Mathematics
Proof of irrationality of √2, √3, √5
Practice questions
After assuming (\sqrt{2}=\frac{p}{q}) in lowest form and getting (p^2=2q^2), why is it correct to write (p=2k)?While proving the irrationality of (\sqrt{3}), what weakness occurs if (a) and (b) in (\sqrt{3}=\frac{a}{b}) are not taken coprime?If (\sqrt{5}) is assumed rational and written as (\sqrt{5}=\frac{x}{y}), which algebraic step correctly leads to (x^2=5y^2)?In the proof for (\sqrt{2}), after getting (q^2=2k^2), which conclusion helps complete the proof?In the proof for (\sqrt{3}), the conclusion (3\mid a) from (3\mid a^2) is based on which principle?If (p) and (q) are coprime, what does obtaining (5\mid p) and (5\mid q) in the proof for (\sqrt{5}) indicate?Which option is the most serious error in the proof of irrationality of (\sqrt{2})?In the proof for (\sqrt{3}), after putting (a=3m), into what form does (a^2=3b^2) change?If a student writes only (\sqrt{5}\approx2.236) to prove rationality or irrationality, why is this argument incomplete?Which structure remains common in the proofs of (\sqrt{2}), (\sqrt{3}), and (\sqrt{5})?If (r) is prime and (\sqrt{r}=\frac{p}{q}) is assumed in lowest form, which equation is obtained after squaring?In the proof for (\sqrt{3}), after getting (b^2=3m^2), what conclusion about (b) is correct?In the proof for (\sqrt{5}), after putting (x=5n), (25n^2=5y^2) is obtained. What is the next correct simplification?In the proof for (\sqrt{2}), when both (p) and (q) are proved even, which statement about (\frac{p}{q}) is correct?Which option states the correct final contradiction in the proof for (\sqrt{3})?If assuming (\sqrt{5}) rational gives (x^2=5y^2), after writing (x=5n), toward which conclusion does the proof move?Which statement shows that the proof of irrationality of (\sqrt{2}) is not based on decimals?In the proof for (\sqrt{3}), if someone writes (a=3b) directly from (3\mid a^2), what is the mistake?Which statement correctly uses the primality of (5) in the proof for (\sqrt{5})?If the square of an integer (n) is even, then (n) is even. How many times is this fact used in the proof for (\sqrt{2})?