Finding the N-th digit of Pi

Here is a very interesting formula for pi, discovered by David Bailey, Peter Borwein, and Simon Plouffe in 1995:
Pi = SUMk=0 to infinity 16-k [ 4/(8k+1) – 2/(8k+4) – 1/(8k+5) – 1/(8k+6) ].
The reason this pi formula is so interesting is because it can be used to calculate the N-th digit of Pi (in base 16) without having to calculate all of the previous digits!

Moreover, one can even do the calculation in a time that is essentially linear in N, with memory requirements only logarithmic in N. This is far better than previous algorithms for finding the N-th digit of Pi, which required keeping track of all the previous digits!

Presentation Suggestions:
You might start off by asking students how they might calculate the 100-th digit of pi using one of the other pi formulas they have learned. Then show them this one…

The Math Behind the Fact:
Here’s a sketch of how the BBP formula can be used to find the N-th hexadecimal digit of Pi. For simplicity, consider just the first of the sums in the expression, and multiply this by 16N. We are interested in the fractional part of this expression. The numerator of a given term in this sum is 16N-k, and it can be evaluated very easily mod (8k+1) using a binary algorithm for exponentiation. Division by (8k+1) is straightforward via floating point arithmetic. Not many more than N terms of this sum need be evaluated, since the numerator decreases very quickly as k gets large so that terms become negligible. The other sums in the BBP formula are handled similarly. This yields the hexadecimal expansion of Pi starting at the (N+1)-th digit. More details can be found in the Bailey-Borwein-Plouffe reference.

The BBP formula was discovered using the PSLQ Integer Relation Algorithm. However, the Adamchik-Wagon reference shows how similar relations can be discovered in a way that the proof accompanies the discovery, and gives a 3-term formula for a base 4 analogue of the BBP result.

How to Cite this Page: 
Su, Francis E., et al. “Finding the N-th digit of Pi.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
David Bailey, Peter Borwein, and Simon Plouffe. “On the rapid computation of various polylogarithmic constants”, Math. Comp. 66(1997), 903-913.

Victor Adamchik and Stan Wagon, “A simple formula for pi”, Amer. Math. Monthly 104(1997), 852-855.

Fun Fact suggested by:
Arthur Benjamin

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