Question

Hey there !! I've been trying to find the derivation of e^iπ + 1 = 0 (Euler's theroem) at a lot of web pages. However I'm not able to !
Please someone derive it !
don't use maclurin series or calculus ! I know the derivation through those methods but it's not a proper way to prove it ! thank you please help.

Answer 1

As it is provided not to use Maclaurin series . So , I'm gonna use De Moivre's Theorem , which is as follows :-

• ${ \bf e^{i \theta} = \cos \theta + i \: \sin \theta}$

Put , $\sf \theta = \pi$ . So , that ;

${ : \implies \quad \sf e^{i \pi} = \cos ( \pi ) + i \: \sin ( \pi )}$

${ : \implies \quad \sf e^{i \pi} = - 1 + i \times 0}$

${ : \implies \quad \sf e^{i \pi} = - 1 + 0}$

${ : \implies \quad { \pmb { \orange { \bf { \underline { \underline { \therefore \quad e^{i \pi} + 1 =0}}}}}}}$

Henceforth , Proved The Euler's Identity !!!!

Used Concepts :-

• $\bf \sin ( \pi ) = 0$

• $\bf \cos ( \pi ) = - 1$

• De Moivre Theorem

Answer 2

Step-by-step explanation:

Well, there is the isomorphism between the rotation group in two dimensions, SO(2), and the group of unitary 1x1 matrices, U(1), also known as the multiplicative group of complex numbers of absolute value 1. But that, at best, is again a mathematical explanation (which I used in the interpretation above). I am sure there are many other interesting ways to look at this, but for the moment I cannot think of any.