Pi must repeat

If Pi has infinite numbers following the decimal, at same point those numbers must fit into a sequence that has already been done, for whatever length of numbers.  Mathemetecians need only figure out how many numbers are repeated and for how long and in what places of the sequence.

>>38

It's transcendental, that's why.

>>38

Because Ferdinand von Lindemann proved in 1822 that it isn't.

That's how.

So what exactly does transcendental mean? In simple terms?

{x {({Sin[360/x]}^2) + {1 - Cos[360/x]}^2}^{1/2}}/2

>>43
http://en.wikipedia.org/wiki/Transcendental_number

>>43
That it's fucking weird.

>>43
a number is transcendental if it is not the root of any polynomial with integer coefficients.

>>46

Welcome to the big and beautiful bestiary of modern mathematics, where complicated creatures eclipse our ken.

Could pi be the root of a polynomial with radical coefficients?  Anyone disproved that?

If pi was something else, what would happen?

So what's the pattern in Pi? If there's none how can it be truly random?

Was Jesus an extraterrestrial? If not, why so come?

>>38
There's a proof somewhere that pi is not an algebraic number (root of a finite polynomial).

Nice trolling RedCream, I lol'd

>>49


The roots of such a polynomial would be roots of some other polynomial with integer coefficients.

For example, the roots of x^2 - (sqrt(2))x + 1 are also roots of x^4 + 1.

>>51
It's not random in any rigorously defined mathematical sense.  It's completely deterministic; there's dozens of compact summation formulas that will give you pi to any precision you want.

"How do we know pi isn't some fraction of a square root, like 20sqrt(2)/9 ?"
Wtf?
Why are people with no knowledge of math posting on a math/science message board?

http://library.wolfram.com/examples/quintic/people/Lambert.html
Just go try to learn stuff on your own before posting on the internet.

>>54

Is that always true though, proof?

OK, what about non-polynomials, where the variable has any of negative, fractional, or radical exponents?

>>56
to get knowledge out of churlish cranky queers

>>58

That is a good question.  Pi is not the root of any "polynomial" equation where the polynomials have powers of x that are fractions, either positive or negative.

For radical exponents, I believe the question is open.  That means we don't know the answer. 

If you allow *arbitrary* exponents, pi is a root of a zillion different equations.   For example, pi is a root of the equation x^q - 2 = 0, where q = log(2)/log(pi). 

If you allow the "polynomials" to have infinitely many terms, you can find equations of which pi is a root.  For example, it is the smallest positive root of
x - x^3 / 6 + x^5 / 120 - x^7 / 5040 + ... .   But this isn't a polynomial.

I hope this does more to clear up the situation than it does to confuse it.

>>47
almost - it's not a solution to any non-zero polynomial with coefficients in Q. but it's essentially with integer coefficients, since you can multiply through by something to get all integers.

>>60

for those interested, that's the Taylor/Maclaurin series for sinx.

>>62
We're not.

>>63
Jeepers, harsh dude

>>61


That's obvious, but I'd imagine it'd not be a solution to any finite polynomial with radical co-efficients, but proving that would be presumably harder.

>>65
If you accept that pi is transcendental, it isn't hard at all. The radicals are a subset of the algebraic numbers, which are (by definition) an algebraically closed field. If there was a polynomial with algebraic number coefficients which had pi as a root, then there are two possibilities:
(1) Pi is an algebraic number; contradiction, as pi is transcendental
(2) The algebraic numbers have a nontrivial algebraic extension; also a contradiction, as they are algebraically closed.

Of course, the hard part is the proof that pi is transcendental; if I recall correctly the idea is to show that i*pi (and therefore pi itself) must be transcendental because e is transcendental and e^(i*pi) = -1 is rational.

>>66
Ah, one more thing: it has to be shown that the algebraic numbers only contain elements which are in fact algebraic over the rationals. This is necessary for the "algebraic number or transcendental number" dichotomy used in (1).

>>66

Yes, it should follow from the Gelfond-Schneider theorem and the transcendentality of e.  But I believe the way Lindemann showed it was by some variation of Liouville's method, by showing that there are rational approximations of pi that are too accurate.

>>39
>>40

*facepalm* You mean people with autism.