"A claim has been made: √2 can be written as a fraction. Today your group is the jury. You won't be told whether the claim is true — you're going to test it until it either survives or breaks."
Square 1 through 8. Note even/odd for both \(n\) and \(n^2\).
Conjecture: "If \(n^2\) is even, then \(n\) must be ___."
Prove instead: "if \(n\) is odd, then \(n^2\) is odd" — let \(n=2k+1\) and expand.
Suppose \(\sqrt2=\frac{p}{q}\), with \(p,q\) sharing no common factor.
Square both sides. Rearrange to \(p^2=\) (something with \(q\)).
What does that force about \(p\)? Write \(p=2k\).
Substitute \(p=2k\) into \(p^2=2q^2\). Simplify.
What does this force about \(q\)?
Compare to how \(p,q\) were defined. What's gone wrong?
Write the closing verdict on √2, in your own words.
Retry the exact same trial on √4. Where does it break down?
Generalize the rule to any prime. Try √3 and √5. Which whole numbers N give an irrational √N?