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
When proving the irrationality of \(\sqrt{3}\) by the contradiction method, which assumption is made at the start?ExpertLevel 17If \(\sqrt{3}=\frac{p}{q}\) is assumed, where p and q are integers, why must \(\frac{p}{q}\) be taken in lowest terms in the proof by contradiction of its irrationality?ExpertLevel 17In the proof by contradiction for the irrationality of \(\sqrt{3}\), assume \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. Which conclusion from \(p^2=3q^2\) is essential for the proof?ExpertLevel 17In the proof by contradiction that \(\sqrt{2}\) is irrational, assume \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion contradicts this assumption?ExpertLevel 17A student says, “\(\sqrt{3}\) is rational because it can be written as \(1.732\).” What is the main flaw in this reasoning?ExpertLevel 17A student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. On squaring, he gets \(p^2=2q^2\). He says that \(q\) is even because the right-hand side is even. What should be the first correct conclusion in this argument?ExpertLevel 17A student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q\ne0\), and concludes from \(p^2=2q^2\) that both \(p\) and \(q\) are even. If the student did not assume \(p\) and \(q\) to be coprime, why is this conclusion not automatically a contradiction?ExpertLevel 17In the proof by contradiction that \(\sqrt{3}\) is irrational, suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. From \(p^2=3q^2\), which conclusion is necessary?ExpertLevel 17Which statement can be a wrong but tempting answer in the proof of (\sqrt{3})?ExpertLevel 65In the proof of (\sqrt{2}), which statement is a middle step rather than the final conclusion?ExpertLevel 65In the proof of √3, which statement is a middle step rather than the final conclusion?ExpertLevel 65What is the most essential exam caution in proving irrationality of (\sqrt{2})?ExpertLevel 65If \(n\) is an integer and \(3\mid n^2\), which conclusion used in the proof of the irrationality of \(\sqrt{3}\) is certainly true?ExpertLevel 65If the contradiction proof of \(\sqrt{2}\) succeeds, what is the final logical conclusion?ExpertLevel 65While starting a proof by contradiction for the irrationality of \(\sqrt{2}\), Riya assumes that \(\sqrt{2}=p/q\), where \(p\) and \(q\) are integers. Which additional condition on \(p\) and \(q\) is necessary to make the proof valid?ExpertLevel 17A student calls \(\sqrt{2}\) irrational after seeing its decimal form 1.414213... . Which property must be proved to justify the conclusion?ExpertLevel 17A student claims that \(\sqrt{2}+\sqrt{3}\) is rational. Which argument correctly proves that the claim is wrong?ExpertLevel 17Aarav assumes that \(\sqrt{3}=\frac{m}{n}\), where \(m\) and \(n\) are coprime. After obtaining \(m^2=3n^2\), he says that divisibility of \(m^2\) by 3 does not prove that \(m\) is divisible by 3. Which fact corrects Aarav’s error?ExpertLevel 17In the proof of √3, after taking m = 3k, which step from m² = 3n² proves n divisible by 3?ExpertLevel 65Why is only (r^2) being even not the final contradiction in the proof of (\sqrt{2})?ExpertLevel 17