Your favorite mathematical theorems

[micromass]

A theorem in mathematics can be beautiful in a lot of different ways. The result can be surprising or satisfying. Or perhaps the proof is very elegant and beautiful. Or maybe the theorem nice because it can be applied to other mathematics, or physics or engineering, or anything.

So give here the theorems you find especially beautiful and be sure to say why you like it so much. There is no wrong answer!

 

[jhae2.718]

I am particularly fond of the Cayley transformation,$$\begin{align*}C&=(1-Q)(1+Q)^{-1} = (1+Q)^{-1}(1-Q)\\Q&=(1-C)(1+C)^{-1} = (1+C)^{-1}(1-C)\end{align*}$$where $C$ is an orthogonal matrix and $Q$ is a skew-symmetric matrix.

Why?

1. The right hand side commutes.
2. The forward transformation is the same form as the reverse; you just need to swap $C$ and $Q$.
3. It is an extremely useful result for passively parameterizing rigid-body attitude.
4. It can be used to transform rotation into a cross-product operation.

Cayley, Arthur. "Sur quelques propriétés des déterminants gauches." Journal für die reine und angewandte Mathematik 32 (1846): 119-123.

 

[micromass]

I am particularly fond of the Cayley transformation,$$\begin{align*}C&=(1-Q)(1+Q)^{-1} = (1+Q)^{-1}(1-Q)\\Q&=(1-C)(1+C)^{-1} = (1+C)^{-1}(1-C)\end{align*}$$where $C$ is an orthogonal matrix and $Q$ is a skew-symmetric matrix.

Why?

1. The right hand side commutes.
2. The forward transformation is the same form as the reverse; you just need to swap $C$ and $Q$.
3. It is an extremely useful result for passively parameterizing rigid-body attitude.
4. It can be used to transform rotation into a cross-product operation.

Cayley, Arthur. "Sur quelques propriétés des déterminants gauches." Journal für die reine und angewandte Mathematik 32 (1846): 119-123.

And even more interesting: it works in infinite dimensions too. You can use it to prove a very general spectral theorem.

 

[anorlunda]

I'm a physics buff. I particularly love Noether's Theorem and Liouville's Theorem because they have immense importance in physics, yet they were derived in pure math ignorant of the physical significance.

I also delight in the Schrödinger equation because it ties knowledge to energy. There can be no knowledge without energy expenditure. That is stunning.

 

[PeroK]

I have always liked:

1. The Remainder Theorem
2. The Binomial Theorem
3. The Bolzano-Weierstrass Theorem

It seems that so much rests on them. They are like perfect foundations.

 

[wukunlin]

The squeeze theorem.

For some reason it reminds me of piping bags
http://www.papstar-products.com/papstar_pe/prodpic/100-Piping-bag-2-6-l-55-cm-x-26-5-cm-transparent-12488_b_0.JPG

 

[leroyjenkens]

Pythagorean theorem. Not only because it has helped me so much with problems I was probably supposed to solve in a more difficult manner, but also because it's the only mathematical theorem I can think of.

 

[micromass]

Pythagorean theorem. Not only because it has helped me so much with problems I was probably supposed to solve in a more difficult manner, but also because it's the only mathematical theorem I can think of.

The Pythagorean theorem is indeed one of those theorems which are extremely beautiful. The result is totally unexpected in my opinion: I see no a priori reason or explanation for why the sides of a rectangular triangle should behave in this way. The proofs of the theorem are also quite nice. There are a lot of proofs too (not all of which are that rigorous).

http://www.cut-the-knot.org/pythagoras/

 

[Tosh5457]

Stokes[/PLAIN] theorem and residue theorem. They're so important that they contain much of the information of vector calculus and complex analysis, respectively.

 

[BOAS]

I don't know if it has a nice name, but it was the first theorem I worked through largely on my own (had done others in class), and the feeling of satisfaction was great.

"The sum of the first n positive integers is $$\frac{n(n+1)}{2}$$"

 

[micromass]

Stokes[/PLAIN] theorem and residue theorem. They're so important that they contain much of the information of vector calculus and complex analysis, respectively.

