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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?

MrYouMath
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  • If so why not $e^{2\cdot3\cdot5...p_{2017}\pi i}-1=0$? – Michael Rozenberg Oct 02 '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 02 '17 at 14:53
  • Hmm ... good point but I would guess that it is too complicated and long :D. – MrYouMath Oct 02 '17 at 14:53
  • Also, yours is a consequence of the original (square both sides), so that may be a reason to advocate for the original. – Randall Oct 02 '17 at 14:54
  • I like $e^{-i\pi}+1=0$ better. Beauty is in the eye of the beholder. – copper.hat Oct 02 '17 at 14:54
  • @copper.hat: Why do you like this form better? So I am not the only one that is having a different number one :). – MrYouMath Oct 02 '17 at 14:55
  • @MrYouMath: More symbols :-). – copper.hat Oct 02 '17 at 14:57
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    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." –  Oct 02 '17 at 15:03
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    @Sassatelli: A major drawback is that $\pi$ is missing. – MrYouMath Oct 02 '17 at 15:04
  • @MrYouMath How is it missing? Let $\alpha = \pi$. – Randall Oct 02 '17 at 15:07
  • @Randall: Yes you have to do something, but $\alpha$ is not so interesting as $\pi$ (remember we are talking about subjective sense of beauty). – MrYouMath Oct 02 '17 at 15:13
  • @MrYouMath okay, then I guess as my own personal preference in beauty, I prefer equations involving functions (multiple bits of information) over solitary constants (only one bit of information). – Randall Oct 02 '17 at 15:22
  • This is related to the pi vs. tau debate. Some people consider $e^{i\tau}=1$ to be the most elegant version of this identity. – littleO Oct 02 '17 at 16:37

4 Answers4

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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 $-1$).

Especially Lime
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  • Very good point. – MrYouMath Oct 02 '17 at 15:04
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    Then $ e^{i\theta}=\cos\theta + i\sin\theta $ tells much more than that identity – user577215664 Oct 02 '17 at 15:13
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    @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 02 '17 at 15:15
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    @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 02 '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 02 '17 at 17:19
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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.

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

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

Community
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CopyPasteIt
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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.

Lwins
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  • 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 02 '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 02 '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 02 '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 02 '17 at 15:39
  • @MrYouMath I respect your opinion. – Lwins Oct 02 '17 at 15:41