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(An answer because this is a little long for a comment) Here is a proof which does not use coprimality, but is essentially the same: Suppose that $a/b = \sqrt{2}$ where $a$ and $b$ are integers. Then $a=\sqrt{2}b$ and so $a^2 = 2 b^2$ where both sides are integers. Let $n$ be the number of factors of 2 in $a$ and $m$ be the number of factors of two in $b$. ...

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You are correct in that the coprimality of $a$ and $b$ is not used in its full strength. It is adequate to merely assume that they are not both even. But since people are so used to reducing a rational number to lowest terms, making this assumption improves the readability of the proof. The unnecessary stronger assumption of coprimality is not used, but it ...

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I wrote an article that explains mathematical induction in the easiest way possible using an easy-to-understand example. It was published in Medium's #1 Math Publication, so I guess it must be pretty good. Feel free to take a look. https://medium.com/cantors-paradise/a-comprehensible-introduction-to-mathematical-induction-aef1a61eedcf The used example is: ...

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I think proofs can be/should be taught as early as possible. I am not sure at what age exactly but as soon as they can understand the concepts maybe around 12-13 years old. So I keep giving them bits and pieces of it and if they are able to pickup the concepts then I teach them fully. One of the things that we need to do is to coherently represent all the ...

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