Mathematics
Proof of irrationality of √2, √3, √5
Practice questions
(\sqrt{2}) is assumed rational and written as (\sqrt{2}=\frac{p}{q}). If (p) and (q) are coprime, what is the correct next conclusion from (p^2=2q^2)?While proving the irrationality of (\sqrt{3}), (p^2=3q^2) is obtained. Which conclusion about (p) with reason is correct?If (\sqrt{5}=\frac{p}{q}) is assumed in lowest form and (p^2=5q^2) is obtained, in which form should (p) be written?In the proof of (\sqrt{2}), after putting (p=2k), which equation follows from (p^2=2q^2)?In the proof of (\sqrt{3}), after putting (p=3k), (9k^2=3q^2) is obtained. What is the correct form of (q^2)?In the proof of (\sqrt{5}), after putting (p=5k), (25k^2=5q^2) is obtained. Which conclusion is correct?Which option is a wrong conclusion in the proof of (\sqrt{2})?If assuming (\sqrt{3}) rational makes both (p) and (q) divisible by (3), which fact is proved false?Where is the fact that (5) is prime used in the proof of irrationality of (\sqrt{5})?In the proof of (\sqrt{2}), after getting (q^2=2k^2), which reasoning is correct?Which option gives the correct order of the proof of (\sqrt{3})?In the proof of (\sqrt{5}), if (p=5k) and (q=5r) are obtained, which conclusion is correct?If a student stops at only (p^2=2q^2) while proving (\sqrt{2}) irrational, why is the proof incomplete?In the proof of (\sqrt{3}), why can we not directly say (q) is divisible by (3) from (p^2=3q^2)?Which statement is wrong in the proof of (\sqrt{5})?Why are (p) and (q) taken as coprime in all three proofs?Which option correctly states a difference between the proofs of (\sqrt{2}) and (\sqrt{3})?If assuming (\sqrt{5}) rational proves both (p) and (q) divisible by (5), what does this contradict?In the proof of (\sqrt{2}), both (p) and (q) are found even. Why is this called a contradiction?Which option is the correct short proof idea for the irrationality of (\sqrt{3})?