Mathematics
Proof of irrationality of √2, √3, √5
Practice questions
If someone writes (q^2) is even directly from (p^2=2q^2) in the proof of (\sqrt{2}), why is this a rushed step?In the proof of (\sqrt{5}), both (p) and (q) being divisible by (5) exposes what issue?Which statement gives the correct basis for writing (p=3k) in the proof of (\sqrt{3})?Which option gives the correct final reason in the proof of (\sqrt{2})?A student says (\sqrt{5}) is rational because (5) is rational. What is the correct correction?If in the proof of (\sqrt{2}=\frac{p}{q}), both (p) and (q) get common factor (2), which initial statement becomes false?In the proof of (\sqrt{5}), which step comes just before concluding that (q) is divisible by (5)?Which statement is a wrong method in the proofs of (\sqrt{2}), (\sqrt{3}), and (\sqrt{5})?If from (p^2=2q^2), (p=2k) and then (q=2r) are obtained, which proof does this complete?Which option completes the proof of (\sqrt{3})?Which option correctly tells the purpose of writing (p=5k) in the proof of (\sqrt{5})?If assuming (\sqrt{2}) rational gives a contradiction, what is the correct conclusion?Which option is a wrong simplification in the proof of (\sqrt{3})?Which statement gives both the correct conclusion and reason in the proof of (\sqrt{5})?Which option is the most useful precaution while writing an irrationality proof in an exam?If in the proof of (\sqrt{3}), both (p) and (q) are divisible by (3), what can be said about (\frac{p}{q})?Which option gives the correct common structure of all three proofs?Which statement correctly completes the proof of (\sqrt{2}), (\sqrt{3}), or (\sqrt{5}) in the final line?In the proof of (\sqrt{2}), when both (a) and (b) are proved even, which conclusion is most appropriate?Which idea works similarly in the proofs of (\sqrt{3}) and (\sqrt{5})?