beautiful math demonstrations

[perceval]

I read a book recently that mentioned cantor's proof that there were "more" real number than natural numbers.

i was taught this demonstration in college some 15 years ago (... oh my god...) and i remember it perfectly even though i never used it since, because its such a beautiful yet simple idea.


Did you have that kind of experience in math ? which proof do you remember as beautiful ?

Now that i think of it i can recall one other beautiful proof:

Fermat's little theorem by group theory: the proof by group theory is absolutly elementary(when you have the group theory basis of course): you can't forget it after you've seen it once . Compare to the pedestrian proof ("counting necklace proof" in the url)

An exercise that also left me a durable memory is the van der monde determinant .
Again, if you try with a brute force method you can be in a world of pain yet a simple argument gives you the beautiful simple answer. This matrix is even useful in real life for a bunch of stuff!

Unfortunatly, after college proof became more and more convoluted with a succession of very abstract lemmas ... and i turned to physics but i would have liked to be able to work with this kind of math more. Unfortunatly those maths are more than a century old.

[cata]

http://mathoverflow.net/questions/8846/proofs-without-words

[perceval]

http://mathoverflow.net/questions/8846/proofs-without-words

nice link !

[Solomon]

Did you have that kind of experience in math ? which proof do you remember as beautiful ?

The ones I could solve by myself.

[bayu]

It needs a little number theory to enjoy it:

It is easy to see, that if you got a number >2 which is prime and can be written as the sum of two squares, then it is congruent 1 modulo 4.

The other direction is more interesting:

If you have a prime number which is congruent 1 modulo 4, then it is always the sum of two squares.

This theorem goes back to Fermat. I have seen 2 proofs (by Euler and by Gauss) in the literature going over pages. They're beautiful. And then there is a proof in one sentence
http://people.mpim-bonn.mpg.de/zagier/f ... lltext.pdf

It took another genius to find it
http://en.wikipedia.org/wiki/Zagier

By far my favorite.

[flOvermind]

My favourite "simple proof" is the proof to the fact that a triangle inside a half-circle always has a 90° angle.

There are two ways to prove this:
The complicated way involves a lot of juggling around with angle sums. It's easy to do in principle, but rather involved.
The simple way just exploits some symmetry. It's one of these proofs that are hard to find, but once you see it, it's really obvious

[jts]

When I was thinking about this, the proof that the rational numbers are dense in the real numbers came to mind - I think that what made it memorable is that it's intuitive once you start thinking about it, but the proof is tricky. The sandwich theorem was the first non-geometry proof I ever saw, and that was memorable. There are two cool deductions of the volume of a sphere from cones - one by rearranging the slices of the cone, one by assuming the sphere is made up of cones with their vertices at the center.

I also dimly remember being impressed by at least two different proofs involving sets of measure zero, but I can't recall the details or the names of the sets at the moment.

[cyclops]

I like the proof of the number of primes being infinite.

[Redundant]

I like the proof of the number of primes being infinite.

Which one is your favorite? I'm a big fan of one by Erdos that by counting square-free numbers puts a lower bound on the number of primes less than n.

[Bill Spight]

My favourite "simple proof" is the proof to the fact that a triangle inside a half-circle always has a 90° angle.

There are two ways to prove this:
The complicated way involves a lot of juggling around with angle sums. It's easy to do in principle, but rather involved.
The simple way just exploits some symmetry. It's one of these proofs that are hard to find, but once you see it, it's really obvious

My favorite simple proof is of the pons asinorum, or that the opposite angles of an isosceles triangle are equal. The key step of the proof is to flip the triangle over on itself.

As for the proofs about the right triangle, I am not sure which ones you mean. Here is one that is probably different. Complete the circle and draw a diameter from the vertex of the supposed right angle. Draw lines from the intersection to the other vertices. It is easy to show that the resulting quadrilateral is a rectangle.

Edit: Or rotate the half-circle 180 degrees to form a quadrilateral with itself. Then flip the quadrilateral over onto itself.

[cyclops]

I like the proof of the number of primes being infinite.

Which one is your favorite? I'm a big fan of one by Erdos that by counting square-free numbers puts a lower bound on the number of primes less than n.

Well if there are a finite number of them you can multiply them and add one to the product. You have constructed another prime.

I also like the derivation of the formula for the sum of a finite geometric series.
and then, maybe more physics, that any massive body has one unique point inside or outside through with a suspending line must go.

Last time an employee asked me how much is 1000 times 165. I was surprised but answered one hundert sixty five thousand and wrote down 165000. When I wanted it to explain it to him I felt fell silent.

On elliptic curves I like the proof that the constructive multiplication is indeed associative.

Anyone knows a intuitive proof why the volume of a pyramide is indeed 1/3 of base times height?

edit: to fall, fell, fallen

[flOvermind]

As for the proofs about the right triangle, I am not sure which ones you mean. Here is one that is probably different. Complete the circle and draw a diameter from the vertex of the supposed right angle. Draw lines from the intersection to the other vertices. It is easy to show that the resulting quadrilateral is a rectangle.

That's the one I meant

The bottom triangle is a 180 degree rotated version of the top triangle, so the sides are parallel. Because both diagonals of the quadrilateral are diameters of the circle, their length is equal. And any quadrilateral with parallel sides and equal diagonals is a rectangle

[HermanHiddema]

Anyone knows a intuitive proof why the volume of a pyramide is indeed 1/3 of base times height?

How about:

If the base is square, and the top of the pyramid has right angles, then you can put six of them together (tips touching) to make a cube with sides equal to the base. The volume of one pyramid is 1/6th of that of the cube, and since the height is 1/2 of the side length of the cube, the volume is 1/3 of base times height.

Stretching a 3d object is any direction will scale the volume linearly, so the formula holds for any other pyramid.

[cyclops]

My favorite simple proof is of the pons asinorum, or that the opposite angles of an isosceles triangle are equal. The key step of the proof is to flip the triangle over on itself.

As for the proofs about the right triangle, I am not sure which ones you mean. Here is one that is probably different. Complete the circle and draw a diameter from the vertex of the supposed right angle. Draw lines from the intersection to the other vertices. It is easy to show that the resulting quadrilateral is a rectangle.

Edit: Or rotate the half-circle 180 degrees to form a quadrilateral with itself. Then flip the quadrilateral over onto itself.

If you connect the third point of the triangle to the center of the circle you have divided the triangle into two isosceles triangles. Flip them as in Bill's pons asinogwat towards each other and you have a rectangle!

[Bill Spight]

My favorite simple proof is of the pons asinorum, or that the opposite angles of an isosceles triangle are equal. The key step of the proof is to flip the triangle over on itself.

As for the proofs about the right triangle, I am not sure which ones you mean. Here is one that is probably different. Complete the circle and draw a diameter from the vertex of the supposed right angle. Draw lines from the intersection to the other vertices. It is easy to show that the resulting quadrilateral is a rectangle.

Edit: Or rotate the half-circle 180 degrees to form a quadrilateral with itself. Then flip the quadrilateral over onto itself.

If you connect the third point of the triangle to the center of the circle you have divided the triangle into two isosceles triangles. Flip them as in Bill's pons asinogwat towards each other and you have a rectangle!

You mean two flips to make a rotation, right?