Mathematics
Proof of irrationality of √2, √3, √5
Practice questions
In the proofs of (\sqrt{3}) and (\sqrt{5}), which prime factors appear respectively instead of (2)?Which opening sentence is most complete for proving the irrationality of (\sqrt{2})?In the proof for (\sqrt{5}), if both (x) and (y) turn out divisible by (5), what can be said about (\gcd(x,y))?If no contradiction appears while proving (\sqrt{3}) irrational by assuming it rational, which condition is probably missing?Which option directly conflicts with (p) and (q) being coprime in the proof for (\sqrt{2})?In the irrationality proof of (\sqrt{5}), what idea is hidden in moving from (5\mid x^2) to (x=5m)?Which statement correctly generalizes the proof of irrationality of (\sqrt{3})?If someone writes (q^2=4k^2) after putting (p=2k) in (p^2=2q^2), where is the mistake?In the proof for (\sqrt{2}), if both (p) and (q) are even, by which number can the fraction be further reduced?Which option shows the correct order for proving the irrationality of (\sqrt{3})?If (a) and (b) are coprime but the proof gives (a=3m) and (b=3n), what conclusion follows?How does (5) not being a perfect square help in understanding the irrationality of (\sqrt{5})?In the proof for (\sqrt{2}), when both (p) and (q) turn out even, which initial statement is proved false?Which option gives the correct reasoning to reach (q) in the proof for (\sqrt{3})?If someone says (\sqrt{2}) is irrational because (2) is even, what is the correct correction?While taking (x) and (y) coprime in the proof for (\sqrt{5}), what must be kept in mind?If (3\mid a) and (3\mid b), what contradiction arises with assuming (\frac{a}{b}) in lowest form?Which option correctly explains the parity idea in the proof for (\sqrt{2})?In the proof for (\sqrt{5}), which condition is necessary while taking (5\mid y) from (5\mid y^2)?If (a^2=3b^2) is obtained in proving (\sqrt{3}) irrational, why is it correct to say (a^2) is a multiple of (3)?