Mathematics
Proof of irrationality of √2, √3, √5
Practice questions
At which point is coprimality used decisively in the proof for (\sqrt{2})?Which option gives the correct final sentence for proving the irrationality of (\sqrt{5})?In the proof for (\sqrt{3}), after putting (a=3k), what becomes clear from (b^2=3k^2)?If (\sqrt{4}) is used instead of (\sqrt{2}), why will the same contradiction proof not apply?In the proof for (\sqrt{5}), if both (x) and (y) are divisible by (5), which statement would be false?Which option correctly states the role of (3) in the proof of irrationality of (\sqrt{3})?In the proof for (\sqrt{2}), if both (p) and (q) are even, what does writing (p=2m) and (q=2n) show?Which option gives an incorrect conclusion about (x) from (x^2=5y^2) in the proof for (\sqrt{5})?What is the best exam tip related to the irrationality of (\sqrt{2}), (\sqrt{3}), and (\sqrt{5})?If assuming (\sqrt{3}=\frac{p}{q}) gives (p^2=3q^2), what is the main purpose of showing divisibility by (3) for both (p) and (q) in the proof?If (\sqrt{2}=\frac{p}{q}) is assumed in lowest form and (p^2=2q^2) is obtained, which sequence is most logical to reach the contradiction?In the irrationality proof of (\sqrt{3}), which reasoning is strongest while writing (3\mid p) from (3\mid p^2)?If (\sqrt{5}) is assumed rational as (\sqrt{5}=\frac{a}{b}), which condition about (a) and (b) is essential for the proof?In the proof for (\sqrt{2}), if someone writes (q^2=4k^2) after putting (p=2k), what is the correct correction?If (p=3r) has been proved in the irrationality proof of (\sqrt{3}), which step is correct to conclude about (q)?In the proof for (\sqrt{5}), after (5\mid a) is proved from (a^2=5b^2), (a=5t) is written. What does this indicate?Which statement directly conflicts with the coprimality of (p) and (q) in the proof for (\sqrt{2})?If a student writes (\sqrt{3}\approx1.732) and treats it as proof of irrationality, what is the main weakness?While proving (\sqrt{5}) irrational, both (a) and (b) turn out divisible by (5). What is its effect on (\gcd(a,b))?If (\sqrt{8}) is considered instead of (\sqrt{2}), what is the best short reason that (\sqrt{8}) is irrational?