Euler's Formula

The five most important numbers in mathematics all appear in a single equation!

Presentation Suggestions:
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, 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.

The Math Behind the Fact:
Introduce the "imaginary number" i, a number with the property that i²=-1. Make sure students understand that, say, i⁵=i. Take the Taylor series of e^t and plug "it" in (that's "i*t"). Since e^t 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).

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!

How to Cite this Page:
Su, Francis E., et al. "Euler's Formula." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References: any basic complex analysis text, such as by Ahlfors or Churchill. Keywords:    complex analysis, calculus Subjects:    calculus, analysis Level:    Easy Fun Fact suggested by:   Joshua Sabloff Suggestions? Use this form. 4.10 current rating Click to rate this Fun Fact...     *   Awesome! I totally dig it!     *   Fun enough to tell a friend!     *   Mildly interesting     *   Not really noteworthy and see the most popular Facts! Get the Math Fun Facts iPhone App!

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