Mathematics
Proof of irrationality of √2, √3, √5
Practice questions
While proving the irrationality of (\sqrt{2}), if (\sqrt{2}=\frac{p}{q}) is assumed in lowest form, which reasoning from (p^2=2q^2) is most accurate?In the proof of (\sqrt{3}), what is the proper basis for writing (p=3k) from (p^2=3q^2)?If (\sqrt{5}=\frac{p}{q}) is assumed in lowest form and (p=5k) is obtained in the proof, what is the next important aim?In the proof of (\sqrt{2}), after putting (p=2r), which correct simplification is obtained from (p^2=2q^2)?If (p=3r) and (q=3s) are obtained in the proof of (\sqrt{3}), what is the most appropriate contradiction?Which step is wrong in order in the proof of (\sqrt{5})?If (p) and (q) are coprime, which result would most directly contradict this?In the proof of (\sqrt{2}), if someone says that (p^2=2q^2) immediately makes both (p) and (q) even, what is the correct comment?In the proof of (\sqrt{3}), (p^2=3q^2) gives (p=3k) and then (q=3r). Why does this make the original assumption false?Which option shows an algebraic mistake in the proof of (\sqrt{5})?What completes the proofs of (\sqrt{2}), (\sqrt{3}), and (\sqrt{5})?If assuming (\sqrt{3}) rational gives (p^2=3q^2), which is the correct chain until (q) is proved divisible by (3)?In the proof of (\sqrt{5}), (p) is proved divisible by (5) from (p^2=5q^2). What does this depend on?Which option states the final contradiction in the proof of (\sqrt{2}) in correct language?If a student writes (3=\frac{p}{q}) from (\sqrt{3}=\frac{p}{q}), what is the correct correction?In the proof of (\sqrt{5}), if both (p) and (q) are proved divisible by (5), which conclusion is the most logical?Which statement gives the correct basis for proving (q) even in the proof of (\sqrt{2})?If (r) is prime and (r\mid x^2), what is its correct use in the proofs of (\sqrt{3}) and (\sqrt{5})?In the proof of (\sqrt{3}), after getting (q^2=3k^2), why can (q=3r) be written?Which option is the biggest common misconception in the proofs of (\sqrt{2}), (\sqrt{3}), and (\sqrt{5})?