1. Re: Interesting mathematical statements

Originally Posted by glittergal96
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!
Pretty pumped to do this in galois theory this sem (apparently it's related to the $A_{60}$ group from algebra?).

2. Re: Interesting mathematical statements

Missed and fixed.

Next bizarre truth

$\sum_{n=1}^{\infty}\frac{n^3}{2^n} = 26$

Supposedly first proven by Bernoulli.

Here is the Polylogarithmic evaluation of the infinite sum.

$\mathrm {Li}_s(z) = \sum_{k=1}^\infty \frac{z^k}{k^s}$

$Taking s = -3 and z = \frac{1}{2} yields our desired sum.$

$The particular case of s = -3 has the form: \mathrm {Li}_{-3}(z) = \frac{z(1+4z+z^2)}{(1-z)^4}$

$Plugging in z = \frac{1}{2} evaluates our series to be exactly 26.$

$\noindent Please post Bernoulli's solution on this thread if it exists and you happen to find it.$
Idk how Bernoulli did it, but you can easily evaluate such sums (replace 3 with any positive integer in fact) by repeatedly applying the operator (z d/dz) to the geometric series, which we already know how to sum. Then just chuck in 1/2.

3. Re: Interesting mathematical statements

@realisenothing

Yep, solvability of a polynomial equation is related to a certain group theoretic property. The symmetric group S_n has this property iff n < 5.

(And A_60 fails to have this property as it is structurally pretty much the same as S_5.)

4. Re: Interesting mathematical statements

So, to be more explicit about the geometric series approach to that summation:

$S(z):=\sum_{n\geq 1}n^3z^n=\left(z \frac{d}{dz}\right)^3 f(z)=zf'+3z^2f''+z^3f'''$

where

$f(z)=(1-z)^{-1}$

is the geometric series in the unit disk (which is easy to differentiate).

This gives us the rational function expression for S.

5. Re: Interesting mathematical statements

An Identity that comes from the proof of Euler's Partition Theorem:

$\prod_{k=1}^\infty (1+x^k) \equiv \prod_{k=1}^\infty \frac{1}{1-x^{2k-1}}$

In other words...

$(1+x)(1+x^2)(1+x^3)\dots \equiv \frac{1}{(1-x)(1-x^3)(1-x^5)\dots}$

6. Re: Interesting mathematical statements

Its Christmas why r u guys doing maths.

7. Re: Interesting mathematical statements

Radius of convergence will just be 1.

8. Re: Interesting mathematical statements

Originally Posted by turntaker
Its Christmas why r u guys doing maths.
if you think this is surprising, then you're in for a shocker if you knew me IRL.

9. Re: Interesting mathematical statements

if you think this is surprising, then you're in for a shocker if you knew me IRL.
Life isn't all about maths though...
-------------------------------------
Brahmagupta's formula:

$\\ If the side lengths of any cyclic quadrilateral is known, then the area of the quadrilateral is \\ A= \sqrt {(s-a)(s-b)(s-c)(s-d)} \\ where s is the semiperimeter of the quadrilateral.$

10. Re: Interesting mathematical statements

Originally Posted by leehuan
Life isn't all about maths though...
-------------------------------------
Brahmagupta's formula:

$\\ If the side lengths of any cyclic quadrilateral is known, then the area of the quadrilateral is \\ A= \sqrt {(s-a)(s-b)(s-c)(s-d)} \\ where s is the semiperimeter of the quadrilateral.$
No. Almost all of it is though. :PPPPPP

$Brahmagupta's formula becomes Heron's formula for the area of a triangle when one of the three sides degenerates.$

11. Re: Interesting mathematical statements

A must know for the Ext 2 student:

Fundamental Theorem of Algebra
$\\ Every non-constant single-variable polynomial always has at least one root.$

12. Re: Interesting mathematical statements

This has appeared elsewhere on this forum before:

Expressions for the Golden Ratio
$\\ \varphi =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\ddots } } }$

$\\ \varphi =\sqrt { 1+\sqrt { 1+\sqrt { 1+\sqrt { 1+\dots } } } }$

13. Re: Interesting mathematical statements

If you square the first 9 numbers with only 1's in them:

$\\ { 1 }^{ 2 }=1\\ { 11 }^{ 2 }=121\\ { 111 }^{ 2 }=12321\\ { 1111 }^{ 2 }=1234321\\ { 11111 }^{ 2 }=123454321\\ { 111111 }^{ 2 }=12345654321\\ { 1111111 }^{ 2 }=1234567654321\\ { 11111111 }^{ 2 }=123456787654321\\ { 111111111 }^{ 2 }=12345678987654321$

14. Re: Interesting mathematical statements

$Ramanujan's continued fractions for algebraic numbers in terms of e.$

$\sqrt[4]{5}\sqrt{\frac{1+\sqrt{5}}{2}}- \sqrt{\frac{1+\sqrt{5}}{2}} = \frac{e^{-\frac{2\pi}{5}}}{1+\frac{e^{-2\pi}}{1+\frac{e^{-4\pi}}{1+\frac{e^{-6\pi}}{1+\ddots}}}}$

