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    -insert title here- Paradoxica's Avatar
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    Interesting mathematical statements







    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Ancient Orator leehuan's Avatar
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    Re: Interesting mathematical statements

    Except why is this in non school lol
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    Ancient Orator leehuan's Avatar
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    Re: Interesting mathematical statements

    Last edited by leehuan; 23 Dec 2015 at 9:41 PM.
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    -insert title here- Paradoxica's Avatar
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    Re: Interesting mathematical statements

    Quote Originally Posted by leehuan View Post
    Except why is this in non school lol
    Well it's not exactly a question asking thread, and it's more for entertainment.



    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Executive Member KingOfActing's Avatar
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    Re: Interesting mathematical statements

    Quote Originally Posted by leehuan View Post
    The sequence is conjectured to always diverge to 1
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    Ancient Orator leehuan's Avatar
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    Re: Interesting mathematical statements

    Quote Originally Posted by Paradoxica View Post
    Well it's not exactly a question asking thread, and it's more for entertainment.
    You could always have posted it under just maths then or extracurricular

    Fermat's Last Theorem


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    Re: Interesting mathematical statements

    Quote Originally Posted by KingOfActing View Post
    The sequence is conjectured to always diverge to 1
    I know lol. This is one of the biggest mind gobbling problems to pure mathematicians apparently; WHY?

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    -insert title here- Paradoxica's Avatar
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    Re: Interesting mathematical statements

    Quote Originally Posted by leehuan View Post
    I know lol. This is one of the biggest mind gobbling problems to pure mathematicians apparently; WHY?
    The thing is, we don't even know how to begin to approach this problem. Paul Erdos has already commented on this problem.
    Looks like we'll just have to wait for the next
    Euler/Erdos/Tao/Gauss/Noether/Polya/Hilbert/Russell/Lagrange/Riemann/Hardy/Poincare/Fermat/Grothendieck/Newton/Leibniz/Weierstrass/Cauchy/Descartes/Dirichlet/Cantor/Fibonacci/Jacobi/Ramanujan/Hamilton/Godel/Pascal/Apollonius/Laplace/Liouville/Eisenstein/Banach/Peano/Bernoulli/Viete/Fourier/Huygens/Chebyshev/Lebesgue/Turing/Cardano/Minkowski/Littlewood/Legendre/Birkhoff/Lambert/Poisson/Wallis/Tarski/Frege/Hausdorff/Neumann/Galois
    to come around and resolve the problem.

    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Ancient Orator leehuan's Avatar
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    Re: Interesting mathematical statements

    Quote Originally Posted by Paradoxica View Post
    Ok link me the proof lol

    I'm worried about a 1+2+3+4+...=-1/12 here

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    Re: Interesting mathematical statements

    If you have a small ball in 3 dimensional space, it is possible to decompose it as a union of a finite number of sets, which can be moved by rotations and translations such that the pieces never overlap and such that the final object constructed is an arbitrarily large ball.

    Colloquially, one can cut a pea into a finite number of pieces and reassemble it into something the size of the sun.

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    Executive Member KingOfActing's Avatar
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    Re: Interesting mathematical statements

    Quote Originally Posted by leehuan View Post
    Ok link me the proof lol

    I'm worried about a 1+2+3+4+...=-1/12 here
    It's another zeta regularisation, I'm pretty sure.
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    Loquacious One turntaker's Avatar
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    Re: Interesting mathematical statements

    1+1=2



    Last edited by turntaker; 23 Dec 2015 at 10:44 PM.
    "I have crippling depression" -Mahatma Gandhi
    Quote Originally Posted by Katsumi View Post
    lol

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    Re: Interesting mathematical statements

    amazing
    "I have crippling depression" -Mahatma Gandhi
    Quote Originally Posted by Katsumi View Post
    lol

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    Re: Interesting mathematical statements

    Can we keep our posts restrained to at least MX2 level and not making bad usages of mathematics lmao

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    Executive Member KingOfActing's Avatar
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    Re: Interesting mathematical statements

    Zeta regularisations are important

    1 + 2 + 3 + ... =/= -1/12, but is rather 'assigned' that value
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    Re: Interesting mathematical statements

    -1 x -1 = 2
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    "I have crippling depression" -Mahatma Gandhi
    Quote Originally Posted by Katsumi View Post
    lol

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    -insert title here- Paradoxica's Avatar
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    Re: Interesting mathematical statements

    Quote Originally Posted by leehuan View Post
    Ok link me the proof lol

    I'm worried about a 1+2+3+4+...=-1/12 here


    Quote Originally Posted by KingOfActing View Post
    It's another zeta regularisation, I'm pretty sure.
    Correct, although I don't actually fully understand regularisation yet.











