Interesting mathematical statements (1 Viewer)

Paradoxica · Dec 24, 2015

glittergal96 said:
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
$$\noindent $ \eta(s) = \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k^s} \Rightarrow \zeta(s) - \eta(s) = 2\sum_{k=1}^\infty (2k)^{-s} = 2^{1-s}\sum_{k=1}^\infty k^{-s} = 2^{1-s}\zeta(s) \Rightarrow \Rightarrow \zeta(s) = \frac{\eta(s)}{1-2^{1-s}} \Rightarrow \zeta(0) = -\eta(0) = -\frac{1}{2}$

$\zeta '(s) = \frac{\eta '(s)(1-2^{1-s}) - \eta(s)2^{1-s}\log{2}}{(1-2^{1-s})^2} \Rightarrow -\zeta '(0) = \eta'(0) + \log{2}$

$\eta '(s) = \sum_{k=1}^\infty (-1)^{k-1}\frac{\log{k}}{k^s} \Rightarrow \eta '(0) = \sum_{k=1}^\infty (-1)^{k-1}\log{k}$

$$\noindent$2\eta '(0) = 2\sum_{k=1}^\infty \log(2k) - 2\sum_{k=1}^\infty \log(2k+1) =\left ( -\sum_{k=1}^\infty \log(2k-1) + \sum_{k=1}^\infty \log(2k) \right ) - \left ( -\sum_{k=1}^\infty \log(2k) + \sum_{k=1}^\infty \log(2k+1) \right ) = \sum_{k=1}^\infty \log \left (\frac{2k}{2k-1} \right ) + \sum_{k=1}^\infty \log \left (\frac{2k}{2k+1} \right ) = \sum_{k=1}^\infty \log \left ( \frac{4k^2}{4k^2 -1} \right ) = \log \left ( \prod_{k=1}^{\infty} \frac{4k^2}{4k^2 -1} \right ) = \log{\frac{\pi}{2}} \Rightarrow \eta '(0) = \frac{1}{2}\log{\frac{\pi}{2}}$

$-\zeta '(0) = \frac{1}{2}\log{\frac{\pi}{2}} + \log{2} = \log{\sqrt{2\pi}}$

$\therefore \infty ! = \sqrt{2\pi}$

If you do not know where I got the product identity from, recall the 1995 HSC paper, in which we proved said identity.

glittergal96 · Dec 24, 2015

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

The function $\zeta(s)$ is defined by the Dirichlet series $\sum_n n^{-s}$ 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.)

glittergal96 · Dec 24, 2015

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

People should post them in the undergrad marathon!

InteGrand · Dec 24, 2015

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/Casim...f_Casimir_effect_assuming_zeta-regularization .

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

Paradoxica · Dec 24, 2015

$Borwein Integral$

$\int_0^\infty \frac{\sin{x}}{x} dx = \frac{\pi}{2}$

$\int_0^\infty \frac{\sin{x}}{x}\frac{\sin{\frac{x}{3}}}{\frac{x}{3}} dx = \frac{\pi}{2}$

$\int_0^\infty \frac{\sin{x}}{x}\frac{\sin{\frac{x}{3}}}{\frac{x}{3}}\frac{\sin{\frac{x}{5}}}{\frac{x}{5}} dx = \frac{\pi}{2}$

$\vdots$

$\int_0^\infty \frac{\sin{x}}{x}\frac{\sin{\frac{x}{3}}}{\frac{x}{3}}\frac{\sin{\frac{x}{5}}}{\frac{x}{5}} \dots \frac{\sin{\frac{x}{11}}}{\frac{x}{11}}\frac{\sin{\frac{x}{13}}}{\frac{x}{13}} dx = \frac{\pi}{2}$

$$\noindent$\int_0^\infty \frac{\sin{x}}{x}\frac{\sin{\frac{x}{3}}}{\frac{x}{3}}\frac{\sin{\frac{x}{5}}}{\frac{x}{5}} \dots \frac{\sin{\frac{x}{11}}}{\frac{x}{11}}\frac{\sin{\frac{x}{13}}}{\frac{x}{13}}\frac{\sin{\frac{x}{15}}}{\frac{x}{15}} dx = \frac{467807924713440738696537864469}{935615849440640907310521750000}\pi = \frac{\pi}{2}-\frac{6879714958723010531\pi}{935615849440640907310521750000} \approx \frac{\pi}{2} - 2.31 \times 10^{-11}$

glittergal96 · Dec 24, 2015

Yes, I know of these experiments. I just haven't learned the physics well enough to understand the connection with the zeta function. (So I definitely cannot give a satisfactory answer to your question without copying/pasting someone elses response lol.)

