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anilv
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Example of a non-constructive proof
Claim: There exist irrational numbers p and q such that p^q is a rational number.
Proof: Consider the irrational number x = sqrt(2) [easy to prove this is irrational]. Now consider the number y = x^x. If y is rational, we're done. If not, then y is irrational. In that case, consider the number y^x.
y^x = (x^x)^x = x^(x times x) = sqrt(2) ^ (sqrt(2) times sqrt(2) ) = sqrt(2) ^ (2) = 2 [a rational number]
Therefore, there must exist irrational numbers p and q such that p^q is rational. This proof is valid even though an explicit pair of numbers with this property was never found.
_________________ (DefQon1) use a Rhino reconstruotereatarerer - (Pasta) I need to figure out how to get rid of this UrQa Suqqa Ukuk (Bluenoser) Put your finger in your mouth and gag reflex should do the rest
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Thu Apr 05, 2012 3:31 pm |
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anilv
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Re: Example of a non-constructive proof
Lemma: sqrt(2) is irrational.
Proof: Suppose sqrt(2) = p/q for some integers p and q that do not share prime factors. Then
2 = p^2 / q^2 2 times q^2 = p^2
So p^2 is even. This implies p is even as well since an odd number squared is odd. We can write p = 2r for some integer r.
sqrt(2) = 2r / q 2 = 4 r^2 / q^2 q^2 = 2 r^2
Therefore q^2 is even, so q is also even by the same reasoning. But we stipulated that p and q shared no prime factors, so this is a contradiction. Therefore, sqrt(2) cannot be written as the quotient of two integers, i.e. it is irrational.
_________________ (DefQon1) use a Rhino reconstruotereatarerer - (Pasta) I need to figure out how to get rid of this UrQa Suqqa Ukuk (Bluenoser) Put your finger in your mouth and gag reflex should do the rest
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Thu Apr 05, 2012 3:33 pm |
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Swift.Kill
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Re: Example of a non-constructive proof
Enk would u please explain that math xD? Calcytron
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Thu Apr 05, 2012 4:26 pm |
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anilv
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Re: Example of a non-constructive proof
The first post proves that certain numbers exist without giving any way of actually producing an example of them. The second post is just a supporting proof of one of the facts used in the first post.
The possibility of non-constructive proofs is a big deal philosophically. Most people aren't comfortable with the idea that you can prove something exists without being able to actually give an example of it.
_________________ (DefQon1) use a Rhino reconstruotereatarerer - (Pasta) I need to figure out how to get rid of this UrQa Suqqa Ukuk (Bluenoser) Put your finger in your mouth and gag reflex should do the rest
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Thu Apr 05, 2012 4:33 pm |
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kilua
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Re: Example of a non-constructive proof
This is beautiful, here it is explained in a video i found after i looked it up in the interwebs. skipped to where he explains it: http://www.youtube.com/watch?feature=pl ... nHI#t=354s
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Thu Apr 05, 2012 4:50 pm |
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anilv
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Re: Example of a non-constructive proof
Nice link. The guy does the same exact proof (of course, it's pretty standard).
_________________ (DefQon1) use a Rhino reconstruotereatarerer - (Pasta) I need to figure out how to get rid of this UrQa Suqqa Ukuk (Bluenoser) Put your finger in your mouth and gag reflex should do the rest
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Thu Apr 05, 2012 5:58 pm |
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thijs12b
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Re: Example of a non-constructive proof
Screw you guys!
_________________ My Characters (to avoid confusion): Xenophanes, Protagoras, Error Message, Aku'Qa and Unfriendly
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Thu Apr 05, 2012 6:12 pm |
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playerboy345
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Re: Example of a non-constructive proof
Screw me too!
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Fri Apr 06, 2012 4:02 am |
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Imra15
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Re: Example of a non-constructive proof
don't forget to screw me
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Fri Apr 06, 2012 5:25 am |
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Jesus 2.0
Joined: Sat Dec 18, 2010 3:11 pm
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Re: Example of a non-constructive proof
I feel left out now.
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Fri Apr 06, 2012 6:48 am |
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kilua
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Re: Example of a non-constructive proof
I feel sad for you people who dislike math
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trevor54 wrote: im only a noob Calypso wrote: adum is the best.
Samantha is second.
Calypso is third.
no wai!
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Fri Apr 06, 2012 7:10 am |
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zhuang281
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Re: Example of a non-constructive proof
Enks, that was one of the questions I had to prove in my final maths exam for Year 12
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Fri Apr 06, 2012 7:17 am |
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Shawn
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Re: Example of a non-constructive proof
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Fri Apr 06, 2012 10:27 am |
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a jedi master
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Re: Example of a non-constructive proof
In college you tend to hate actual numbers, because then you have to carry them with you through all the derivatives/integrals
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Fri Apr 06, 2012 11:00 am |
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anilv
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Re: Example of a non-constructive proof
a jedi master wrote: In college you tend to hate actual numbers, because then you have to carry them with you through all the derivatives/integrals This is what I try to tell my students but some of them just refuse to work in terms of variables. Algebra, people!
_________________ (DefQon1) use a Rhino reconstruotereatarerer - (Pasta) I need to figure out how to get rid of this UrQa Suqqa Ukuk (Bluenoser) Put your finger in your mouth and gag reflex should do the rest
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Fri Apr 06, 2012 11:05 am |
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