    Last edited by Paradoxica; 23 Dec 2015 at 10:59 PM.
    leehuan likes this.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Rambling Spirit
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    Re: Interesting mathematical statements

    I like the 1 + 2 + 3 + 4 + … = -1/12 result, and find it also quite amazing that this is used in physics and gives some experimentally verifiable results. There's a lot of 'weird' stuff like this in this series of lectures on Mathematical Physics by Carl Bender that can be found on YouTube.

    Also, I think this thread should be in the maths Extracurricular Topics forum.
    leehuan likes this.

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    -insert title here- Paradoxica's Avatar
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    Re: Interesting mathematical statements

    Quote Originally Posted by glittergal96 View Post
    If you have a small ball in 3 dimensional space, it is possible to decompose it as a union of a finite number of sets, which can be moved by rotations and translations such that the pieces never overlap and such that the final object constructed is an arbitrarily large ball.

    Colloquially, one can cut a pea into a finite number of pieces and reassemble it into something the size of the sun.
    Only if I accept the axiom of choice. : PPPPPPP
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Re: Interesting mathematical statements

    Quote Originally Posted by Paradoxica View Post
    Only if I accept the axiom of choice. : PPPPPPP
    Even if you don't accept the axiom of choice (which is a bit limiting, but some minority of mathematicians don't), you would not be able to prove that such a reassembling of the pea into the sun is impossible. (Because the axiom of choice is consistent with the other axioms of set theory.)

    This is still pretty unintuitive.

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    -insert title here- Paradoxica's Avatar
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    Re: Interesting mathematical statements

    Quote Originally Posted by glittergal96 View Post
    Even if you don't accept the axiom of choice (which is a bit limiting, but some minority of mathematicians don't), you would not be able to prove that such a reassembling of the pea into the sun is impossible. (Because the axiom of choice is consistent with the other axioms of set theory.) This is still pretty unintuitive.
    it was a joke -_-
    I find that people who reject the axiom of choice are on the same level as those who reject the law of the excluded middle. Half of mathematics is based upon contradiction.

    Got the proof for derivative of zeta at zero












    If you do not know where I got the product identity from, recall the 1995 HSC paper, in which we proved said identity.
    Last edited by Paradoxica; 24 Dec 2015 at 10:55 AM.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Re: Interesting mathematical statements

    With all of these zeta function things, people often get confused by what is meant by these divergent sums having values.

    The function is defined by the Dirichlet series only where it converges, which is the half-plane where the real part of s exceeds 1.

    Elsewhere in the complex plane (apart from at s=1), the zeta function is defined by analytic continuation. In other words, there is a unique "nice" function on the complex plane that extends the series where it converges.

    So at points like s=0 it is not like that series is equal to zeta(0), it is just that that sum diverges in the traditional sense and hence is an undefined object. We might as well use that sum notation to instead represent the "nice function" that the convergent sum extends to.

    In this sense the statement is more like: If the sum of the natural numbers had to be defined to be something, -1/12 would be a natural value for it to have.

    Definitely a lot of the reason that non-mathematicians find this so interesting is that they view the sum as actually converging to -1/12 in a more traditional sense which is of course nonsense. The amount that this fact is thrown around colloquially does not help.

    (I can't say I know much about how this fact crops up in physics though.)

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    Re: Interesting mathematical statements

    Some of these statements are pretty fun to prove and not too difficult btw.

    People should post them in the undergrad marathon!

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    Rambling Spirit
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    Re: Interesting mathematical statements

    I think these kinds of divergent sums (and integrals for that matter) come up all in the time in quantum physics. For example, the 1+2+3+4+... one comes up in calculating the Casimir Force in 1D in Quantum Electrodynamics. The 13 + 23 + 33 + ... one comes up in the 3D version of this calculation (its value is ζ(-3) = 1/120 using analytic continuation of the Riemann-Zeta function). As far as I know, experiments have now been done and agree with predictions to a good extent. Here's a derivation on Wiki of the 3D Casimir effect that uses ζ(-3):

    https://en.wikipedia.org/wiki/Casimi...regularization .

    Is it just a coincidence that using these regularised sums happens to give apparently physically sensible answers??

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    Re: Interesting mathematical statements













    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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