Approximating $\pi$ with least digits Do you a digit efficient way to approximate $\pi$? I mean representing many digits of $\pi$ using only a few numeric digits and some sort of equation. Maybe mathematical operations also count as penalty.
For example the well known $\frac{355}{113}$ is an approximation, but it gives only 7 correct digits by using 6 digits (113355) in the approximation itself. Can you make a better digit ratio?
EDIT: to clarify the "game" let's assume that each mathematical operation (+, sqrt, power, ...) also counts as one digit. Otherwise one could of course make artifical infinitely nested structures of operations only. And preferably let's stick to basic arithmetics and powers/roots only.
EDIT: true. logarithm of imaginary numbers provides an easy way. let's not use complex numbers since that's what I had in mind. something you can present to non-mathematicians :)

Here is a site that focuses on numerical computation of rational approximations of $\pi$:
http://www.isi.edu/~johnh/BLOG/1999/0728_RATIONAL_PI/
Also, using a truncated form of the continued fraction will give nice approximations.  The first few fractions given are $3$, $22\over7$, $333\over106$, $355\over113$, $103993 \over33102$ and $104348\over33215$.  These numerators and denominators are given by the OEIS sequences A002485 and A002486 respectively.
You may be interested in this page.  It states that $355 \over 113$ is the "best" (efficiency-wise) rational approximation with the denominator less then 30,000.
