Euler's Identity

θ

1.26

m

6

scale

Euler's identity,

π



+1=0

, has been called the most beautiful equation in mathematics. It unites the most basic numbers of mathematics:

e

(the base of the natural logarithm),

i

(the imaginary unit =

-1

),

π

(the ratio of the circumference of a circle to its diameter), 1 (the multiplicative identity), and 0 (the additive identity) with the basic arithmetic operations: addition, multiplication and exponentiation—using each once and only once! The identity is a particular example of the more general Euler formula:

θ

e

=cos(θ)+isin(θ)

. The series expansion of

z

e

is

∞

∑

n=0

n

z

n!

with partial sums

S

m

=

m

∑

n=0

n

z

n!

.

This Demonstration starts with Euler's identity but lets you experiment with the more general formula for various values of

θ

. You can see how quickly the series expansion of

iθ

e

converges by varying the number of terms in its partial sums with

m

. The green polygonal line joins the points

S

0

,

S

1

, …,

S

m

where

z=

iθ

e

. You can zoom in with the scale, which is logarithmic. When you zoom in sufficiently close, the center of the picture switches from the origin to the point being approximated.