What does e^(pi*i) equal?

I don't think that there is really anything wrong with concepts in mathematics having nothing to do with the real world. Just look at my username.

Without getting too far into set theory, one of the consequences of the axiom of choice is known as the Banach-Tarski Paradox, which says that you can take one unit ball, subdivide it into finitely many subsets, move them around, and recombine them to form two unit balls. In other words, you start with a solid ball, cut it up, move the pieces around, and recombine it to be two solid balls the same size as the first. Jesus had nothing on Banach or Tarski.

The trick is, you are splitting the ball up into non-measurable sets (essentially, pieces whose volume is undefined). Most people probably have some intuitive understanding of what a set is, and there are definitely some real-world parallels to simple sets (you could consider a bag of apples to be like a set of apples, for example) but that is just scratching the surface. The full concept "set", in mathematics, has no real-world parallel, and most certainly does not match most peoples' intuition. In real life it is not possible to cut up a solid ball and re-assemble the pieces to be 2 solid balls of the same size and density as the first, but in mathematics it is possible. That is because non-measurable sets do not exist in the real world.

As for whether or not something like a non-measurable set can be said to "exist" or not if it has no real-world parallel, I think that calling that a philosophical problem is giving the question more credit than it deserves. It's just a question of semantics. If you define "exists" to mean, "exists in the real world", then, no, it does not exist. If you define "exists" to mean, "at least one example can be derived from the laws of mathematics", then, yes, it does "exist". It is a theoretical concept, and nothing more.

Personally, I think that people who insist that their model of mathematics perfectly mirror reality should spend less time worrying about that and more time doing math, but that's just me. My personal opinion is that philosophy is what people argue about when they don't understand the actual concepts but want to sound smart anyway. I've always felt that the beauty of mathematics was that it was disconnected from the "real world"; it is abstract and theoretical and you actually have to think in order to solve real problems, rather than just doing an experiment and seeing what happens.

As far as e^(i*pi), that is clearly -1. I would be lying if I told you that I remembered how to prove it. However, I do specifically remember the moment in time that I understood the proof. I always hated that stuff -- I always found abstract algebra to be much more interesting.

I'm not a mathematician, so forgive my ignorance.

I have no doubt that the imaginary unit i with property of i^2 = -1 exists. In my (naive) world i has certain (non-trivial) non-contradicting properties, and hence it exists simply because you can apply logic to it. My (naive) question is: how many of those "i"s exist.

Restating the question in a more precise language: If you know i, can you find a j which share all those properties i has, and yet is different ?
This might sound stupid, but one candidate which shares all properties i has, would be j = -i. And then clearly j is not equal to i. Are there any else ?

Quote: MangoJ

I'm not a mathematician, so forgive my ignorance.

I have no doubt that the imaginary unit i with property of i^2 = -1 exists. In my (naive) world i has certain (non-trivial) non-contradicting properties, and hence it exists simply because you can apply logic to it. My (naive) question is: how many of those "i"s exist.

Restating the question in a more precise language: If you know i, can you find a j which share all those properties i has, and yet is different ?
This might sound stupid, but one candidate which shares all properties i has, would be j = -i. And then clearly j is not equal to i. Are there any else ?

By the fundamental theorem of algebra, there can certainly be no more than two.

Well yes, they can't be more than 2 - in set C.

But what about in other, larger sets ? For example in quaternions there is i, j, k which all share the same properties of the i in C - and yet they are very different. And it's even not clear (at least for me) if the quaternion's i is identical to the imaginary unit of the complex numbers.

Quote: MangoJ

Well yes, they can't be more than 2 - in set C.

But what about in other, larger sets ? For example in quaternions there is i, j, k which all share the same properties of the i in C - and yet they are very different. And it's even not clear (at least for me) if the quaternion's i is identical to the imaginary unit of the complex numbers.

I guess I'm not completely sure what you mean by "all the properties of i".

But, x^2 + 1 = 0 has no more than two solutions in any field. C is algebraically complete, so extending it is not going to change anything. This is what I mean when I say, by the fundamental theorem of algebra, there can't be more than 2. So, if you include "being a root of x^2 + 1" as a property of i, then there can be no more than two.

So, is Hamilton wrong by his reflection on http://en.wikipedia.org/wiki/Quaternion ?
Stating that i^2 = j^2 = k^2 = -1 would then imply that i, j and k are each either the imaginary unit or it's negative by the fundamental theorem of algebra ?
But then ijk = -1 would clearly be wrong. Such an obvious mistake...