$\frac{\sqrt{6\sqrt{3}}-(1+\sqrt{3})}{4} = \frac{e^{-\frac{2\pi}{3}}}{1+\frac{e^{-2\pi}+e^{-4\pi}}{1+\frac{e^{-4\pi}+e^{-8\pi}}{1+\frac{e^{-6\pi}+e^{-12\pi}}{1+\ddots}}}}$

$\sqrt{\sqrt{2} -1} = \frac{\sqrt{2}e^{-\frac{\pi}{4}}}{1+\frac{e^{-2\pi}}{1+e^{-2\pi}+\frac{e^{-4\pi}}{1+e^{-4\pi}+\frac{e^{-6\pi}}{1+e^{-6\pi}\ddots}}}}$

15. Re: Interesting mathematical statements

Very famous result on what e actually is:

$e=\lim _{ n\rightarrow \infty }{ { \left( 1+\frac { 1 }{ n } \right) }^{ n } }$

Still famous but not as famous result on what e is, esp amongst HSC students:

$e=\sum _{ n=0 }^{ \infty }{ \frac { 1 }{ n! } }$

16. Re: Interesting mathematical statements

Originally Posted by leehuan
Very famous result on what e actually is:

$e=\lim _{ n\rightarrow \infty }{ { \left( 1+\frac { 1 }{ n } \right) }^{ n } }$

Still famous but not as famous result on what e is, esp amongst HSC students:

$e=\sum _{ n=0 }^{ \infty }{ \frac { 1 }{ n! } }$
$e^x \equiv \sum_{r=0}^\infty \frac{x^r}{r!}$

$\lim_{n \rightarrow \infty} \left (1+\frac{x}{n} \right )^n \equiv e^x$

$\noindent By expanding the first statement and expanding the second statement using the binomial theorem for n = \infty, one can prove the two statements above are identical.$

17. Re: Interesting mathematical statements

Tbh isn't the first statement really just a Taylor series?

18. Re: Interesting mathematical statements

Originally Posted by leehuan
Tbh isn't the first statement really just a Taylor series?
Yeah, but sometimes it's actually more convenient to define functions by their power series representation and then prove other things about them.

19. Re: Interesting mathematical statements

Originally Posted by InteGrand
Yeah, but sometimes it's actually more convenient to define functions by their power series representation and then prove other things about them.
$The only other things we have to go off of are:$

$e^x is the fundamental solution to the differential equation y' = y$

$e^x is it's own derivative$

$e^x is the solution for the equation x = \int_1^{e^x} \frac{\mathrm{d}t}{t}$

The first and second statements are nearly equivalent down to the family of solutions for any differential equation.

The third statement is somewhat convoluted.

20. Re: Interesting mathematical statements

987654321 is divisible by 9
987654312 is divisible by 8 (this particular one gives 123456789 btw)
987654213 is divisible by 7
987653214 is divisible by 6
987643215 is divisible by 5
987543216 is divisible by 4
986543217 is divisible by 3
976543218 is divisible by 2
876543219 is divisible by 1

0123456789 * 2 = 246913578 (a permutation of 0123456789)
And likewise up to 100; excluding multiples of 3

21. Re: Interesting mathematical statements

There is approximately a 50% chance of two people sharing the same birthday in a room of 23 people

22. Re: Interesting mathematical statements

Originally Posted by Soulful
There is approximately a 50% chance of two people sharing the same birthday in a room of 23 people
This is, of course, assuming that every birthday date is identically probable, and that leap years do not exist, and that birthdays are treated as truly random. Under these conditions, a group of 70 people have a 99.9% chance of two people sharing a birthday.

23. Re: Interesting mathematical statements

Certainly not a pure mathematician, but Cauchy's Integral Formula stands out for me.

Then I moved to stats.

24. Re: Interesting mathematical statements

Originally Posted by AAEldar
Certainly not a pure mathematician, but Cauchy's Integral Formula stands out for me.

Then I moved to stats.
That's not a bad thing, there are unsolved problems in stats applicable to the real world such as p-values, error values, correlation vs. causation, interpolation of data, etc.
Most of these are relevant to the sciences, as statistical methods are required for collecting any information. Even the medical fields and the humanities require it.

But I digress.

$\noindent The curious improper integral \int_{-\infty}^{\infty} \frac{1+x}{1+x^2} \mathrm{d}x is divergent, yet if you evaluate it normally as a limit towards infinity, it converges to exactly \pi .$

$Conclusion: \pi = \infty$

25. Re: Interesting mathematical statements

There is a lot of highly theoretical mathematics here.
Does anyone have more real-world mathematical statements?
(The birthday problem was a good one)

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