Stokes' Theorem is extremely elegant and very important. It has a lot of consequences such as Green's theorem, the divergence theorem, the fundamental theorem of algebra, Brouwer's fixed point theorem, the invariance of domain theorem, a lot of complex analysis theorems, etc. It's definitely one of the most important theorems out there. Too bad that most books don't give a nice derivation of the theorem.

The residue theorem is also very nice. Complex analysis has a lot of beautiful theorems. For exampel, the Cauchy integral formula is very nice too: http://en.wikipedia.org/wiki/Cauchy's_integral_formula

I don't know if it has a nice name, but it was the first theorem I worked through largely on my own (had done others in class), and the feeling of satisfaction was great.

"The sum of the first n positive integers is $$\frac{n(n+1)}{2}$$"

This was the theorem that Gauss discovered when he was about 5 years old:

This is attributed to an early school lesson when the teacher thought he would keep the class busy whilst he popped out for something. He set the test of adding all the whole numbers from 1-100. By the time he reached the door, Gauss had the answer.Gauss imagined the problem as 1 + 2 + 3 +...+98 + 99 + 100, but then he wrote the numbers underneath but in reverse order. 100 +99 + 98...+3 + 2 + 1. So each 100 pairs of vertical numbers added up to 101 so the total was 10100 but this is twice the true answer as each number is included twice. The total is therefore 5050. This lead to the general formula that the sum of consecutive numbers from 1 to n is n(n+ 1) ÷ 2.

Another very nice proof of this result is geometrical: http://www.mathsisfun.com/algebra/triangular-numbers.html

The generalization of this result would be: what is the sum

$$1^2 + 2^2 + 3^2 + ... + n^2$$

or in general, what is the sum of $$1^k + 2^k + 3^k + ... + n^k$$

This is not at all obvious. The results for the first few $k$ are:

$$\begin{eqnarray*}<br /> 1 + 2 + 3 + 4 + 5 + ... + n & = & \frac{1}{2}n^2 + \frac{1}{2}n\\<br /> 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + ... + n^2 & = & \frac{1}{3}n^3 + \frac{1}{2}n^2 + \frac{1}{6}n\\<br /> 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + ... + n^3 & = & \frac{1}{4}n^4 + \frac{1}{2}n^3 + \frac{1}{4}n^2\\<br /> 1^4 + 2^4 + 3^4 + 4^4 + 5^4 + ... + n^4 & = & \frac{1}{5}n^5 + \frac{1}{2}n^4 + \frac{1}{3}n^3 - \frac{1}{30}n\\<br /> 1^5 + 2^5 + 3^5 + 4^5 + 5^5 + ... + n^5 & = & \frac{1}{6}n^6 + \frac{1}{2}n^5 + \frac{5}{12}n^5 - \frac{1}{12}n^2\\<br /> 1^6 + 2^6 + 3^6 + 4^6 + 5^6 + ... + n^6 & = & \frac{1}{7}n^7 + \frac{1}{2}n^6 + \frac{1}{2}n^5 - \frac{1}{6}n^3 + \frac{1}{42}n\\<br /> 1^7 + 2^7 + 3^7 + 4^7 + 5^7 + ... + n^7 & = & \frac{1}{8}n^8 + \frac{1}{2}n^7 + \frac{7}{12}n^6 - \frac{7}{24}n^4 + \frac{1}{12}n^2\\<br /> 1^8 + 2^8 + 3^8 + 4^8 + 5^8 + ... + n^8 & = & \frac{1}{9}n^9 + \frac{1}{2}n^8 + \frac{2}{3}n^7 - \frac{7}{15}n^5 + \frac{2}{9}n^3 - \frac{1}{30}n\\<br /> 1^9 + 2^9 + 3^9 + 4^9 + 5^9 + ...+ n^9 & = & \frac{1}{10}n^{10} + \frac{1}{2}n^9 + \frac{3}{4}n^8 - \frac{7}{10}n^6 + \frac{1}{2}n^4 - \frac{3}{20}n^2\\<br /> 1^{10} + 2^{10} + 3^{10} + 4^{10}+ 5^{10} + ... + n^{10} & =& \frac{1}{11}n^{11} + \frac{1}{2}n^{10} + \frac{5}{6}n^9 - n^7 + n^5 - \frac{1}{12}n^3 + \frac{5}{66}n<br /> \end{eqnarray*}$$

The question is to find a pattern for the general case. We can see some parts of the pattern, but finding the general case is not at all easy. If you want a tough challenge, you can try it.

