Mathematics
Proof of irrationality of √2, √3, √5
Practice questions
After assuming (\sqrt{3}=\frac{p}{q}) and squaring, (p^2=3q^2) is obtained. What is the correct conclusion about (p)?If (\sqrt{5}) is rational, how should it be correctly written?Which general statement applies to the proofs of (\sqrt{2}), (\sqrt{3}), and (\sqrt{5})?In the proof for (\sqrt{2}), after taking (\frac{p}{q}) in lowest form, both (p) and (q) turn out even. What does this disprove?If (p^2=5q^2) and (p=5t), which final conclusion about (q) is correct?Which statement is not part of a sufficient argument for proving (\sqrt{2}) irrational?Which point in the proof of (\sqrt{3}) depends on (3) being prime?If (x) and (y) are coprime, which situation is impossible?After assuming (\sqrt{5}) rational, which sequence is most correct?Why is the proof for (\sqrt{2}) not complete by only writing that (p^2) is even?Which option gives the correct basis for writing (p=3r) in the proof for (\sqrt{3})?If a student writes (\sqrt{5}=\frac{5}{1}), what is the most correct correction?In the proofs of (\sqrt{2}), (\sqrt{3}), and (\sqrt{5}), in what form is the rational assumption taken?In the irrationality proof of (\sqrt{3}), why is (p^2) divisible by (3) from (p^2=3q^2)?Which statement gives the correct contradiction at the end of the proof for (\sqrt{5})?If (n) is odd, then (n^2) is odd. In which proof is this fact used directly?Why is the condition (q\neq0) necessary in the proof for (\sqrt{3})?Which option is only an incomplete hint for the irrationality of (\sqrt{5}), not a full proof?If (p) and (q) are coprime and then (2\mid p), (2\mid q) are proved, what type of result is this?After assuming (\sqrt{2}) rational, (2q^2=p^2) is written. What is the correct use of this equation?