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 September 11th, 2012, 05:58 PM #1 Senior Member   Joined: Jan 2010 Posts: 324 Thanks: 0 Proof for there does not exist a unique x that is an ... Proof of: There does not exist a unique x that is an element of the rationals such that x squared equals 3. Would contradiction be the way to do this or trichotomy like how you would for there exists a unique x >0 such that x squared equals 2?
 September 11th, 2012, 06:33 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Proof for there does not exist a unique x that is an .. Personally, I would use reductio ad absurdum (contradiction) for this.
 September 11th, 2012, 06:44 PM #3 Senior Member   Joined: Nov 2011 Posts: 595 Thanks: 16 Re: Proof for there does not exist a unique x that is an .. Hi, Ok I have never been very good at this so please don't take this for granted, might be wrong, just an attempts here. So if $\sqrt{3}$ is rational ,there exists p and q with gcd(p,q)=1 such that $\frac{p}{q}=\sqrt{3}$ implies that $p^2=3q^2$. Since 3 is prime $p^2$ can not divide 3, so $p^2$ divides $q^2$ which means there exists an integer a such that $q^2=ap^2$ which means $3a=1$ which is absurd (a is an integer). I guess this shows generally (IF it is correct....) that $\sqrt{p}$ with p prime is always irrational.
 September 11th, 2012, 07:03 PM #4 Senior Member   Joined: Jan 2010 Posts: 324 Thanks: 0 Re: Proof for there does not exist a unique x that is an .. Mark, elaborate. I would do x squared does not equal three (he hasnt defined square root so i cant use it in the proof). Then do what?
 September 11th, 2012, 08:18 PM #5 Senior Member   Joined: Nov 2011 Posts: 595 Thanks: 16 Re: Proof for there does not exist a unique x that is an .. So is there something wrong in the proof I wrote? Would be nice to at least acknowledge you saw it, especially when people spend a bit of their time to try to help I hope it is not just a problem that "he hasnt defined defined square root" because you can of course simply start the proof saying: Let us assume a rational x=p/q such that $x^2=3$ so that $\frac{p^2}{q^2}=3$ and then catch back above....
 September 11th, 2012, 08:20 PM #6 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Proof for there does not exist a unique x that is an .. I would begin by assuming there is a rational number whose square is 3: $\frac{p^2}{q^2}=3$ where p and q are co-prime, i.e., $\frac{p}{q}$ is in lowest terms. This implies: $p^2=3q^2$ which means $p^2$ must be a multiple of 3, but since p is squared, it must be a multiple of 9, which implies $q^2$ must also be a multiple of 9, and so our assumption that p and q are co-prime has been contradicted.
September 11th, 2012, 08:34 PM   #7
Senior Member

Joined: Nov 2011

Posts: 595
Thanks: 16

Re: Proof for there does not exist a unique x that is an ..

I think this one also works!
Just when you say
Quote:
 which means must be a multiple of 3, but since p is squared, it must be a multiple of 9, which implies must also be a multiple of 9
actually I think it does not imply $q^2$ is a multiple of 9 since there is a three already multiplying $q^2$ but at least $q^2$ has to be a multiple of 3 which is a common factor with 9 so the conclusion of absurdity is safe here!

 September 11th, 2012, 08:46 PM #8 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Proof for there does not exist a unique x that is an .. Yes, you are right, I meant to type a 3, but typed 9 instead! I'm glad you caught that!
 September 11th, 2012, 09:04 PM #9 Senior Member   Joined: Nov 2011 Posts: 595 Thanks: 16 Re: Proof for there does not exist a unique x that is an .. No problem And both proofs would work the same for any prime number instead of three
 September 11th, 2012, 09:08 PM #10 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Proof for there does not exist a unique x that is an .. Yes, and hopefully we are not as stricken as the followers of Pythagoras...

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