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 :)
 A: With pure rational approximations, there are sharp limits (related to Roth's Theorem) in terms of how far you can go.
More generally, this will depend strongly on the operations allowed. For example,
$$
\log(0-1)/\sqrt{0-1}
$$
has 5 operations and 4 digits and is exact.
A: Let me throw in Clive's suggestion to look at the wikipedia site. If we allow for logarithm (while not using complex numbers), we can get 30 digits of $\pi$ with
$\frac{\operatorname{ln}(640320^3+744)}{\sqrt{163}}$
which is 13 digits and 5 operation, giving a ratio of about 18/30=0.6.
EDIT: Here is another one I found on this site:
$\ln(31.8\ln(2)+\ln(3))$
gives 11 digits of $\pi$ with using 5 numbers and 4 operations.
A: 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.
A: Apparently, I'm not the first to ask ->
http://www.contestcen.com/pi.htm
However, they often don't count operations. Surely, but defining the set of allow operations (+, -, *, /, sqrt, pow, ...?) one could set up an information "entropy" measure to make it a fair game :)
A: (2143/22)^(1/4) = 3.14159265258
ignoring the parentheses & "^" since they don't get written when written with indices and grouping that's 9 digits/operations to get 9 digits. so 1:1 but I was happy I found it nonetheless
