Euler's Identity - Maple Help

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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 =

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