The quaternions are not a field.

Quote: AxiomOfChoice

The quaternions are not a field.

The quaternions are the largest associative extension of the reals that is a division algebra, so it is as far as you can go unless you want to go non-associative. In that case, you can go to the Octonians, but that's the last step.

http://en.wikipedia.org/wiki/Octonion

Cool stuff. I did my Ph.D. dissertation (in part) on quaternion algebras over other types of fields (there was some Galois cohomology in there).

Mustang Sally's Vi Hart is a fascinating woman.

Pi

I must say, I have never been impressed by Vi Hart. I feel that she is famous for being a woman who does math, rather than for doing math. Which is really a shame, because there are some very, very good female mathematicians who have made significant contributions to their fields.

Quote: AxiomOfChoice

I must say, I have never been impressed by Vi Hart. I feel that she is famous for being a woman who does math, rather than for doing math. Which is really a shame, because there are some very, very good female mathematicians who have made significant contributions to their fields.

Fair enough. What I find fascinating is that she externalizes her internal monologue in much the same way that my mind works, verbalizing the creative randomness I feel when I'm in "the zone". For me, the 2am mind meandering is not connected to the verbal/expressive parts of my brain. I can do it riffing with another person, but can't usually maintain a stream-of-consciousness-spew out loud. So, whether I keep a pad and paper or a little recorder sitting there, the best I get is some unconnected datapoints that have no apparent relationship when I try to apply the insight. She is connected to and verbalizes the process of thought itself (at least how my brain works), in a jumping, cross-cutting, non-linear way that I can not just follow, but extrapolate into new connections for myself. I'm really enjoying tuning in to her.

pi versus 2*pi doesn't really matter. does it? The latter seems more aesthetic in usage, but that's all to say about it. This doesn't mean anything is wrong with pi, it just says that the historical choice which quantity of the circle to call pi could have been chosen more wisely. But that is too easy to say in retrospect.

Historically "bad choices" do happen all the time. If electrical charges were defined (possibly in the early raise of chemnistry) the other way around, electrons would carry the positive charge, and would make some concepts, for example electric currents in metals, aesthetically more appealing.

Other choices in math could be the gamma function, where a shift of 1 in it's argument would not only identify the faculty function with the gamma function (restricted to all natural numbers). It would also make the integral representation more elegant "t^x exp(-t)" instead of "t^(x-1) exp(-t)".
And particular other relations such as "gamma(x) * gamma(1-x) = pi / sin(pi*x)" would be of the more appealing form "gamma(x) * gamma(-x)".

This all *could* be revised, but the problem arising from a simultaneous mixture of old and new conventions would make life pretty hard. Maybe in some future cultures people will laugh about us how "stupid" our math were....

I have to say, I thought that the whole pi vs tau thing was stupid, until I read a very good argument in favor of tau and I was converted immediately. I don't have the link handy, but if I find it I'll post it later.

I may be opinionated but that doesn't mean that I don't have an open mind!

In quantum physics you come across a lot of laws containing factors of 2*pi (with the 2 often the only "magic number") that from a physicist point it is natural to consider 2*pi more fundamental than pi. In fact for any kind of calculation resulting in a "naked" pi you rather check again where you might have lost factors of 2.

Quote: teliot

w = (sqrt(2)/2)*(1 + i). For this (and all) eight roots of unity, w^4 = -1.Can you write out the other 3 eighth roots?

w^4 = -1 = exp( (1+2k)*pi*i )
k=0 -> w = (sqrt(2)/2)*(+1 + i)
k=1 -> w = (sqrt(2)/2)*(-1 + i)
k=1 -> w = (sqrt(2)/2)*(-1 - i)
k=2 -> w = (sqrt(2)/2)*(+1 - i)



We used to use cepstrums at work. It is the result of taking the Inverse Fourier transform (IFT) of the logarithm of the estimated spectrum of a signal. There was cepstrums, saphe, liftering, cepstral and quefrency which corresponded to the concepts of spectrums, phase, filtering, spectral, and frequency. We never worried about these quantities were "real", just if they were useful in describing the world. A colleague was one of the first people to use this analysis in the question of the grassy knoll in 1963.

It's like the sqrt(2). If you ask someone what it is, then it a self defining number "It's a number that if you multiply it by itself you get 2". If you say the answer to x^2=2 is sqrt(2), you are really saying nothing. The definition of the sqrt(2) is that it is the answer to that equation. You might also give a decimal approximation, and say that it is irrational. But the definition of the square root of 2 is just a tautology.