Mathematics
Proof of irrationality of square root 2 and square root 3
√2 और √3 की अपरिमेयता का प्रमाण
In Class 9 Mathematics, this Number Systems topic explains how to prove that √2 and √3 are irrational numbers. Students use proof by contradiction: they assume a square root can be written as a fraction in lowest terms, then apply prime-factor and divisibility properties to show that the assumption leads to an impossibility. The lesson strengthens understanding of rational and irrational numbers, factors, parity, and the logic of mathematical proof, while helping learners present each step clearly and accurately.
Practice questions
In the proof of (\sqrt{2}), which condition is necessary along with assuming (\sqrt{2}=\frac{m}{n})?MediumLevel 17In a proof by contradiction, Arjun assumes that \(\sqrt{2}=\frac{m}{n}\), where \(m\) and \(n\) are coprime. He obtains \(m^2=2n^2\) and says, “\(m\) is even, so \(\sqrt{2}\) is irrational.” What is missing from his argument?MediumLevel 17If \(\sqrt{3}\) is assumed to be \(\frac{p}{q}\) in lowest terms, where \(p\) and \(q\) are coprime integers, what follows from \(p^2=3q^2\)?MediumLevel 17Which option gives the correct reasoning from (p²) to (p) in the proof of (√3)?MediumLevel 17On what basis does the final conclusion in irrationality of (√2) come?MediumLevel 17On what basis does the final conclusion in irrationality of (\sqrt{3}) come?MediumLevel 17In the proof of (\sqrt{2}), if a student writes directly from (m^2=2n^2) that (n) is even, what is the correct comment?MediumLevel 17In the proof of (\sqrt{3}), if a student directly writes from (p^2=3q^2) that (q) is divisible by (3), what is the correct comment?MediumLevel 17What is the main similarity in the proofs of (\sqrt{2}) and (\sqrt{3})?MediumLevel 17What is the main difference between the proofs of (\sqrt{2}) and (\sqrt{3})?MediumLevel 17A student claims that \(\sqrt{3}\) is rational because its square, \(3\), is a rational number. Which statement about this argument is correct?MediumLevel 17Which statement correctly describes the main idea used in the proof that \(\sqrt{2}\) is irrational?MediumLevel 17Which option gives the correct squared relation used in the proof of \(\sqrt{2}\)?MediumLevel 17Riya says, “
\(\sqrt{3}=1.732\ldots\), so it is irrational because its decimal expansion is non-terminating.” What is the main error in Riya’s reasoning?MediumLevel 17If assuming (√2) rational gives a contradiction, which conclusion is correct?EasyLevel 17If assuming √3 is rational breaks the coprime condition, which conclusion is correct?MediumLevel 17Suppose \(\sqrt{3}=\frac{m}{n}\), where \(m\) and \(n\) are coprime integers. Which conclusion proves a contradiction to this assumption?MediumLevel 17In the proof by contradiction for the irrationality of \(\sqrt{2}\), what is the main purpose of assuming \(\sqrt{2}=\frac{p}{q}\) in lowest terms?MediumLevel 17Which option is a wrong start in the proof of \(\sqrt{2}\)?MediumLevel 17If \(\sqrt{2}\) is assumed to be rational and written as \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, why does a contradiction arise in the proof?MediumLevel 17