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Prove If x^2 is irrational then x is irrational 
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#1
Jan612, 02:29 AM

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1. The problem statement, all variables and given/known data
Prove If x^2 is irrational then x is irrational. I can find for example π^2 which is irrational and then π is irrational but I don't know how to approach the proof. Any hint? 


#2
Jan612, 04:44 AM

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#3
Jan612, 06:49 AM

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#4
Jan612, 06:59 AM

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Prove If x^2 is irrational then x is irrational
Wouldn't it suffice to observe that [itex]\frac{p^2}{q^2}[/itex] is a reduced rational number since p and q are coprime? Which would imply that [itex]x^2[/itex] is rational as well, which contradicts the original assumption of the irrationality of [itex]x^2[/itex].
In other words, the negation of the proposition [itex]x^2 \notin \mathbb{Q} \Rightarrow x \notin \mathbb{Q}[/itex] leads to a contradiction. Hence the proposition is true. 


#5
Jan612, 07:09 AM

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#6
Jan612, 08:25 AM

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The only thing I can think of more fundamental than that would be to fully primefactorise [itex]p^2[/itex] and [itex]q^2[/itex], but this just leads to a more messy yet no more convincing argument. 


#7
Jan612, 08:39 AM

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#8
Jan612, 10:39 AM

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RGV 


#9
Jan612, 10:44 AM

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Thanks everybody!



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