Euler's Identity - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim

Euler's Identity

Main Concept

Euler's identity is the famous equality e i π + 1 = 0, where:


e is Euler's number ≈ 2.718


i is the imaginary number; i 2= 1


This is a special case of Euler's formula: e i x = cosx + isinx, where x = π:

e i π = cosπ + isinπ

e i π = 1 + i0

e i π + 1 = 0 

Visually, this identity can be defined as the limit of the function 1 +i πnn as n approaches infinity. More generally, e z can be defined as the limit of  1 +znn as n approaches infinity.


For a given value of z, the plot below shows the value of 1 +znn as n increases to infinity, as well as the sequence of line segments from 1 +znk to 1 +znk+1. Each additional line segment represents an additional multiplication by 1 +zn. For z = πi , it can be seen that the point approaches 1.

Click Play/Stop to start or stop the animation or use the slider to adjust the frames manually. Choose a different value of z to see how the plot is affected. Use the controls to adjust the view of the plot.


z =

More MathApps