Could someone explain to me why e^(pi*i)=-1? I understand how to do the math with Taylor series, and I know that it comes from Argand diagrams, blah, blah, blah, but what I don’t understand is:
Why is it that pi (which has to do with circles) and e (which has to do with compound interest) have this deep connection? Clearly the way the Taylor series work out isn’t a coincidence, but why is e the magic number for rotating a vector on the complex plane instead of, say, phi?
Come to think of it, why is it the case that the probability of any two random whole numbers being coprime is (pi^2)/6? What do whole numbers have to do with plane geometry?!?
First extend e^x, cosx, and sinx into the complex plane by taking the taylor series of them; for example the taylor series of e^x is sigma{infinity,n=0]x^n/n! but we are extending it into the complex plane so we use a complex variable, z.
e^z = 1 + z + z^2/2! + z^3/3! …
sinz = z - z^3/3! + z^5/5! -/+ …
cosz = 1 - z^2/2! + z^4/4! +/- …
if z = ix, then the Euler-Moivre equation is: e^(ix) = cosx + isinx
If we replace x with pi we get:
e^(ipi) = cos(pi) + i(0)
e^(ipi) = -1
this would explain why pi is used, to simplify the trigonmetric identity, and to get rid of the imaginary part (i’m sure there is another mathematical reason for this), but where e comes out of is beyond me. This probably however has something to do with the fact that INT{1,x)1/x,dx = 1 where x = e, and lim x-> 0 (x + 1)^(1/x) = e, since this is pretty much the basis for e.
The best answer thatI can give, is that it demonstrates the relationship between the trignometric functions (which involve pi) and the exponential function (which involves e). In fact, in complex analysis, they’re two different variations of the same thing.
2*pi is the period which will cause a trignometric function to have a slope of 1 at x = 0, and e is the exponential base which will cause an exponential function to have a slope of 1 at x = 0. We chose to do our trig in radians (hence bringing in pi), and to use the particular number e in our compound interest calculations, to take advantage of this.
One way e is defined is as the base of the logarithm that results from integrating 1/x. The properties of e, (like e=(1+1/n)[sup]n[/sup] as n goes to infinity) and its relation to compond interest follow from this integral definition. Extending the definition to the complex numbers:
log(z) = integral(from 1 to z) [ dw/w ]
where z and w are complex, and the integral is over any path in the complex plane which does not pass through the pole at the origin and connects the points (1+0*i) and z.
Evaluate log(-1) by using this integration formula. The result of this integration is the same regardless of the path, so choose a path that makes integration as simple as possible. The path joining (1+0i) and (-1+0i) which makes integration easiest is the semicircle with center at the origin and radius of 1. From trigonometry, the variable w along this arc is
w=cos(theta)+i*sin(theta)
as theta ranges from 0 to pi radians. Doing the change of variables with
dw=-sin(theta)+i*cos(theta)
the integral simplfies to:
log(-1)=integral(from 0 to pi) [i*d(theta)]=i*pi
and since e is the base of this log: -1=e[sup]i*pi[/sup].
So the e in the identity follows from the integral definition of log(z). The pi/circle connection is from integrating over the semicircular path. The i comes from requiring integration in the complex plane in order to connect +1 to -1 while avoiding the pole at the origin.
This is what bugs me. You say that the result is independent of the path, and then you say that the pi comes from the path you choose. It seems to me that the reason the semicircular path is easier is because pi is already in there; ie that you’ve swapped cause and effect.
I wasn’t quite correct in saying that the integral is independent of path, but it is if the path never crosses the positive real axis or the origin. However if the path does cross the positive real axis the result is changed by integer multiples of 2pii.
Anyway that doesn’t really answer your last question. If you don’t cross that axis the result will always be i*pi. The semicircular path does not change result of integral, it just make doing the integration extremely easy.
If you use some other path like half a square you will still get i*pi, but the integral won’t be something you can so easily find. So maybe I should have said the pi comes from the fact that using a semicircular path makes the integration trivial.
1/137 (or, more precisely, 1/137.035989559) is the fine structure constant, or alpha, defined as q[sup]2[/sup]/(hbar*k[sub]e[/sub]c) , where q is the electron charge, hbar is Dirac’s constant (Plank’s constant over 2pi), k[sub]e[/sub] is the constant in Coulomb’s Law, and c is the speed of light. There’s no known mathematical justification for it having that value, although many physicists (myself included) think that there ought to be. matt_mcl was just showing off his obvious sophistication and general coolness by demonstrating knowledge of it.
Nothing to add to this thread but I just noticed it is Chronos 1000th post and this is my 1000th post. I was hoping to have something fundamentally intelligent to say on such a milestone and yet I can’t think of anything
My tables say the fine structure constant is closer to 1/137.03602, but along the line of the original question, what is the significance of this being equal to
(pi+e)^{sqrt(2)*[phi^(gamma^2)-pi]}
which involves several of the most important mathematical constants: pi, e, sqrt(2), phi ( 1.618033989…), and gamma (Euler’s constant: 0.5772156649…). Or is it just a coincidence?
I don’t get why the fact that these theoretical constants can be lumped together to make 1/137.03602 has any significance whatsoever. They could just as easily equal 67.3925 or 1/56.6789. They have to equal something.
Whether or not it has significance is up for debate. What makes it cool is that the fine structure constant can be written in terms of these other “fundamental” numbers, not that those numbers can be arranged to equal 1/137. If we’d come across 67.3925 or 1/56.6789 in a physical relationship, we’d think that’s cool too.
But 1/137 is the “fine structure constant”, right? There’s where I’m losing you. What does the fine structure constant do, apart from being the answer to that equation?
I don’t recall exactly what the Fine Structure constant does, but I think it’s something relating to atomic energy level spacings. And Manlob, were you bluffing with that relation of yours? I may have miskeyed something, but I calculate that to be approximately -23.9837. By the way, I got my value from the built-in constant library on my calculator; my copy of the CRC Handbook of Chemistry and Physics (admittedly an old edition) gives the value as 1/137.03606, but with the last two places uncertain. RM Mentock’s table is probably from an up-to-date edition of the same source (god, I love that book!), so I’d be inclined to trust him.
IIRC, it has to do with measuring the electromagnetic and gravitational forces that dictates how elementary particles and light particles interact.
Eddison said the reciprocal of the fsc (1/136 in his time) times 2^256 (I think) is the exact number of protons in the universe, I don’t know where this number came from, and I doubt its significance. Of course Eddison was not exactly the best mathematician. He made “proofs” that the fsc was exactly 136, then a few years later when it was said to be 137, he made up yet another proof saying it was yet again exactly 1/137.
