Is there a way to prove Euler’s Identity without using trigonometry?

[RZ923]

Is there a way to prove Euler’s Identity, $e^{πi} + 1 = 0$, without using trigonometry?

[nastya]

Hi @RZ923!

To begin with, Euler’s Identity ($e^{i\pi}+1=0$) is directly related to complex numbers - a topic that is much broader than just trigonometry. These two topics are connected and can be used almost everywhere! The simplest and most common interpretation of complex numbers is a geometric one: vectors in a coordinate system. This interpretation is directly connected with trigonometry: its general form is Euler's formula: $e^{ix}=\text{cos }x + i\cdot\text{sin }x$ (Euler’s Identity is Euler's formula for $x=\pi$).

Euler’s Identity is not just a simple, short, and nice identity that comes from nowhere - it is a huge topic that covers lots of beautiful and interesting information and takes years to find out. Everything is connected here, with trigonometry especially.

[RZ923]

@nastya Thanks