What are your Fav Theorems.

[bloodhound]

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]

[Dave]

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.

[bloodhound]

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

[Tesseract]

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

[Dave]

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].

[bloodhound]

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.

[fourier jr]

- 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

[MandrakeRoot]

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

[JaKiri]

e^ipi = -1

 

It's fun and fruity!

[MandrakeRoot]

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

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

 

Mandrake

[JaKiri]

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.

[NSX]

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

[JaKiri]

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

[MandrakeRoot]

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

[MandrakeRoot]

Hey how do you guys make the math signs ?

 

Mandrake

[Sayonara]

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

 

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

[YT2095]

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

 

"If anything CAN go wrong, it Will!"

[bloodhound]

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

[MandrakeRoot]

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 ?

[MandrakeRoot]

Yes bloodhound, it does indeed

 

Mandrake

[dryan]

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/

[Sayonara]

[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]

[MandrakeRoot]

Hey thanks. Thought it could take the text also.

 

Mandrake

[NSX]

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.

[zaphod]

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