Continued Fraction

Expand a number and its best rational approximations

[3; 7, 15, 1, 292, 1, 1, 1, …]
Continued fraction
[3; 7, 15, 1, 292, 1, 1, 1, …]
As a fraction
3141592653589793 / 1000000000000000
Value
3.1415926536
Convergents (best rational approximations)
kpₖ / qₖDecimalError
03/130.142
122/73.14285714290.00126
2333/1063.1415094340.0000832
3355/1133.14159292040.000000267
4103993/331023.1415926530.000000000578
5104348/332153.14159265390.000000000332
6208341/663173.14159265350.000000000122
7312689/995323.14159265360.0000000000291
Truncated: the number has more terms than shown. Raise the term count to see more.
Convergence of rational approximationsPlot of each convergent pₖ/qₖ against its index, oscillating toward the dashed line at the input value as the approximations improve.3.1415933/122/7333/106355/113103993/33102104348/33215208341/66317312689/9953201234567convergent index k