{"id":305,"date":"2019-06-26T23:31:34","date_gmt":"2019-06-26T23:31:34","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=305"},"modified":"2019-10-18T21:44:19","modified_gmt":"2019-10-18T21:44:19","slug":"eulers-formula","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/eulers-formula\/","title":{"rendered":"Euler’s Formula"},"content":{"rendered":"\n

The five most important numbers in mathematics all appear in a single equation!
e<\/em>i<\/em>*Pi<\/sup> + 1 = 0.
In fact, this is a special case of the following formula, due to Euler:
e<\/em>i<\/em>t<\/sup> = cos(t) + i<\/em>*sin(t).<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
This is a good Fun Fact to use after introducing complex numbers, as it gives some intuition about polar coordinates on C. However, a more interesting use is after teaching the Taylor series of e<\/em>, sin , and cos. See below. You could do the following in class or on a Taylor series homework and then give the Fun Fact as the case where you set t=Pi.<\/p>\n\n\n\n

The Math Behind the Fact:<\/strong>
Introduce the “imaginary number” i<\/em>, a number with the property that i<\/em>2<\/sup>=-1. Make sure students understand that, say, i<\/em>5<\/sup>=i<\/em>. Take the Taylor series of e<\/em>t<\/sup> and plug “it” in (that’s “i<\/em>*t”). Since e<\/em>t<\/sup> converges absolutely everywhere, have them rearrange the resulting series into two series: one with an i in each term, and one with no i’s. What are these two series? Yes, cos(t) and i*sin(t).<\/p>\n\n\n\n

This formula demonstrates a remarkable connection between analysis (in the form of the Taylor series of e, sin, and cos) and geometry (the polar coordinates in C). Heck, it’s a remarkable connection between e, sin and cos!<\/p>\n\n\n\n

How to Cite this Page:<\/strong> 
Su, Francis E., et al. “Euler’s Formula.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

References:<\/strong>
Any basic complex analysis text, such as by Ahlfors or Churchill.<\/p>\n\n\n\n

Fun Fact suggested by:<\/strong>
Joshua Sabloff <\/p>\n","protected":false},"excerpt":{"rendered":"

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