Suggestions of mathematical theorems

[mapleleafs]

Can any of you guys suggest the names of some good (not too complicated), mathematical theorems? For example, I mean something along the lines of Pythagorean theorem, or Heron's theorem. I have this school assignment where I have to pick a theorem and prove it and so on, but I'm having a tough time just deciding which theorem to do.

Suggestions much appreciated.

[Martin]

Fermat's last theorem:

x^n + y^n = z^n has no integer solutions for all n > 2 and x,y,z != 0.

[Martin]

Prove that the number of primes are infinite. Easy, but cool. Or prove the chain rule for differential equations.

EDIT: Better one. Prove that between any two non equal irrational numbers there is an irrational number, and between every two non-equal rational numbers, there is a rational number.

EDIT: sorry, fixed that one ^^

[mapleleafs]

Just for clarification, when you say "between" two such numbers, you mean the "difference between" the two numbers right?

[Martin]

Prove:
If a and b are irrational, and a < b, then there exists an irrational c such that a < c < b
If a and b are rational, and a < b, then there exists a rational c such that a < c < b

[Martin]

Proof of infinite primes:

We know that the list of primes is not empty (since we know that 3, 5, 7, 11, 13, ... ) are prime.

Let p be the set of all known primes.

So assume that there are only n primes. Multiply all of these together creating M = p1 x p2 x ... x pn.

We know that every prime number divides M. Add 1 to M, and now no prime number divides it (since dividing by any prime would give a remainder of 1).

Therefore, M+1 is prime, or has prime factors not in P. Since this works for any n, there are infinite primes.

[Cervantes]

The Goldbach Conjecture:

Prove that every even number greater than 2 can be written as the sum of two primes.

Or prove that the above statement is not true. Either way.

[Hikaru79]

You guys are evil Looks like fun.

Here's two more great ideas. Prove the Riemann Hypothesis or the Poincare Conjecture. If you do, you'll probably get a pretty good mark

[Martin]

On the more easy and serious side.

Prove that 0.9... is exactly 1 and not just really close (it's true, I promise).

[Naveg]

Martin wrote:

On the more easy and serious side.

Prove that 0.9... is exactly 1 and not just really close (it's true, I promise).

No it's not true, but i won't give away the solution. Once infinity is involved in a proof, the proof is pretty well null. This question must be expressed as a limit

[Martin]

There's absolutely no question that it's true.

EDIT: So as not to make 3 posts in a row, here are 54 proofs of the Pythagorean Theorem. http://www.cut-the-knot.org/pythagoras/index.shtml

[Martin]

A bunch of proofs.

1
Let x = 0.9...
10x = 9.9...
10x - x = 9x = 9
.: x = 1

2
0.99... = 9/10 + 9/100 + 9/1000 + ...
let s = 9 * (1/10 + 1/100 + ... + 1/(10^n) + ...)

We know that the sum of a geometric series of the form
S = a + ar + ar^2 + ar^3 + ... + ar^n + ...
is S = a/(1 - r)

Let a = 9/10 and r = 1/10 then
S = 9/10 / (1 - 1/10) = 9/10 / 9/10 = 1.

3
1/3 = 0.333...
2/3 = 0.666...
3/3 = 0.999...

I've got more if you're not convinced

[md]

The last proof certainly isn't true; 1/3 does not equal 0.33... it may be close but it's not equal. As for the other proofs, though they seem pretty convincing they only work because you're using an infinite number of 9's; and infinities are cheating.

[Martin]

Infinity's not cheating. You don't believe in limits eh?

Another one.

Number theory tells us that there must be an infinite number of terms between 0.9... and 1, or they're equal. And these numbers are...?

The last one is true, 1/3 is indeed equal to 0.333...

[Albrecd]

Quote:

Once infinity is involved in a proof, the proof is pretty well null

It's used in converting repeating decimals into fractions in the same context as his first example (9.9... - 0.9...)