of a solar year) and musical tuning systems (approximating frequency ratios with small integers).
Enter a decimal number and choose how many terms to expand, and this calculator shows its continued fraction representation [a0; a1, a2, …].
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