Except why is this in non school lol
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.
Except why is this in non school lol
Last edited by leehuan; 23 Dec 2015 at 9:41 PM.
2015 HSC
Mathematics 2U
2016 HSC
Advanced English, Drama, Maths 3U, Maths 4U, Music 2, Extension Music, Chemistry, Software Design & Development
Currently studying
Advanced Mathematics (Hons)/Computer Science @ UNSW (2017– )
You could always have posted it under just maths then or extracurricular
Fermat's Last Theorem
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.
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.
2015 HSC
Mathematics 2U
2016 HSC
Advanced English, Drama, Maths 3U, Maths 4U, Music 2, Extension Music, Chemistry, Software Design & Development
Currently studying
Advanced Mathematics (Hons)/Computer Science @ UNSW (2017– )
Can we keep our posts restrained to at least MX2 level and not making bad usages of mathematics lmao
Zeta regularisations are important
1 + 2 + 3 + ... =/= -1/12, but is rather 'assigned' that value
2015 HSC
Mathematics 2U
2016 HSC
Advanced English, Drama, Maths 3U, Maths 4U, Music 2, Extension Music, Chemistry, Software Design & Development
Currently studying
Advanced Mathematics (Hons)/Computer Science @ UNSW (2017– )
Last edited by Paradoxica; 23 Dec 2015 at 10:59 PM.
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.
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.
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.
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.)
Some of these statements are pretty fun to prove and not too difficult btw.
People should post them in the undergrad marathon!
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 1^{3} + 2^{3} + 3^{3} + ... 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??
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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