(Tbh, this sort of thing has always interested me less than mathematics, but I might read a bit in the near future to see if I can give a good answer.)

glittergal96 · Dec 24, 2015

Classic one.

For arbitrary polynomial equations of degree 1,2,3,4 over the complex numbers, we can find the solutions in terms of the coefficients, the basic operations of arithmetic, and radicals (taking roots of some degree).

For degree 5 and greater, one can prove that this is not generally possible!

leehuan · Dec 24, 2015

glittergal96 said:
Classic one.

For arbitrary polynomial equations of degree 1,2,3,4 over the complex numbers, we can find the solutions in terms of the coefficients, the basic operations of arithmetic, and radicals (taking roots of some degree).

For degree 5 and greater, one can prove that this is not generally possible!

Lol proof that there is no 'quintic formula' would interest people.

leehuan · Dec 24, 2015

Another classic that any E4 MX2 student knows:

$\frac {\pi}{4} = 1 - \frac {1}{3} + \frac {1}{5} - \frac {1}{7} + \dots$

Paradoxica · Dec 24, 2015

glittergal96 said:
Classic one.

For arbitrary polynomial equations of degree 1,2,3,4 over the complex numbers, we can find the solutions in terms of the coefficients, the basic operations of arithmetic, and radicals (taking roots of some degree).

For degree 5 and greater, one can prove that this is not generally possible!

Abel–Ruffini theorem. Yeah, this was one of the things that jump started the development of modern mathematics. If this had never been proven, we wouldn't have computers, since the advent of modern logic, information theory and computational mathematics arose from the mathematical revolution of the 1900s.

But I digress.

$\sum_{n=1}^{\infty} \tan^{-1} \frac{1}{F_{2n+1}}=\frac{\pi}{4}$

$Where $F_m$ is the $ \mathrm {m}^{\mathrm {th}}$ Fibonacci number.$

leehuan · Dec 24, 2015

Classics and more classics:

Typically, the amount of petals on a plant is a Fibonacci number due to it's relationship with the golden mean.

Paradoxica · Dec 24, 2015

leehuan said:
Classics and more classics:

Typically, the amount of petals on a plant is a Fibonacci number due to it's relationship with the golden mean.

Extending on this fact, the petals are typically arranged so that each new petal is separated from the previous one by the golden angle.

glittergal96 · Dec 24, 2015

leehuan said:
Lol proof that there is no 'quintic formula' would interest people.

This proof would need to be pretty long to make any sense to HSC students, or even early undergrads. A decent amount of abstract algebra needs to be developed for this proof.

Paradoxica said:
Abel–Ruffini theorem. Yeah, this was one of the things that jump started the development of modern mathematics. If this had never been proven, we wouldn't have computers, since the advent of modern logic, information theory and computational mathematics arose from the mathematical revolution of the 1900s.

But I digress.

$\sum_{n=1}^{\infty} \tan^{-1} \frac{1}{F_{2n+1}}=\frac{\pi}{4}$

$Where $F_m$ is the $ \mathrm {m}^{\mathrm {th}}$ Fibonacci number.$

In what sense do you think the Abel-Ruffini theorem jump-started modern mathematics? Galois theory is beautiful but it isn't nearly as central as you make it out to be here. (Also, what does it have to do with what was done in the early 1900s and the mathematics that led to the invention of computing?)

leehuan · Dec 24, 2015

Another cliche, but possibly interesting to the MX2 student.

