What did the mathematician say when he was asked to comment on the usefulness of calculus? -"It has its limits..."
It's a sad show of intellect when one laughs at a math joke. LOL... great joke. http://www.scienceteecher.com/have_your_pi.htm I own that t-shirt... sadly, I wear it. It was a prize, along with a trophy and newspaper exposure for winning a calculus tourny at UWG. Please Register or Log in to view the hidden image!
Absane, if that shirt isn't completely a joke, how does one go about proving that statement... I'm serious. Is it a number close to e, or necessarily e.
It's an approximation, good to about 7 decimal places. another: e^(Pi*sqrt(163))=262537412640768744 <--an integer!!! (I lie, but it's good to many decimals)
I heard of e^(i)(Pi) + 1 = 0 in a biography on richard feynman. He called it "the most remarkable formula in math" in one of his childhood notebooks. I would like to see how to derive that result. That all three strange yet familiar numbers, e, i, and Pi, could combine with 1 and make zero....I'll agree that it's remarkable.
Absane, that's really terrible. Not maths, but still... A proton walks into a bar and asks for a pint of beer and a pint of gin. The barman looks at him and says "Are you sure?" to which the proton replies "Yes, I'm positive"
Just an approximation. Though the shirt is great for an ego boost around people Please Register or Log in to view the hidden image!
What's slightly less amazing, yet pretty close, is i<sup>i</sup>. Figure that one out Please Register or Log in to view the hidden image!
Shmoe is of course correct, but it may help some to expand his reply a little. e^x can be expressed in a power series. The terms of that power series can be collected into two separate group to make two new series. One of these new series is the power series for cosine and the other, except for a factor of "i" for sin, or something like that. (Not being a mathematician, I am neither as careful nor as concise as Shmoe.) When Shmoe's z=pi or 0 or any multiple of pi, the sin(z) is zero, but for greatest effect on people, z = pi is used to get the cos(z) = -1 and thus "kill" the +1 of the equation, making the interesting equation that envolves: 0,1,e,i and pi.
You take e<sup>ix</sup> and write out the Taylor expansion... you see for even powers, i will be 1 or -1... and for odd powers, you will get i or -i. You factor out the i and you see you are left with cos(x) + i*sin(x), where you recognize the Taylor series for cos(x) and sin(x).
e^(i)(Pi) + 1 = 0 The Euler formula shows that e^it = cos t + i sin t If we put t=pi, and use the fact that cos pi = -1, sin pi = 0, we get e^(i pi) = -1 Re-arranging e^(i pi) + 1 = 0
Like Shmoe and absane, you are also correct, but anyone who knows "Euler formula" or in absanes's case what a "Taylor Series" is, I do not think needs help with Feymann's "most interesting" equation. Thus, for them, I think my answer, which does not make esoteric (for them) references is more useful. You need, IMHO, to consider the knowledge the question asker probably has when answering. - Not a bad point to keep in mind, again MHO.
I agree with you, Billy T. But I doubt this poster knows what a power series is, either. You have to start somewhere, and you can't give a whole course in complex analysis in a couple of posts. So, I started with the Euler formula, which is fairly fundamental to the whole idea of complex numbers, I think.
I wouldn't think that's the case, alyosha appears to be self studying from apostol and had mentioned working through a more basic book iirc. I tend to lean towards minimal information for a few reasons. Not knowing exact backgrounds, it's tough to know when to stop and you can always supply more information on furthur request. There's also heaps of references available in texts and online (mathworld, planetmath, etc.) where someone can look up at least the basic meanings of terms they don't understand. I also prefer figuring stuff out as much as possible on my own, so my own preference on the receiving end is to be pointed in the right direction. Finally, I'm lazy, so it's less effort this way.
