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 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 !

How to Cite this Page: 
Su, Francis E., et al. “Pi Approximations.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

Fun Fact suggested by:
Francis Su

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