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I think that e^(i*pi) + 1 = 0 is massively overrated. Because the beauty that people find is largely based on the symbols that are present and because it doesn

I think that e^(i*pi) + 1 = 0 is massively overrated. Because the beauty that people find is largely based on the symbols that are present and because it doesn't really say anything meaningful beyond itself. It's not even a formula or an equation - it's a result. It's like being impressed by cos(pi) = -1. The more general formula e^(ix) = cosx + i*sinx is a lot more impressive. Both because of it's implications for the rest of mathematics and aesthetically. If you look at the way the vectors at 38:00-39:20[1] cancel out to always land on the unit circle - I can get behind calling that beautiful. [1] https://youtu.be/ZxYOEwM6Wbk?t=2280 https://youtu.be/ZxYOEwM6Wbk?t=2280

jleahy 5y ago
Even e^(ix) = cosx + i*sinx is an obvious result of the definition i^2= - 1.
- sobriquet9 5y ago
  In hindsight, and if you know Mclaurin series. Complex numbers were known and used for many years before Euler found this obvious result.
- red_trumpet 5y ago
  If you already know the power series expansions of cos and sin then yes. But then you put the burden of proof into showing that the power series of cos and sin indeed define the same functions as the trigonometric definition.
- japanuspus 5y ago
  I do not think this is true: you can introduce the complex numbers (including the imaginary unit `i`) without defining exponentiation or introducing `e`. As a consequence, no properties of `e` can be deduced from the definition of the imaginary unit. In particular, the result is not "obvious".
spuz 5y ago
The first line of the article is "Beauty, they say, is in the eye of the beholder" and here your comment seems to relate directly to this line and the rest of the article and yet someone still found reason to downvote you for having your own point of view. Come on HN, seriously...
- bobthechef 5y ago
  > "Beauty, they say, is in the eye of the beholder" Incidentally, I reject this claim. I take beauty to be objectively real and not a subjective reaction to something. The latter I would call taste. (A materialist would of course object because for him, the world is just a bunch of boring atoms bumping into each other, though he will never explain how his mind, just another boring bunch of jostling atoms, is capable of entertaining such subjective delusions and why they can exist in his mind-as-jostling atoms but not in the world "out there", more jostling atoms.) That some variation in opinion exists about what is beautiful (and in what way) does not disprove the claim. Not everyone is equally discerning, some have perverse tastes, etc.
  - teucris 5y ago
    Can you define beauty then?
tonyarkles 5y ago
The lecture in the sequences and series portion of calculus where the prof took the Taylor series of e^x and then substituted x=i*theta into the resulting polynomial and all of the terms magically divided themselves into the Taylor series’s for sin and cos was a jaw dropping moment for me. I was in my second year of EE, and it completely changed my relationship with a ton of practical material for me, eg the Fourier transform. On the other hand, I also took vector calculus that year but it wasn’t until the following year when I took the E&M course (electric and magnetic fields) where I truly appreciated surface integrals, path integrals, etc. One of the biggest Aha! moments there was really appreciating how properly structuring the problem could dramatically simplify the computations. Eg if you can make it symmetric you can often cause integrals to disappear implicitly without having to compute them at all, or you can take a dramatically simpler path through a vector field and end up at the same place without a ton of complex computation. Absolutely beautiful :D
ska 5y ago
> Because the beauty that people find is largely based on the symbols that are present The identity involves the five most fundamental constants and the four most fundamental operations - and nothing else. That is why a lot of people find it satisfying.

jleahy 5y ago

Even e^(ix) = cosx + i*sinx is an obvious result of the definition i^2= - 1.

sobriquet9 5y ago
In hindsight, and if you know Mclaurin series. Complex numbers were known and used for many years before Euler found this obvious result.
red_trumpet 5y ago
If you already know the power series expansions of cos and sin then yes. But then you put the burden of proof into showing that the power series of cos and sin indeed define the same functions as the trigonometric definition.
japanuspus 5y ago
I do not think this is true: you can introduce the complex numbers (including the imaginary unit `i`) without defining exponentiation or introducing `e`. As a consequence, no properties of `e` can be deduced from the definition of the imaginary unit. In particular, the result is not "obvious".

spuz 5y ago

The first line of the article is "Beauty, they say, is in the eye of the beholder" and here your comment seems to relate directly to this line and the rest of the article and yet someone still found reason to downvote you for having your own point of view. Come on HN, seriously...

bobthechef 5y ago
> "Beauty, they say, is in the eye of the beholder" Incidentally, I reject this claim. I take beauty to be objectively real and not a subjective reaction to something. The latter I would call taste. (A materialist would of course object because for him, the world is just a bunch of boring atoms bumping into each other, though he will never explain how his mind, just another boring bunch of jostling atoms, is capable of entertaining such subjective delusions and why they can exist in his mind-as-jostling atoms but not in the world "out there", more jostling atoms.) That some variation in opinion exists about what is beautiful (and in what way) does not disprove the claim. Not everyone is equally discerning, some have perverse tastes, etc.
- teucris 5y ago
  Can you define beauty then?

tonyarkles 5y ago

The lecture in the sequences and series portion of calculus where the prof took the Taylor series of e^x and then substituted x=i*theta into the resulting polynomial and all of the terms magically divided themselves into the Taylor series’s for sin and cos was a jaw dropping moment for me. I was in my second year of EE, and it completely changed my relationship with a ton of practical material for me, eg the Fourier transform. On the other hand, I also took vector calculus that year but it wasn’t until the following year when I took the E&M course (electric and magnetic fields) where I truly appreciated surface integrals, path integrals, etc. One of the biggest Aha! moments there was really appreciating how properly structuring the problem could dramatically simplify the computations. Eg if you can make it symmetric you can often cause integrals to disappear implicitly without having to compute them at all, or you can take a dramatically simpler path through a vector field and end up at the same place without a ton of complex computation. Absolutely beautiful :D

ska 5y ago

> Because the beauty that people find is largely based on the symbols that are present The identity involves the five most fundamental constants and the four most fundamental operations - and nothing else. That is why a lot of people find it satisfying.