Wasan Pi Series Calculator
Approximate pi using the accelerated series inspired by the Japanese wasan mathematician Takebe Katahiro. Enter a term count to see the approximation and its error.
Input
Enter a number of terms to approximate pi using the accelerated series inspired by the wasan mathematician Takebe Katahiro. It uses the fact that arcsin of one half equals pi over six for a unit circle.
The term count is an integer from 1 to 60. More terms bring the approximation closer to true pi.
Result
Pi approximation using 6 terms
3.14157121469
Error versus true pi
0.00002143889965
Terms used
6
True pi (reference)
3.14159265359
Convergence by term
As each term is added, the partial sum approaches the square of pi over six and the approximation converges to true pi. Up to the first 30 terms are shown.
| Term | Term value | Pi approximation | Error |
|---|---|---|---|
| 1 | 0.25 | 3 | 0.14159265358979 |
| 2 | 0.020833333333 | 3.122498999199 | 0.01909365439059 |
| 3 | 0.002777777778 | 3.138470965295 | 0.00312168829475 |
| 4 | 0.000446428571 | 3.141030313221 | 0.00056234036908 |
| 5 | 0.000079365079 | 3.141485090117 | 0.00010756347261 |
| 6 | 0.000015031265 | 3.14157121469 | 0.00002143889965 |
The series expands the square of arcsin t as a power series with t set to one half. Each coefficient comes from the previous one by recurrence, and the approximation is pi about six times the square root of the partial sum S. Each term shrinks by roughly one quarter so it converges quickly.
How it works
- This calculation uses the fact that arcsin of one half equals pi divided by six for a unit circle. Takebe Katahiro expanded the square of arcsin as a power series to obtain a high precision approximation of pi.
- Each term of the series is derived from the previous one by a simple recurrence, and the partial sum S converges toward the square of pi over six. The approximation is recovered as pi is about six times the square root of S.
- Each term shrinks by roughly a factor of one quarter, so the series converges far faster than the plain Leibniz series. This reflects the acceleration techniques developed by Edo era wasan.
- Wasan, the mathematics of Seki Takakazu and Takebe Katahiro, was a uniquely Japanese tradition that studied pi and series acceleration independently. This tool reconstructs that idea in an original form for educational approximation.
- The more terms you add, the faster the error against true pi shrinks. The table lets you watch how the approximation and error change term by term.
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Wasan Pi Series Calculator