How does this proof show the irrationality of $\sqrt{2}$ ?
I am new to proofs and don't really understand the logic used here.
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4Which step don't you understand? – Wojowu Aug 09 '15 at 22:26
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1I just think I didn't understand the proof strategy in general. I've never done proofs before. – yung sprint Aug 09 '15 at 22:36
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This is probably too broad to answer, but this specific problem is a proof by contradiction where we assume a hypothesis and then show that a logical consequence of that hypothesis leads to some contradiction, thus showing the hypothesis was false – pancini Aug 09 '15 at 22:52
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1Yes! I just did some research on the topic. – yung sprint Aug 09 '15 at 23:59
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1Thanks to everyone who answered/commented. – yung sprint Aug 09 '15 at 23:59
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"Suppose it weren't. Then everything goes horribly wrong and we get a contradiction. Since we assume maths is consistent - can't derive contradictions - we must have made a faulty assumption or a faulty step of reasoning. But you can see for yourself that our steps of reasoning are all correct, so we must have made a faulty assumption. We made only one assumption: that $\sqrt{2}$ is rational. That assumption is faulty, so in fact $\sqrt{2}$ is irrational."
Compare with solving a sudoku. Every so often you make a step as follows: "I have a choice between a 1 or a 3 for this square. Suppose it were a 1. Then I follow through the consequences and find two 2's in the same row. Therefore it must have been a 3 originally after all, not a 1."
Patrick Stevens
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1The sudoku analogy was helpful. I've never done proofs before, but I could relate to that. – yung sprint Aug 09 '15 at 22:37
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As Wojowu said, you will get better results if you specify the parts you have difficulty understanding. However, this being a relatively simple proof, I will try to explain the proof in its entirety.
This is called a proof by contradiction. Basically, you assume the opposite–that it is rational. You then arrive at a contradiction, thus proving that it is not rational.
The key point of this proof is that $p$ and $q$ $\textbf{must not}$ have any common factors. By reading the proof, you will see that both $p$ and $q$ must be divisible by $2$, meaning that they $\textbf{do}$ have a common factor of 2. Since this defies the original assumption. This proof by contradiction shows that $\sqrt{2}$ cannot be written as a ratio of coprime integers. Therefore, it must not be rational.
EDIT: I would recommend searching common types of proof to familiarize yourself with them. I would assume you are studying them and will encounter a lot of proofs henceforth.
Here is a small list to get started. Feel free to search beyond this list to learn all about proofs!
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1Thanks for explaining proof by contradiction and for the references. – yung sprint Aug 09 '15 at 22:42