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© 1999-2010 by Francis Edward Su

From the Fun Fact files, here is a Fun Fact at the Easy level:

# Euler's Formula

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

ei*Pi + 1 = 0.

In fact, this is a special case of the following formula, due to Euler:

eit = cos(t) + i*sin(t).

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 et and plug "it" in (that's "i*t"). Since et 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!

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.
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