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 i^{2}=1.
Make sure students understand that,
say, i^{5}=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>.

