There’re totally two kinds of Euler’s formula:
The 1st one:
a topological invariance——F + V = E + 2, where F is the number of faces, V the number of vertices, and E the number of edges.
A cube, for example, has 6 faces, 8 vertices, and 12 edges, and satisfies this formula.
The 2nd one:
The formula is still valid if x is a complex number.
The original proof is based on the Taylor series expansions of e^x and of sin x and cos x for real numbers x:
The three equations above yields what was shown in figure1.
Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates.
PS: Complex plane can be considered as a modified Cartesian plane.