Mathematics
Proof of irrationality of √2, √3, √5
Practice questions
What is the root cause of contradiction in the proofs of (\sqrt{2}), (\sqrt{3}), and (\sqrt{5})?In the proof of (\sqrt{3}), what is the basis for writing (q=3r) after getting (q^2=3k^2)?In the proof of (\sqrt{5}), if (p=5k) and (q=5r), how can (\frac{p}{q}) be reduced?Which option states the final contradiction in the proof of (\sqrt{2}) most correctly?If a student writes (3=\frac{p}{q}) directly from (\sqrt{3}=\frac{p}{q}), what is the correct correction?In the proof of (\sqrt{5}), writing (p=5k) from (p^2=5q^2) depends on which condition?In the proof of (\sqrt{2}), if (a=2k) and (b=2r), which statement about (\gcd(a,b)) is correct?If (r) is prime and (r\mid x^2), how is it used in the proofs of (\sqrt{3}) and (\sqrt{5})?Which statement leaves the proof of (\sqrt{5}) incomplete?In the proof of (\sqrt{2}), which option is a correct but incomplete conclusion?Which option disturbs the logical order in the proof of (\sqrt{3})?In the proof of (\sqrt{5}), if (p=5k) and (q=5r) are obtained, which initial condition breaks?In the proof of (\sqrt{2}), (\frac{a}{b}) is in lowest form. If (a=2m) and (b=2n), which conclusion is most suitable?Which option is a wrong algebraic simplification in the proof of (\sqrt{3})?Which final reason is most accurate in proving (\sqrt{5}) irrational?Which statement tells the real role of squaring in the proofs of (\sqrt{2}), (\sqrt{3}), and (\sqrt{5})?If in the proof of (\sqrt{3}), both (p) and (q) are divisible by (3), what is the effect on the lowest form of (\frac{p}{q})?In the proof of (\sqrt{2}), if (a^2) is even, why must (a) be even?Which option correctly states the difference between the proofs of (\sqrt{2}) and (\sqrt{5})?Which deeper idea is common in the proofs of (\sqrt{3}) and (\sqrt{5})?