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Prove $\sqrt{3}$ is irrational. (Proof by contradiction).

Let $\sqrt{3}$ be a rational number in simplest form $\frac pq$.

So squaring both sides of $\sqrt{3}=\frac pq$ we get $3=(\frac {p}{q})^2$ which translates to $3=\frac{p^2}{q^2}$.

Multiply both sides of the equation by $q^2$ yields $3q^2=p^2$. Now $p^2$ is taken to be divisible by 3 and thus an odd number, $p$ is also odd because any odd number squared is also odd.

So let $p=3s$ where s is an integer. Then $3q^2=(3s)^2 = 3q^2=9s^2$. Dividing both sides of the equation by 3 leaves us with $q^2=3s^2$.

Here is is taken that $q^2$ is divisible by 3 and is odd and so is $q$.

Therefore both $q \text{ and}\; p$ have a common factor of being odd and divisible by 3, proving that the $\sqrt{3}$ is irrational.

Are there any gaps that I could improve on?

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    The part "$p^2$ is divisible by $3$ and therefore an odd number" is not true, $6^2$ is divisible by $3$ but not odd.2012-11-11
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    You might also want to simplify matters by assuming, for the sake of contradiction, that there exist $p,q \in \mathbb{Z}$ such that $\sqrt{3} = \frac{p}{q} \text{ and}\; gcd(p,q) = 1.$2012-11-11
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    @AndréNicolas - so does $6$ itself :)2012-11-11
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    True! But I wanted to quote the OP exactly.2012-11-11
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    @Jake +1 for showing your work!2012-11-11
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    possible duplicate of [Critiques on proof showing $\sqrt{12}$ is irrational.](http://math.stackexchange.com/questions/305559/critiques-on-proof-showing-sqrt12-is-irrational)2013-05-17
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    @DominicMichaelis Surely critiques are localised to the specific question, no? Jake cannot gain a critique of his proof by looking at critiques of user21154's proof...2013-05-17
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    @user1729 their proof nearly looks quiet the same. I voted to close mainly to shorten the unanswered questions. As it seems here is no work left to done I thought linking to a pretty similiar question2013-05-17
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    @DominicMichaelis Why not upvote an answer then?! This is a perfectly good question which does not deserve to be closed! (Although I do understand your point, as it is an old question. But still, a simple upvote is much neater and more elegant!)2013-05-17
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    @user1729 closing a question doesn't mean that it is bad. I think those check my prof questions are most times to localized. I did not found any answer extremely nice and wanted to clean up more so I need to save some upvotes :)2013-05-17
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    @DominicMichaelis Maybe I am a bit loose with my upvotes... but anyway, I do agree that "check my proof" questions are too localised, but then of course we could close it as such, no? (And perhaps say that this question is similar to the linked one, in that they make the same error.)2013-05-17
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    @User1729 yeah maybe that have been the better way to do so2013-05-17

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