Mathematics
Proof of irrationality of √2, √3, √5
Practice questions
If (\sqrt{2}) is assumed rational and written as (\sqrt{2}=\frac{p}{q}), where (p) and (q) are coprime, where does the contradiction mainly come from?Which statement is most essential in proving the irrationality of (\sqrt{3})?If (\sqrt{5}=\frac{a}{b}) and (a,b) are coprime, what is the first correct conclusion from (a^2=5b^2)?In the proof that (\sqrt{2}) is irrational, what conclusion is obtained after putting (p=2k)?Why is the assumption (\sqrt{3}=\frac{m}{n}) finally proved wrong when (m,n) are taken coprime?If a student writes that (\sqrt{5}) is rational because (5) is a whole number, what is the correct correction?Why is it necessary to take (\frac{p}{q}) in lowest form while proving the irrationality of (\sqrt{2})?Which argument directly proves that (\sqrt{3}) cannot be rational?If (a^2) is divisible by (5), what conclusion about (a) is correct in the proof for (\sqrt{5})?After assuming (\sqrt{2}) rational and writing (p^2=2q^2), which condition must not be forgotten?Which type of square root is proved irrational in proofs like those for (\sqrt{2}), (\sqrt{3}), and (\sqrt{5})?If (\sqrt{3}=\frac{p}{q}) and (p=3r), what correct equation follows next?In the proof for (\sqrt{5}), if (a=5k), what is proved about (b)?Which option is an incorrect step in the proof of (\sqrt{2}) being irrational?If (r) is prime and (r\mid x^2), what conclusion is used in proving the irrationality of (\sqrt{r})?Which option gives the correct order of proof for the irrationality of (\sqrt{5})?Why is (p) considered even when (p^2) is even in the proof for (\sqrt{2})?If (\sqrt{3}) were rational, in which form would it be correctly written?Which statement is common to the proofs of (\sqrt{2}), (\sqrt{3}), and (\sqrt{5})?Which option correctly states the contradiction in the proof of (\sqrt{3})?