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Euler’s identity

$e^{i\pi }+1=0$

Geometric interpretation

Any complex number $z=x+iy$ can be represented by the point $(x,y)$ on the complex plane. This point can also be represented in polar coordinates as $(r,\theta )$, where $r$ is the absolute value of $z$ (distance from the origin), and $\theta$ is the argument of $z$ (angle counterclockwise from the positive x-axis). By the definitions of sine and cosine, this point has cartesian coordinates of $(r\cos \theta ,r\sin \theta )$, implying that $z=r(\cos \theta +i\sin \theta )$. According to Euler’s formula, this is equivalent to saying $z=re^{i\theta}$.

Euler’s identity says that $-1=e^{i\pi }$. Since $e^{i\pi }$ is $re^{i\theta }$ for $r$ = 1 and $\theta =\pi$ , this can be interpreted as a fact about the number −1 on the complex plane: its distance from the origin is 1, and its angle from the positive x-axis is $\pi$ radians.

Euler’s identity - Wikipedia