# Why does A/B= Root 2?



## Clever-Fox (Oct 3, 2013)

Just a thought, but why is it that this equation, A/B= Root 2, is true? I understand the math behind it, but it still boggles my mind...


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## jastius (Oct 3, 2013)

isn't it the reverse of veita's formula? where:


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## Clever-Fox (Oct 9, 2013)

Not sure... Although, I've never seen _that_ formula before...


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## Vertigo (Oct 10, 2013)

Tell me more; what are A and B and does 'root 2' mean square root of 2?


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## Dozmonic (Oct 10, 2013)

It's a proof by unique factorisation that root 2 is irrational

From Wikipedia:

*Proof by unique factorization*

 An alternative proof uses the same approach with the fundamental theorem of arithmetic which says every integer greater than 1 has a unique factorization into powers of primes.


Assume that 
	

	
	
		
		

		
		
	


	




 is a rational number. Then there are integers _a_ and _b_ such that _a_ is coprime to _b_ and 
	

	
	
		
		

		
		
	


	




. In other words, 
	

	
	
		
		

		
		
	


	




 can be written as an irreducible fraction.
The value of _b_ cannot be 1 as there is no integer _a_ the square of which is 2.
There must be a prime _p_ which divides _b_ and which does not divide _a_, otherwise the fraction would not be irreducible.
The square of _a_ can be factored as the product of the primes into which _a_ is factored but with each power doubled.
Therefore by unique factorization the prime _p_ which divides _b_, and also its square, cannot divide the square of _a_.
Therefore the square of an irreducible fraction cannot be reduced to an integer.
Therefore the square root of 2 cannot be a rational number.
 This proof can be generalized to show that if an integer is not an  exact kth power of another integer then its kth root is irrational. The  article quadratic irrational gives a proof of the same result but not using the fundamental theorem of arithmetic.


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## Velocius quam lucem (Oct 10, 2013)

Then what the proof is really saying is that A/B != to root 2. (!= meaning not equal)

This may get some wheels turning:

From the equality √2  = a/b   it follows that 2 = a Squared/b Squared, or a Squared = 2 * b Squared.  So the square of "a" is an even number since it is two times something. From this we can see that "a" itself is also an even number. Why? Because it can't be odd; if "a" itself was odd, then a * a would be odd too. An odd number times odd number is always odd.  (try it out)

I have always found it interesting that 1 over the square root of 2 is equal to the square root of 2 over 2, or 1/√2 = √2/2. (It can also be written as 2 to the 1/2 power over 2, or 1 over 2 to the 1/2 power).


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