Great Theorems of Mathematics

PixyMisa

Persnickety Insect
PixyMisa
Joined
Dec 9, 2002
Messages
16,343
Location
Sunny Munuvia
Just wondered what people consider the coolest theorems in mathematics. They may be the ones that shook up the field, or the ones that made a complex question simple.

I'll nominate Godel's incompleteness theorem* for the first category, and Cantor's diagonalisation argument for the second. The proof that the square root of 2 is irrational is another good one.

What other proofs have made you think, hey, that's really neat!?

* I know he had more than one.
 
Fermat's Last Theorem and the Fundamental Theorem of Arithmetic.

The first is easy to understand and crazy hard to prove. The second is just awesome as a basis.
 
The infinity of the primes should be considered. It's a simple and ingenious proof -- over twenty three hundred years old!
 
PixyMisa said:
Just wondered what people consider the coolest theorems in mathematics. They may be the ones that shook up the field, or the ones that made a complex question simple.

I'll nominate Godel's incompleteness theorem* for the first category, and Cantor's diagonalisation argument for the second. The proof that the square root of 2 is irrational is another good one.

What other proofs have made you think, hey, that's really neat!?

* I know he had more than one.
Click to expand...

I have no idea how much use it is, but the I like the proof that an irrational number raised to the power of an irrational number can be rational.
 
nescafe said:
That is pretty neat. Linky?
Click to expand...

root(2)^root(2) is either rational or irrational. If it's rational, QED.
If it's irrational, then consider (root(2)^root(2))^root(2), which is also one irrational number raised to the power of another.
(root(2)^root(2))^root(2) = root(2)^(root(2)*root(2)) = root(2)^2 = 2. QED.
 
AvalonXQ said:
root(2)^root(2) is either rational or irrational. If it's rational, QED.
If it's irrational, then consider (root(2)^root(2))^root(2), which is also one irrational number raised to the power of another.
(root(2)^root(2))^root(2) = root(2)^(root(2)*root(2)) = root(2)^2 = 2. QED.
Click to expand...

Short and sweet. Very nice.
 
As a Statistics teacher, I like the Central Limit Theorem. I also like sample size calculations - where margin of error determines sample size, regardless of population size. Pretty cool how a sample of ~1000 can give a small margin of error of a vote, regardless of how many people are in the voting population. Of course, it all depends on how well the sample has been selected.
 
maddog said:
As a Statistics teacher, I like the Central Limit Theorem. I also like sample size calculations - where margin of error determines sample size, regardless of population size. Pretty cool how a sample of ~1000 can give a small margin of error of a vote, regardless of how many people are in the voting population. Of course, it all depends on how well the sample has been selected.
Click to expand...

...and that people in the sample don't lie about who or what they voted for.
 
Hard to choose just one...

Pi is both irrational and transcendent.

The quantity of prime numbers is infinite.

But the Reimann Hypothesis ... I can barely wrap my mind around it, except in the most general way. It's like watching a cricket match -- I know that something significant is going on, but my only understanding of the action comes from watching the scoreboard.

The Riemann hypothesis implies results about the distribution of prime numbers that are in some ways as good as possible. Along with suitable generalizations, it is considered by some mathematicians to be the most important unresolved problem in pure mathematics.
Click to expand...
 
AvalonXQ said:
root(2)^root(2) is either rational or irrational. If it's rational, QED.
If it's irrational, then consider (root(2)^root(2))^root(2), which is also one irrational number raised to the power of another.
(root(2)^root(2))^root(2) = root(2)^(root(2)*root(2)) = root(2)^2 = 2. QED.
Click to expand...

This is the canonical example of a "non-constructive proof". It's cool because it proves the existence of something without providing an example.
 
I would never put Fermat's Last Theorem among the coolest; it isn't very useful as a lemma, and the currently accepted proofs are rather inelegant.

I think this is the most beautiful theorem in mathematics:
sum for k=1 to infinity (1/k^s) = product for p a prime (p^s-1)/(p^s).
Proof: http://bit.ly/mHASn
 
You must log in or register to reply here.

Back
Top Bottom