Take your calculator, and enter a number consisting of several 5’s, such as 5555555.\u00a0
Now take the\u00a0reciprocal\u00a0of this number.\u00a0
Now take the sine of this result (in degrees).<\/p>\n\n\n\n
You should get a very interesting number:<\/p>\n\n\n\n
3.14159… x 10-9<\/sup>.<\/p>\n\n\n\n Are you surprised?<\/p>\n\n\n\n Presentation Suggestions:<\/strong> The Math Behind the Fact:<\/strong> Now 1\/55555…5 is a number that is approximately 1.8 x 10-n<\/sup>\u00a0where n is the number of 5’s. To convert from degrees to radians we multiply by Pi and divide by 180. So then the result is a number close to Pi x 10-(n+2)<\/sup>, a very small number in radians, and the small angle\u00a0approximation\u00a0holds, so taking the sine of this number does not change it very much.<\/p>\n\n\n\n How to Cite this Page:<\/strong> Fun Fact suggested by: <\/strong> Take your calculator, and enter a number consisting of several 5’s, such as 5555555.\u00a0Now take the\u00a0reciprocal\u00a0of this number.\u00a0Now take the...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[3,130,8,12],"class_list":["post-329","page","type-page","status-publish","hentry","tag-easy","tag-functions","tag-geometry","tag-other"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/329","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=329"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/329\/revisions"}],"predecessor-version":[{"id":1625,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/329\/revisions\/1625"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=329"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=329"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
The result is quite striking. (If you did not get a surprising result, be sure your calculator is set to take trig functions in “degrees”.) Try it for other numbers consisting of only 5’s. Challenge yourself to figure out why it is true before looking at the explanation below!<\/p>\n\n\n\n
You will find the number in your calculator looks like Pi, scaled by some power of 10. This result depends on the degrees-radians conversion and the “small angle approximation”, which says that sin(A) is approximately A (when A is measured in radians). This approximation is used often in physics and engineering.<\/p>\n\n\n\n
Su, Francis E., et al. “Sine of (1\/55555….).” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n
Dan Kalman <\/p>\n","protected":false},"excerpt":{"rendered":"