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 number theory!

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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