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 = π⋅i , 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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