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Comment by DeathArrow | Hacker News Reader

I find this beautiful:

e^iπ = −1

The more general formula (e^ix = cosx + i*sinx) looks better to me because it defines exponentiation of a complex number as a rotation around a unit circle. It has a nice proof, some cool visualisations and a lot of implications to a bunch of other things in mathematics - I can get behind calling that beautiful.

The special case of x=pi... it's like being excited that sin(pi)=0 or cos(pi)=-1. It doesn't really say anything meaningful or consequential, people like it only because of the symbols it includes. It feels kind of like a math meme that people like to repeat and I can't get behind it.

Maybe it's just not for me and I should just let other people like what they like.

I feel this way as well. In fact every time I have tried to remember the "most beautiful equation" I had to think of it in the context of the unit circle and work it out by assigning pi to x. Otherwise I don't get any wow out of it.

It doesn't really say anything meaningful or consequential

The impressive aspect of that version of the equation is simply the idea that you can obtain a plain old integer (-1) using nothing but simple arithmetic operations on two random-looking transcendentals.

The thing about that is that an imaginary power of a number isn't really simple - it's not like you can do repeated multiplication "i*pi times" and get a result. An imaginary power is defined as cosx + isinx (or the equivalent Taylor series). That's what I meant when I said that all people are impressed by is that cos(pi) = -1 or that sin(pi) = 0.

Yeah! Which of course also means that e and π can be defined in terms of one another.

That is probably the most insightful thing I've read all year. I wonder if there are any subtle implications.

One interesting thing is that it means it's not impossible to think that π+e or πe or π^e or some other combination of the two could be a simpler number (right now most of these numbers have no known/proven properties - they could even be rational for all we know).

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