# Euler's Identity

Euler's Identity

Euler's identity, +1=0, has been called the most beautiful equation in mathematics. It unites the most basic numbers of mathematics: (the base of the natural logarithm), (the imaginary unit = ), (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: =cos(θ)+isin(θ). The series expansion of is with partial sums =.

π

e

i

-1

π

θ

e

z

e

∞

∑

n=0

n

z

n!

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 . You can see how quickly the series expansion of converges by varying the number of terms in its partial sums with . The green polygonal line joins the points , , …, where . 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.

θ

iθ

e

m

S

0

S

1

S

m

z=

iθ

e