Euler’s Formula

The five most important numbers in mathematics all appear in a single equation!
e^i*Pi + 1 = 0.
In fact, this is a special case of the following formula, due to Euler:
e^it = 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 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. <https://www.math.hmc.edu/funfacts>.

References:
Any basic complex analysis text, such as by Ahlfors or Churchill.

Fun Fact suggested by:
Joshua Sabloff