$\int _{ -\infty }^{ \infty }{ { e }^{ -{ x }^{ 2 } }dx } =\sqrt { \pi }$

Paradoxica · Dec 24, 2015

glittergal96 said:
This proof would need to be pretty long to make any sense to HSC students, or even early undergrads. A decent amount of abstract algebra needs to be developed for this proof.

In what sense do you think the Abel-Ruffini theorem jump-started modern mathematics? Galois theory is beautiful but it isn't nearly as central as you make it out to be here. (Also, what does it have to do with what was done in the early 1900s and the mathematics that led to the invention of computing?)

Everything was setup perfectly, the Abel Ruffini theorem was one of the first dominoes that toppled over in mathematical progress. I believe the first domino to fall over was the general solution to cubic polynomial equations. Well perhaps my remark on logic was a bit unfounded, but Information theory was worked on around the same time.

glittergal96 · Dec 24, 2015

Paradoxica said:
Everything was setup perfectly, the Abel Ruffini theorem was one of the first dominoes that toppled over in mathematical progress. I believe the first domino to fall over was the general solution to cubic polynomial equations. Well perhaps my remark on logic was a bit unfounded, but Information theory was worked on around the same time.

It was in the 1800s, how was it one of the first? It was significant in the subject of polynomial equations, and in this specific subject the cubic formula was the first of a sequence of dominoes after a long period of stagnant theory. The theory of polynomial equations is far from being central to mathematics though, and has very little to do with logic/information theory.

InteGrand · Dec 24, 2015

glittergal96 said:
It was in the 1800s, how was it one of the first? It was significant in the subject of polynomial equations, and in this specific subject the cubic formula was the first of a sequence of dominoes after a long period of stagnant theory. The theory of polynomial equations is far from being central to mathematics though, and has very little to do with logic/information theory.

What kinds of things are considered central to mathematics? The theory of real numbers?

glittergal96 · Dec 24, 2015

In car, will reply properly when I get home

. There definitely isn't a single subject you can pinpoint above all others though.

Paradoxica · Dec 24, 2015

InteGrand said:
What kinds of things are considered central to mathematics? The theory of real numbers?

Everything is connected...
It's like an infinitely complicated web that gets more and more complicated as time passes on... So it's likened to brains and the internet and human society in general...
The Riemann Hypothesis has a massive number of connections to areas in modern mathematics, as does the ABC conjecture. I'd provide more detail, but these things are so connected to virtually everything that I struggle to get across the immensity of this network.

glittergal96 · Dec 24, 2015

InteGrand said:
What kinds of things are considered central to mathematics? The theory of real numbers?

Okay, so some things in the last few centuries that are more significant and central to mathematics as a whole (centrality being judged by number of connections with diverse areas of math):

-The foundations of analysis being tightened up by people like Cauchy / Weierstrauss / etc. After this we were able to do analysis in far more general settings, and actually be sure of our conclusions.

-The work on the foundations of mathematics in the early 20th century, including Godel's results. They might have dealt a crippling blow to our ambitions of having a completely satisfactory foundation for mathematics, but at least it led to a greater understanding of how formal systems work.

-Calculus as originally developed by Newton/Leibniz/etc. It might not have been entirely rigorous at the time, but the physical applicability was immediately obvious.

-Point-set topology developed in the 20th century (in its current form), this is super important to many fields.

-The rigorous development of abstract algebra at the end of the 19th century / start of the 20th. This was well after the work of people like Galois/Abel on polynomial equations, and is considerably more general / abstract in its outlook.

-The development of differential geometry and more recently algebraic geometry, which are very different to the classical subject of geometry studied millenia ago.

These are kind of the roots of the core "branches" of modern mathematics. There are other smaller areas like number theory and information theory of course.

If you view mathematics as like a tree, then the listed developments are some of the big thick branches at the bottom, near the foundational trunk. Galois theory is some small offshoots from the algebra branch, that also intersects with some other things like the number theoretic part of the tree. It is harder to classify things like coding and information theory in terms of those core branches, but they are certainly more minor in terms of how much mathematics is related to / depends on them.

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