Mathematics
Proof of irrationality of √2, √3, √5
Practice questions
Which option correctly explains proof by contradiction?If (p^2=3q^2) in the proof for (\sqrt{3}), what is the simple reason that (p^2) is divisible by (3)?Which conclusion should be written at the very end of the proof of irrationality of (\sqrt{2})?In the proof for (\sqrt{5}), while writing (5\mid b) from (5\mid b^2), what must be added?If both (p) and (q) are found divisible by (3) in proving (\sqrt{3}) irrational, which statement about (\frac{p}{q}) is correct?Which option gives the correct basis for proving (q) even in the proof for (\sqrt{2})?In the proof for (\sqrt{5}), what is the correct intermediate conclusion needed to write (a=5k) from (a^2=5b^2)?If (n) is an odd integer, then (n^2) is odd. This fact is especially useful in which proof?In the proof for (\sqrt{3}), after putting (p=3k), (9k^2=3q^2) is obtained. What is the correct simplification?Which statement starts the proof of irrationality of (\sqrt{5}) most clearly?In the proof for (\sqrt{2}), (p^2=2q^2) shows (p^2) is even. If (p) is assumed odd, what problem occurs?If (\sqrt{3}) were rational, what impossible situation would appear at the end of the proof?In the proof for (\sqrt{5}), if someone writes (a=25k) from (5\mid a^2), what is the mistake?Which property of rational numbers is used in the proofs of (\sqrt{2}), (\sqrt{3}), and (\sqrt{5})?If (p) and (q) are coprime, why is it impossible to get both (p=2m) and (q=2n)?Which statement is unnecessary in the proof of irrationality of (\sqrt{3})?In the proof for (\sqrt{5}), both (a) and (b) are found divisible by (5). What type of result is this?If (\sqrt{2}) is assumed rational and finally both (p,q) turn out even, which conclusion is logical?In the proof for (\sqrt{3}), after getting (3\mid p) and then (3\mid q), which statement would be false?What is the best exam formula for proving irrationality of (\sqrt{2}), (\sqrt{3}), and (\sqrt{5})?