Here is the solution: http://en.wikipedia.org/wiki/Faulhaber's_formula
A very nice derivation of the general formula can be found in the following intriguing document which attempts to generalize calculus to discrete situations: https://www.cs.purdue.edu/homes/dgleich/publications/Gleich 2005 - finite calculus.pdf

 

[Matterwave]

I was going to say Stoke's theorem...but it's been taken.

 

[micromass]

I was going to say Stoke's theorem...but it's been taken.

***Stokes'

 

[Matterwave]

***Stokes'

Stoke's this just flows naturally from my keyboard.

 

[jostpuur]

There are some truly horrendous "theorems" out there that do not even satisfy the usual standards for theorems. Examples include the Noether's theorem, Bloch's theorem and Spin-statistics theorem. I have never seen these being formulated in such form that they could be called mathematical theorems, but still everybody insists that they would be theorems. All the necessary definitions and assumptions are missing, and only the claims are present. If people don't know how to formulate ideas as theorems, then they should be called conjectures.

 

[Hepth]

There are some truly horrendous "theorems" out there that do not even satisfy the usual standards for theorems. Examples include the Noether's theorem, Bloch's theorem and Spin-statistics theorem. I have never seen these being formulated in such form that they could be called mathematical theorems, but still everybody insists that they would be theorems. All the necessary definitions and assumptions are missing, and only the claims are present. If people don't know how to formulate ideas as theorems, then they should be called conjectures.

I never knew Noether's theorem was not a "theorem" in peoples eyes? Can you explain that? She was a mathematician, I had always assumed it was quite straight forward.
Heres a newer English translation :

http://arxiv.org/pdf/physics/0503066

 

[WannabeNewton]

I never knew Noether's theorem was not a "theorem" in peoples eyes?

Don't take the bait my friend.

 

[Jorriss]

I never knew Noether's theorem was not a "theorem" in peoples eyes? Can you explain that? She was a mathematician, I had always assumed it was quite straight forward.
Heres a newer English translation :

http://arxiv.org/pdf/physics/0503066

One can also see Arnold's statement of Noether's Theorem in his classical mechanics text.

 

[micromass]

There are some truly horrendous "theorems" out there that do not even satisfy the usual standards for theorems. Examples include the Noether's theorem, Bloch's theorem and Spin-statistics theorem. I have never seen these being formulated in such form that they could be called mathematical theorems, but still everybody insists that they would be theorems. All the necessary definitions and assumptions are missing, and only the claims are present. If people don't know how to formulate ideas as theorems, then they should be called conjectures.

Sure, there are some results in physics which are not mathematically rigorous (or are not presented as such). But Noether's theorem is a horrible example of this. See Arnolds book which gives a rigorous version of it.

 

[td21]

Godel's incompleteness theorem.

 

[Dadface]

Fermat's last theorem. Did Fermat really find a proof? If so it must be much simpler than the proof found by John Wiles.Perhaps there was a proof but a mistake in it. I like to think that Fermat was having a joke.

 

[jostpuur]

I have seen the Noether's theorem in many places, and it hasn't looked like a theorem, but perhaps it is a theorem in some places, and not a theorem in some other places.

 

[HomogenousCow]

I have seen the Noether's theorem in many places, and it hasn't looked like a theorem, but perhaps it is a theorem in some places, and not a theorem in some other places.

You can find a straightforward summary of Noether's theorem on wikipedia.
In fact it follows pretty straightforwardly from first principles in lagrangian mechanics.

