The Math Thread

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Paranormal
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The Math Thread

Post by Paranormal »

Hey guys, move from off-topic. Don't hijack the off-topic with random math and brain-hurting equations.

Here's your own place to post that jazz and non-jazz. :lol:
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Re: The Math Thread

Post by Boingo »

1+1=2

I'm so evil. :D
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Vortex
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Re: The Math Thread

Post by Vortex »

From off-topic:
Taalit wrote:
OnyxIonVortex wrote:
Taalit wrote:2pi doesn't appear more than just pi, in pure mathematics. That's kinda only a physics thing.

Exhibit A: my signature.
Then you haven't seen enough mathematics yet :P it appears a lot in the context of complex numbers, exponentials, logarithms, trigonometry, hiperbolas, spheres, fourier transforms, distributions, hermitian operators, extensions of complex numbers... I'd guess it's something like 95% 2*pi vs. 5% another constant*pi
Maybe *you* have only seen the math where it's mostly 2pi. What do you think of that, huh?
Also, pi in general doesn't occur very often in logarithms or exponentials at all unless you're applying them to complex numbers, so you're kind of just giving an overall topic and then stating all of the things it includes. Also: What do you mean 'extensions of complex numbers'?
Dude, complex numbers are everything! XD the properties of the gamma function arise from it being defined starting from a exponential, and that's why 2*pi appears a lot. The zeta function is almost useless if it isn't defined over the complex plane. Also in statistics, the gauss bell's normalised by dividing by the square root of two*pi. Why? Because the canonical function is e^(-x^2/2), which integral over all the reals gives sqrt(2*pi). The area of a circle is the integral of its boundary with respect to the radius, that's why the 2 in the formula disappears. In formulas for higher dimensional spheres it stays. In group theory there's also an exponential map (you can exponentiate matrices!). The logarithm of a complex function is multivalued, the value that appears is 2pi*i and on it depends the cauchy theorem and all the holomorphic constructions. The differential equations often need complex numbers to work, and 2*pi appears on them as a consequence. But exponentials and logarithms are sometimes related with that constant, because some of its real properties depend on complex numbers. So if you try to delete everything related to complex numbers from your discussion, you end up with nothing (no sines, no cosines, very limited exponentials, unexplainable convergence theorems and restrictions,... and that's just the beginning)

And complex numbers are not just that, all geometry depends on it, you have to use logarithms to calculate distances in projections of hyperbolic spaces, to give an example. Also, things that you normally wouldn't think they are related to complex numbers actually are, like series sums, number theory, primes, real integrals, etc., and that's why k*pi appears so often.

The only thing where 2*pi (equivalently complex number constructions) doesn't appear so often is abstract things like category theory, set theory or algebra and topology, and even in the two last ones it appears from time to time. And of course pi doesn't appear in set theory or category theory either, as far as I know.
By extensions of the complex numbers I meant extensions of the idea of a complex number, that give rise to new algebras, like the quaternions, split-complex, octonions, sedenions, etc.
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Re: The Math Thread

Post by Oleander »

Fine, that is a lot of things. But you could still easily find a ton of other things that don't involve 2pi. My whole point is not that 2pi doesn't happen *often*, it's that other multiples also occur often so there's no real point or use to changing it.
OnyxIonVortex wrote: The zeta function is almost useless if it isn't defined over the complex plane.
No it isn't, for integer values above 1 it gives you the probability of x integers being coprime. That's a pretty good use!
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Anteroinen
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Re: The Math Thread

Post by Anteroinen »

Taalit wrote:Fine, that is a lot of things. But you could still easily find a ton of other things that don't involve 2pi. My whole point is not that 2pi doesn't happen *often*, it's that other multiples also occur often so there's no real point or use to changing it.
OnyxIonVortex wrote: The zeta function is almost useless if it isn't defined over the complex plane.
No it isn't, for integer values above 1 it gives you the probability of x integers being coprime. That's a pretty good use!
But isn't pi always connected to circles (or spheres or whatever) in some peculiar and contrived way? If another constant makes more sense with circles, why use the other one at all?

EDIT: Well, I suppose you don't really accept that, so my argument is silly.
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Re: The Math Thread

Post by Vortex »

Taalit wrote:Fine, that is a lot of things. But you could still easily find a ton of other things that don't involve 2pi. My whole point is not that 2pi doesn't happen *often*, it's that other multiples also occur often so there's no real point or use to changing it.
OnyxIonVortex wrote: The zeta function is almost useless if it isn't defined over the complex plane.
No it isn't, for integer values above 1 it gives you the probability of x integers being coprime. That's a pretty good use!
In that I do agree.
Also, yeah you're right, sorry, how could I forget that DX

But isn't pi always connected to circles (or spheres or whatever) in some peculiar and contrived way? If another constant makes more sense with circles, why use the other one at all?

EDIT: Well, I suppose you don't really accept that, so my argument is silly.
Well, you can say mathematicians like simplicity, but they also have to deal with tradition and with a centuries-old established notation, and hundreds of papers written in that notation. It's about balance between those things, like in many other consensus-like subjects.
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Re: The Math Thread

Post by Oleander »

So, are you in university now? You seem like you know quite a bit about mathematics! Way more than I'd expect someone to know in high school; I thought you were pretty young.
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Re: The Math Thread

Post by Vortex »

Yeah, it is my first year :)
Though pretty much all I know about maths is by searching in the Internet and reading several books, same with physics :P
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Re: The Math Thread

Post by Oleander »

Yeah, me too. I'm a senior in high school, but I've done a lot of independent study. Sometimes I wish there were more accessible resources for 'higher' mathematics, like stuff beyond what's talked about in the first couple semesters of Calculus.
What books have you read outside of school?
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gil2455526
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Re: The Math Thread

Post by gil2455526 »

What you studding on college? Like, what course?
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