Talk:Euler's Formula

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I would disagree that this is Euler's most famous formula. I would plump for either e^(i*pi) = -1 or the F+V -E = 2 for convex polyhedra. Google search brings up these two ... --CatWatcher 17:16, 7 April 2007 (EDT)

e^(i*pi) = -1 is derived from euler's formula by set .Jaques 17:21, 7 April 2007 (EDT)
granted, but I have heard it claimed that this formula is the "e=Mc^2" od mathematics, in that it ties together four of the most infamous numbers in the whole of maths. -1, as for centuries, it was thought that numbers could not be negative, pi, which was always thought to be fractional (hence squaring the circle), i, which was thought to be impossible, but now is termed imaginary, and e, the base of natural logarthims, which underpins calculus.
I would certainly say this was Euler's masterpiece, even though I am a mainly a Graph Theorist, and the other 'Euler's Formula' is the one I am most familiar with.--CatWatcher 18:39, 7 April 2007 (EDT)

Arbitrary definition for using the imaginary square root of -1

Andy, what do you think is "arbitrary" about the square root of -1? Extending a field by the root of an equation is a well understood procedure (and i or -i will lead to the same result....) --AugustO (talk) 14:43, 16 August 2015 (EDT)

The square root of negative one is imaginary; Euler's formula is simply a definition rather than a mathematically derived theorem. It's interesting that the comments above compared this with E=mc2, because in both cases it is simply a redefinition rather than a derivation.--Andy Schlafly (talk) 17:57, 16 August 2015 (EDT)
Mathematically, you can derive this formula by doing standard mathematics (i.e., calculating series) in the field R[x]/[x²+1]. --AugustO (talk) 18:08, 16 August 2015 (EDT)
  • Defining i as the solution of is conceptually not more difficult than defining "-1" as the solution of . The latter problem has baffled mankind for centuries! i and -1 are both somewhat quite imaginary - or at least imaginative - entities!
  • Definition - Theorem - Proof: that's somewhat how modern mathematics work, so, you cannot complain that Euler started with a definition.
  • follows quite straightforward by using .

--AugustO (talk) 14:32, 17 August 2015 (EDT)

I have to think about that further. Raising to an exponential power seems to require an additional assumption.--Andy Schlafly (talk) 14:36, 17 August 2015 (EDT)
Just plug it in and see what happens - I suppose, that is what Euler has done :-)
Granted, it took two centuries to get all the details right (convergence, resorting of infinite sums, etc.), but that's the gist. Straightforward and beautiful. --AugustO (talk) 16:04, 17 August 2015 (EDT)

Just a thought

There is nothing wrong with "arbitrary definitions" in mathematics. Mathematicians will make them up all the time, sometimes with amusing, sometimes with usable results.

The "imaginary number i" is not more artificial than the "number 2": try to think about two without thinking of two objects - or the cipher! The concept of 2 is as imaginary as the concept of i. We use both, because they work, not only in one instance, but in many: the complex numbers can be introduced in various ways, and all of them work hand-in-glove together, always a sign that something is done right! So, we shouldn't bother about the unfortunate naming of i, but follow the applications!

The square root of -1 was introduced as first as a party trick - to solve cubic equations in contests. As so often in mathematics, it suddenly became quite relevant, and nowadays, complex analysis is something every undergraduate in maths or physics has to know! It doesn't matter how you feel about it, it is only important that it works... --AugustO (talk) 18:43, 17 August 2015 (EDT)

"it works ..." at what? Using "i" as an exponent may have value to the extent that any orthogonal coordinates have value, but it could be "j" or "k" to accomplish the same goal.--Andy Schlafly (talk) 18:49, 17 August 2015 (EDT)
  • it works at - e.g. - complex analysis - which allows interesting answers to many questions
  • well, in electrical engineering it generally is j: is this the reason for your confusion?
  • Amusingly, you don't use i as an exponent at first: you put it into the series which defines exp(x), and there it is used as the base with non-negative integers as exponents.
  • we use 2 for two, but ii, β, ||, etc. have accomplished the same goal.
--AugustO (talk) 07:28, 18 August 2015 (EDT)

Just two more thoughts

There are a couple of things tending to sidetrack this discussion. They are red herrings.

  1. The talk of the equation being "the 'e=Mc^2' of mathematics" was a reference to the incredible fame of E=mc^2, not its subject matter. It's like calling someone "the Shakespeare of X" or "the Mozart of Y". As you know, I'm working on putting the fame of E=mc^2 into perspective on the other page. Yes, I'm still working on it. If both equations are "simply a redefinition rather than a derivation", that's a pure coincidence. But in fact neither is a redefinition, something I plan to convince you of.
  2. The talk of the arbitrariness of "i" is a deep question, much deeper than whether you use the symbol "j" or "k" instead. (You're not an electrical engineer by any chance, are you? Oh, right. You are.  :-) The choice of the letter "i" to indicate the second basis vector for complex numbers, the choice of the 16th letter of the Greek alphabet for the ratio of circumference to diameter, and the choice of the symbol "2" for the Peano successor of "1", are completely independent of the subject matter. Whether "i" is ambiguous (it isn't) is a completely different matter. I will explain that when I get the time.

SamHB (talk) 19:55, 17 August 2015 (EDT)