# Mathematical beauty of $e^{i\pi}+1=0$ [closed]

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?

## closed as primarily opinion-based by JMoravitz, copper.hat, I am Back, user223391, Hans LundmarkOct 2 '17 at 15:23

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

• If so why not $e^{2\cdot3\cdot5...p_{2017}\pi i}-1=0$? – Michael Rozenberg Oct 2 '17 at 14:53
• I don't think the prime-ness of 2 has anything to do with why the identity is true, so in my mind it doesn't add much content to think of it that way. – Randall Oct 2 '17 at 14:53
• Hmm ... good point but I would guess that it is too complicated and long :D. – MrYouMath Oct 2 '17 at 14:53
• Personally, I'm still wondering why they don't present it as $e^{i\alpha}=\cos\alpha+i\sin\alpha$ from the beginning: "Tell me the useful stuff right away, please." – user228113 Oct 2 '17 at 15:03
• @Sassatelli: A major drawback is that $\pi$ is missing. – MrYouMath Oct 2 '17 at 15:04

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 $-1$).

• Very good point. – MrYouMath Oct 2 '17 at 15:04
• Then $e^{i\theta}=\cos\theta + i\sin\theta$ tells much more than that identity – Isham Oct 2 '17 at 15:13
• @Isham: Yes it does but it is more complicated and not so compact. In the original form, every letter has a deep meaning (remember that this is just a subjective opinion :D). – MrYouMath Oct 2 '17 at 15:15
• @Isham absolutely, your formula gives much more information, and many people will prefer it for that reason. Conversely, $e^{i\pi}+1=0$ is simpler, and many people will prefer it for that reason. But between $e^{i\pi}+1=0$ and $e^{2i\pi}-1=0$ there's no contest: the latter has less information without the benefit of being any simpler. – Especially Lime Oct 2 '17 at 15:46
• @EspeciallyLime: But the second one has the advantage of the first prime number which happens to be the only even prime number :). – MrYouMath Oct 2 '17 at 17:19

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.

• Very good point. – MrYouMath Oct 2 '17 at 15:05

Looking at the answers and comments I feel compelled to offer this as a compromise,

# $\frac {e^{i \pi} + e^{-i \pi}}{2} = -1$

• +1: For the idea. But too much redundancy in my opinion :). And the fraction line is also not very nice as the $2$ looks so lonely down there :D. – MrYouMath Oct 2 '17 at 15:20
• I like dividing by $2$ - I was averaging to get a compromise! :) – CopyPasteIt Oct 2 '17 at 15:43

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.

• It is not only about the subjective sense of beauty it is also about the meaning that such an identity carries. E.g. the point of Especially Lime is a very good one as it points out that the first one caries more information on it. – MrYouMath Oct 2 '17 at 15:24
• Btw I personally don't like $\tau$ because it is used in many other contexts and the area of a circle would have such an ugly $1/2$ in it :D. But $\pi$ is in general dominantly used for our beloved $\tau$. – MrYouMath Oct 2 '17 at 15:28
• Taking these factors into account seems make the thing a little complex. I can fully understand that a more informative formula is more beautiful, or that the one including lesser but all meaningful symbols is better. However, what should we do if a trade-off occurs? e.g. A more informative one with $5$ symbols vs. a less informative one with $4$ symbols. – Lwins Oct 2 '17 at 15:31
• As this topic is mainly based on opinion there will never be a final answer. But for your last equation, I would say that it introduces the concept of roots (which is positive). But has a redundancy of two symbols $2$ and $i$, hence I (personally) don't think the added complexity does add some value to the identity. – MrYouMath Oct 2 '17 at 15:39
• @MrYouMath I respect your opinion. – Lwins Oct 2 '17 at 15:41