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 both (c) and (d) become even in the proof of (\sqrt{2}), which option correctly indicates infinite descent?ExpertLevel 18If a student assumes that \(\sqrt{2}+\sqrt{3}\) is a rational number \(r\), which conclusion correctly proves a contradiction in this claim?ExpertLevel 18A student says that \(\sqrt{3}\) is irrational because its decimal expansion \(1.732\ldots\) continues endlessly. What is the main error in this argument?ExpertLevel 18In the proof of √3, 3 | h² implies 3 | h. Which broader principle does this illustrate?HardLevel 18In the proof by contradiction that \(\sqrt{3}\) is irrational, which property is used to conclude \(3\mid p\) from \(3\mid p^2\)?ExpertLevel 18Why is it necessary to take the fraction \(\frac{p}{q}\) in lowest terms in the contradiction proof that \(\sqrt{2}\) is irrational?ExpertLevel 18A student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. On squaring, the student gets \(p^2=3q^2\) and concludes that \(p\) is divisible by 3. What is the next essential step to complete the proof?ExpertLevel 18If \(x=\sqrt{2}+\sqrt{3}\), which argument correctly disproves the assumption that \(x\) is rational?ExpertLevel 18Rima says, “\(\sqrt{3}=1.732\); therefore, \(\sqrt{3}\) is rational.” What is the main error in her reasoning?ExpertLevel 18While proving the irrationality of \(\sqrt{2}\) by contradiction, if \(\sqrt{2}=\frac{p}{q}\) where \(p\) and \(q\) are coprime, which conclusion necessarily follows from \(p^2=2q^2\)?ExpertLevel 18A student claims that \(\sqrt{3}\) is rational because its decimal form is 1.732 and \(1.732=\frac{1732}{1000}\). What is the error in the student’s reasoning?ExpertLevel 18Which condition guarantees that the square root of a natural number \(n\) is rational?ExpertLevel 18In the standard proof by contradiction, suppose that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. From \(p^2=3q^2\), which deduction is essential for obtaining the contradiction?ExpertLevel 18Reema says that if \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, then \(p^2=2q^2\) only shows that \(p\) is even; therefore \(\sqrt{2}\) is not proved irrational. Which essential point is missing from Reema's argument?ExpertLevel 18In the proofs of √2 and √3, what should not be treated as the basis of proof?ExpertLevel 18