 

[jostpuur]

Well I have never seen rigor Lagrangian mechanics either, except in my own notes. The physicists talk about minimizing the Lagrangian, but they have no clue of what kind of function space the domain is, or what metric it would have. Of course the domain space must have some metric, because otherwise the Lagrangian couldn't have local minima. I mean how could a mapping

$$L:?\to\mathbb{R}$$

have a local minima, if nobody knows what the domain is, or what metric it would have?

Furthermore, I have also checked with examples, that it is a myth that the Euler-Lagrange equations would always produce minima of the Lagrangian. It is quite common, that the physical solutions are saddle points of the Lagrangian.

It would be very peculiar, if a rigor Noether's theorem could be defined on the framework of the usual Lagrangian mechanics, while it is blatant to all mathematicians of the world that the Lagrangian mechanics itself is nowhere near rigorous.

 

[HomogenousCow]

Well I have never seen rigor Lagrangian mechanics either, except in my own notes. The physicists talk about minimizing the Lagrangian, but they have no clue of what kind of function space the domain is, or what metric it would have. Of course the domain space must have some metric, because otherwise the Lagrangian couldn't have local minima. I mean how could a mapping

$$L:?\to\mathbb{R}$$

have a local minima, if nobody knows what the domain is, or what metric it would have?

Furthermore, I have also checked with examples, that it is a myth that the Euler-Lagrange equations would always produce minima of the Lagrangian. It is quite common, that the physical solutions are saddle points of the Lagrangian.

It would be very peculiar, if a rigor Noether's theorem could be defined on the framework of the usual Lagrangian mechanics, while it is blatant to all mathematicians of the world that the Lagrangian mechanics itself is nowhere near rigorous.

To me rigor is a bottomless well, as long as the formalism is self consistent and agrees with experiment I'm fine with it.

 

[Jorriss]

It would be very peculiar, if a rigor Noether's theorem could be defined on the framework of the usual Lagrangian mechanics, while it is blatant to all mathematicians of the world that the Lagrangian mechanics itself is nowhere near rigorous.

You need to look at better sources, Lagrangian mechanics has a rigorous framework.

 

[micromass]

while it is blatant to all mathematicians of the world that the Lagrangian mechanics itself is nowhere near rigorous.

Blatant to you maybe. I certainly know that it's rigorous enough, and I don't think I'm alone in this.

 

[atyy]

Godel's incompleteness theorem.

I think it's the most amazing theorem in the world. I'm not a mathematician so I'll accept anything that can be made intuitively obvious without proof. As a consequence, I have only read two proofs in my life. Goedel's is one of them (the other was Shannon's).

Stokes[/PLAIN] theorem and residue theorem. They're so important that they contain much of the information of vector calculus and complex analysis, respectively.

Stokes's is beautiful. I have never seen its proof though, since Maxwell's equations make a form of it physically obvious :)

Pythagorean theorem. Not only because it has helped me so much with problems I was probably supposed to solve in a more difficult manner, but also because it's the only mathematical theorem I can think of.

The Pythagorean theorem is beautiful too. Again I have never seen the proof. I believed it after cutting out pieces of paper!

 

[atyy]

The Pythagorean theorem is indeed one of those theorems which are extremely beautiful. The result is totally unexpected in my opinion: I see no a priori reason or explanation for why the sides of a rectangular triangle should behave in this way. The proofs of the theorem are also quite nice. There are a lot of proofs too (not all of which are that rigorous).

http://www.cut-the-knot.org/pythagoras/

What are some examples of the proofs that you don't consider rigourous?

 

[atyy]

I'm a physics buff. I particularly love Noether's Theorem and Liouville's Theorem because they have immense importance in physics, yet they were derived in pure math ignorant of the physical significance.

The usual story told to physicists is that Noether did know the physical importance of the work, and that she started looking at the question because Hilbert had become interested in variational principles in physics (including energy conservation in general relativity). http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html

By Liouville's theorem, do you mean http://en.wikipedia.org/wiki/Liouville's_theorem_(Hamiltonian)? It seems to be about Hamiltonian mechanics, so could it really have been discovered without knowing its significance?