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
If a proof writes \(\sqrt{2}=\frac{m}{n}\) but does not state lowest form, what is the biggest weakness?HardLevel 18In a proof by contradiction for the irrationality of \(\sqrt{3}\), a student obtains \(p^2=3q^2\). Given that \(p\) is divisible by 3, which next step correctly advances the proof?HardLevel 18Suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion is needed to establish the contradiction in the proof of irrationality?HardLevel 18A student says that the decimal expansion of \(\sqrt{2}\) never terminates, so it is irrational. Which of the following arguments rigorously proves this conclusion?HardLevel 18Suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion follows from this assumption and produces a contradiction?HardLevel 18If a student stops after proving only that a is divisible by 3 in the proof of √3, what is the main error?HardLevel 18Which prime-divisibility property is crucial in a proof by contradiction that \(\sqrt{3}\) is irrational?HardLevel 18Rima says, “\(\sqrt{2}\) is irrational because its decimal expansion is infinite.” What is the main flaw in her reasoning?HardLevel 18Which initial assumption is required when proving the irrationality of \(\sqrt{3}\) by the contradiction method?HardLevel 18In the proof of (\sqrt{3}), both (a) and (b) being divisible by (3) breaks which initial assumption?HardLevel 18It is known that \(\sqrt{3}\) is irrational. Which of the following conclusions must be true?HardLevel 18Reena says, “The decimal expansion of \(\sqrt{2}=1.414213\ldots\) is infinite, so it is irrational.” What is the most accurate evaluation of her reasoning?HardLevel 18In irrationality of (\sqrt{2}), which statement does not complete the proof because it is only half of the contradiction?HardLevel 18In irrationality of (\sqrt{3}), which statement alone does not complete the proof?HardLevel 18While proving the irrationality of \(\sqrt{3}\), a student assumes \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime. After obtaining \(a^2=3b^2\), the student states that both \(a\) and \(b\) are divisible by 3. Which reasoning is necessary to justify this conclusion?HardLevel 18Suppose it is claimed that \(s=\sqrt{2}+\sqrt{3}\) is rational. Since \((\sqrt{2}+\sqrt{3})(\sqrt{3}-\sqrt{2})=1\), \(\sqrt{3}-\sqrt{2}=1/s\) would also be rational. Which equation below immediately produces a contradiction from this claim?HardLevel 18Reema says, “\(\sqrt{3}\approx1.732\); therefore, \(\sqrt{3}\) is rational because 1.732 is rational.” What is the correct evaluation of Reema’s argument?ExpertLevel 16While proving the irrationality of \(\sqrt{2}\) by contradiction, after assuming \(p/q\) is in lowest terms, which condition directly contradicts this assumption?ExpertLevel 16In a proof by contradiction, assume that \(\sqrt{3}=\frac{m}{n}\), where \(m\) and \(n\) are coprime integers. If \(3n^2=m^2\) is obtained, which conclusion decisively shows that this assumption is impossible?ExpertLevel 16While proving the irrationality of \(\sqrt{2}\) by contradiction, which condition is essential when assuming \(\sqrt{2}=\frac{p}{q}\)?ExpertLevel 16