Mathematics
Proof of irrationality of √2, √3, √5
Practice questions
In the proof for (\sqrt{3}), if (\frac{p}{q}) is in lowest form but (p=3m) and (q=3n) are obtained, which conclusion is most precise?In the proof for (\sqrt{5}), after showing (a) is divisible by (5) from (a^2=5b^2), which conclusion would be immediately wrong?Which statement would leave the proof of (\sqrt{2}) incomplete?If (r) is a prime number and (\sqrt{r}=\frac{p}{q}) is assumed in lowest form, what is the first general conclusion from (p^2=rq^2)?In the proof for (\sqrt{3}), which shortcut from (p^2=3q^2) to (p=3k) is wrong?After assuming (\sqrt{5}) rational and getting (a^2=5b^2), how does a common factor appear in (a) and (b)?In the proofs of (\sqrt{2}), (\sqrt{3}), and (\sqrt{5}), what changes while the proof method remains the same?If both (p) and (q) are proved even in the proof for (\sqrt{2}), by what can (\frac{p}{q}) be reduced?Why is it necessary to write (q\neq0) while proving the irrationality of (\sqrt{3})?Which option gives the correct basis for divisibility of (b) in the proof for (\sqrt{5})?Which statement correctly moves from (p^2) to (p) in the proof for (\sqrt{2})?If (3\mid p) is obtained from (p^2=3q^2), what is correct about (k) when writing (p=3k)?In the proof for (\sqrt{5}), both (a) and (b) being divisible by (5) breaks which initial condition?While writing the proof for (\sqrt{2}), if someone assumes (\sqrt{2}=\frac{p}{q}) but does not mention lowest form, what problem occurs?Which option gives the most appropriate final sentence for the proof of (\sqrt{3})?If assuming (\sqrt{5}) rational gives (a^2=5b^2), which statement about (a^2) is correct?In the proof for (\sqrt{2}), why is (q) called even after getting (q^2=2k^2)?Which idea is common in the proofs of (\sqrt{3}) and (\sqrt{5})?If someone says (\sqrt{2}) is irrational because (2) is not a perfect square, what correction is appropriate at expert level?In the proof for (\sqrt{5}), putting (a=5k) gives (25k^2=5b^2). What will (b^2) be?