Pi Approximations -- Math Fun Facts
From the Fun Fact files, here is a Fun Fact at the Easy level:
Pi Approximations
Pi is the ratio of the circumference of a circle to   its diameter.  It is known to be irrational and its   decimal expansion therefore does not terminate or repeat.   The first 40 places are:
3.14159 26535 89793 23846 26433 83279 50288 41971...
Thus, it is sometimes helpful to have good   fractional approximations to Pi.   Most people know and use 22/7, since 7*Pi   is pretty close to 22.  But 22/7 is only good   to 2 places.  A fraction with a larger denominator   offers a better chance of getting a more refined estimate.   There is also 333/106, which is good to 5 places.
But an outstanding approximation to Pi is the following:
355/113
This fraction is good to 6 places!  In fact, there is   no "better approximation" among all fractions (P/Q) with   denominators less than 30,000.  [By "better approximation"   we mean in the sense of how close Q*Pi is to P.]
Presentation Suggestions:
Have people verify that 355/113 is a good rational   approximation.  You can also point out that 355/113 is   very easy to remember, since it consists of the digits   113355 in some order!
The Math Behind the Fact:
The theory of continued fractions allows one to   find good rational approximations of any irrational number.   This is covered in an introductory course on number theory !
How to Cite this Page:
Su, Francis E., et al.  "Pi Approximations." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.