Mathematical beauty of $e^{i\pi}+1=0$ I know this is a soft and opinion based question and I risk that this question get's closed/downvoted  but I still wanted to know what other persons, who are interested in mathematics, think about my question.
Whenever people are talking about the most beautiful equation/identity Euler's identity is cited in this fashion:
$$e^{i\pi}+1=0.$$
While I would agree that this is a beautiful identity (see my avatar) I personally always wondered why not 
$$e^{2i\pi}-1 = 0$$
is the most beautiful identity. It has $e$, $i$, $\pi$, $0$ and the number $2$ in it. I prefer it because the number $2$ is the first and at the same time the only even odd prime number. Having the prime numbers, which are in some way the atoms of mathematics, included makes this formula even more pleasant for me. The minus sign seems a little bit "negative" but the good part is that it is displaying the principle of inversion.

So my question is, why is this not the form in which it is most often
  presented?

 A: The main reason is simply that the standard version gives more information. If I know $e^{i\pi}=-1$ then I can deduce $e^{2i\pi}=1$, but not the other way round (knowing $e^{2i\pi}=1$ doesn't tell me whether $e^{i\pi}$ is $+1$ or $-1$).
A: Well said by @copperhat: Beauty lies in the eye of beholder.
I like the form $e^{9i\pi}+1=0$ as $9$ is the first odd composite number. People have different tastes and you cannot force someone to like apples if you do like them. 
One problem I can think of with $e^{2i\pi}-1 = 0$ is that you can sqare both sides of its more elementary counterpart $e^{i\pi}=-1 $ and get you result.
A: Looking at the answers and comments I feel compelled to offer this as a compromise,
$\frac {e^{i \pi} + e^{-i \pi}}{2} = -1$
A: In my opinion, you definitely have authority to define what is beautiful and what is not to yourself. And your post let me think of a constant $\tau = 2\pi$ (see https://tauday.com/), which is thought by some people as a more "beautiful" and more "natural" one rather than $\pi$ since we have seen lots of formulas including $2\pi$.
I personally think that we should be tolerant to different perception of beauty. If you think
$$ e^{\pi i} + 1 = 0 $$
is the greatest, it is fine. For those people who consider
$$ e^{2 \pi i} - 1 = 0$$
as the most fascinating I would say it is totally OK. And in case a person insists that
$$ \sqrt{2} e^{\pi i/2} = 1+i $$
is the best (since it can imply the above two equations) I would not refute because it is more like a personal preference.
