This thread should be used to post your favourite theorems, not only their names, but also what they state.

 

Fundamental Theorem of Arithmetic

 

Each integer greater than 1 can be expressed as a product of primes, and, except for the order in which these primes are written, this can be done in only one way.

 

and

 

Central Limit Theorem

 

If [math]X_1,...,X_n[/math] are independent identically distributed, with mean [math]\mu[/math] and finite variance [math]\sigma^{2}[/math], then [math]\bar X[/math] is approximately [math]

N(\mu ,{{\sigma ^2 } \mathord{\left/

{\vphantom {{\sigma ^2 } n}} \right.

\kern-\nulldelimiterspace} n})

[/math]

for large n, no matter what the distribution of [math]X[/math]

For me, it'd have to be something like the intermediate value theorem, because it seems so inanely pointless, but allows you to prove very useful things.

could u state it please. so that anyone who might be interested would know what your talking about

Euclid's Parallel Postulate

 

Through a point, not on a line, there exists exactly 1 line parallel to the given line.

 

sum (k=1..inf) 1/kn = ?

 

Although others have found that this expression equals PI2 / 6 when n=2, PI4 / 90 when n = 4 and simular solutions for all possible even values of n, no one has discovered an exact value when n is an odd integer (3, 5, 7, ...) (note: when n=1, the sum does not converge, but it does has relations to the gamma constant).

 

Sorry I copied and pasted:

http://www.geocities.com/RainForest/Vines/2977/gauss/euclidpp.html

Certainly.

 

IVT

 

For a function [math]f : [a,b] \to \mathbb{R}[/math] which is continuous, then [math]\forall\, c \in [a,b] \, \exists \, v \in (f(a), f(b))[/math] such that [math]f© = v[/math].

I also quite like the Monotonic Sequence Theorem.

 

Let [math](x_n)[/math] be a sequence which is non-decreasing for [math]n \ge N[/math]. If [math](x_n)[/math] is bounded above, then [math](x_n)[/math] converges, and the limit is the supremum s of the set [math]{x_n:n \in \mathbb{Z}, n \ge N}[/math]. If [math](x_n)[/math] is not bounded above then [math](x_n)[/math] diverges to infinity.

- Heine-Borel Theorem: a subspace of R^n is (with the usual topology) is compact iff it is closed and bounded.

- Bolzano-Weierstrass Theorem: Every bounded infinite set in R^n has an accumulation point

- Sylow's Theorems: http://mathworld.wolfram.com/SylowTheorems.html

- Mean Value Theorem: http://mathworld.wolfram.com/Mean-ValueTheorem.html

Euclids parallel postulate is an axiom and not a theorem.

 

I would say Riesz's Representation theorem : The dual of every hilbert space can be isometrically isomophically embedded into itself.

 

Mandrake

e^ipi = -1

 

It's fun and fruity!

That is not a theorem either, but more a direct result from an identity

exp(i phi) =cos(phi) + i*sin(phi)

 

Mandrake

That is not a theorem either' date=' but more a direct result from an identity

exp(i phi) =cos(phi) + i*sin(phi)

 

Mandrake[/quote']

 

I know it is, but e^i theta isn't as fun and fruity as e^i pi.

I like De Moivre's Theorem. Other than finding complex roots, I like it as a easy(?) way to remember expansions of the sin & cos funx.

 

De Moivre's Theorem

 

If z = r cos(θ) + i r sin(θ), then zⁿ = r cos(nθ) + i r sin(nθ) for all n ∈ N

 

Source: Anton, Rogers. Elementary Linear Algebra Applications Version

I like De Moivre's Theorem. Other than finding complex roots' date=' I like it as a easy(?) way to remember expansions of the sin & cos funx.

 

[b']De Moivre's Theorem[/b]

 

If z = r cos(?) + i r sin(?), then zⁿ = r cos(n?) + i r sin(n?) for all n ? N

 

Source: Anton, Rogers. Elementary Linear Algebra Applications Version

 

De Moivre's is winking at me. It's never done that before

I think the statement of the moivre's theorem is :

If z = rcos(x) + irsin(x), then

z^n = r^n \cos(n x) + i r^n \sin(n x) for all n \in \mathbb{N}

 

Mandrake

Hey how do you guys make the math signs ?

 

Mandrake

There's a thread on using the LaTex maths renderer in the General Maths forum.

 

LaTex is like bb code only more... mathsy.

Murphys Law, relating to the maximum cussedness of matter, Stating that:

 

"If anything CAN go wrong, it Will!"

I think the statement of the moivre's theorem is :

If z = rcos(x) + irsin(x)' date=' then

z^n = r^n \cos(n x) + i r^n \sin(n x) for all n \in \mathbb{N}

 

Mandrake[/quote']

think it works for all rational numbers as well

If [math]z = rcos(x) + irsin(x)[/math], then

[math]z^n = r^n \cos(n x) + i r^n \sin(n x) \; \forall \; n \in \mathbb{N}

[/math]

 

Mandrake

 

PS : why my math formula doesn't show up ?

The only thing to do to have full LaTeX use is to write [ math ], [/math ] around the LaTeX code right ?

Yes bloodhound, it does indeed

 

Mandrake

My favorites:

The Fundamental Theorem of Algebra: An n-th order polynomial has exactly n roots in the complex complex.

For example, [math]x^4-4[/math] has 4 complex roots (Two real, two imaginary).

 

And of course,

The Fundamental Theorem of Calculus

http://archives.math.utk.edu/visual.calculus/4/ftc.9/

[math]If z = rcos(x) + irsin(x)' date=' then

z^n = r^n \cos(n x) + i r^n \sin(n x) \; \forall \; n \in \mathbb{N}

[/math']

PS : why my math formula doesn't show up ?

The only thing to do to have full LaTeX use is to write [ math ], [/math ] around the LaTeX code right ?

You need to use valid syntax. That includes not using invalid syntax, such as trying to write your formula in a sentence.

 

 

If [math]z = rcos(x) + irsin(x)[/math], then [math]z^n = r^n \cos(n x) + i r^n \sin(n x) \; \forall \; n \in \mathbb{N}[/math]

Hey thanks. Thought it could take the text also.

 

Mandrake

ah...drats

 

I tried C&P the theta sign into the quick-reply box. I guess it doesn't work. Thanks to Sayo for the non-winking version of De Moivre's Th'm.

 

I found the Binomial Theorem neat too:

 

The theorem that' date=' for positive integers n,

[img']http://mathworld.wolfram.com/bimg2594.gif[/img]

the so-called binomial series, where are binomial coefficients.

 

It was very neat when we studied it with Pascal's triangle, combinatorics, and the like.

without a doubt Cantor's theorem on the nondenumberability of the continuum.