Pi is the ratio of the circumference of a circle to its diameter. It is known to be\u00a0irrational\u00a0and its decimal expansion therefore does not terminate or repeat. The first 40 places are:<\/p>\n\n\n\n
3.14159 26535 89793 23846 26433 83279 50288 41971…<\/p>\n\n\n\n
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.<\/p>\n\n\n\n
But an outstanding approximation to Pi is the following:<\/p>\n\n\n\n
355\/113<\/p>\n\n\n\n
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.]<\/p>\n\n\n\n
Presentation Suggestions:<\/strong> The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong> How to Cite this Page:<\/strong> Fun Fact suggested by: <\/strong> Pi is the ratio of the circumference of a circle to its diameter. It is known to be\u00a0irrational\u00a0and its decimal...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[28,3,37,10,30],"class_list":["post-265","page","type-page","status-publish","hentry","tag-approximations","tag-easy","tag-irrational","tag-numtheory","tag-pi-formula"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/265","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=265"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/265\/revisions"}],"predecessor-version":[{"id":1565,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/265\/revisions\/1565"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=265"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=265"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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!<\/p>\n\n\n\n
The theory of\u00a0continued fractions<\/em>\u00a0allows one to find good rational\u00a0approximations\u00a0of any irrational number. This is covered in an introductory course on\u00a0number theory!<\/p>\n\n\n\n
Su, Francis E., et al. “Pi Approximations.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n
Francis Su <\/p>\n","protected":false},"excerpt":{"rendered":"