Here is a very interesting formula for pi, discovered by David Bailey, Peter Borwein, and Simon Plouffe in 1995:

Pi = SUM_{k=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 16^{N}. We are interested in the fractional part of this expression. The numerator of a given term in this sum is 16^{N-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