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
Which option correctly distinguishes (q\neq0) and the coprime condition in the proof of (\sqrt{3})?HardLevel 16If \(\sqrt{3}=\frac{p}{q}\) is assumed to be in lowest terms and \(p^2=3q^2\) is obtained, which conclusion is necessary to establish the contradiction?HardLevel 16In the proof of (\sqrt{3}), which skipped step would prevent the final contradiction?HardLevel 16If an answer writes √2 = a/b but does not mention gcd(a,b) = 1, what is the best comment?HardLevel 16Suppose \(r=\sqrt{2}+\sqrt{3}\) is a rational number. Which conclusion from this assumption proves by contradiction that \(\sqrt{2}+\sqrt{3}\) is irrational?HardLevel 16In the proof of (\sqrt{2}), if both (a,b) are even, why can (\frac{a}{b}) be called reducible?HardLevel 16A student says that the decimal expansion of \(\sqrt{3}\) is 1.732... and does not terminate; therefore, \(\sqrt{3}\) is irrational. What is the main error in the student's reasoning?HardLevel 16Which option best describes the role of (2) in the proof of (\sqrt{2})?HardLevel 16Which option best describes the role of (3) in the proof of (\sqrt{3})?HardLevel 16In the proof of \(\sqrt{2}\), if a student ends the proof after only writing \(a\) is even, what is the error?HardLevel 16In the proof of √3, if a student stops after proving only p is divisible by 3, what is the error?HardLevel 16While proving the irrationality of \(\sqrt{3}\) by contradiction, assume \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime. Which conclusion produces a contradiction to this assumption?HardLevel 16Which option gives the correct conclusion and its basis for the proof of (\sqrt{3})?HardLevel 16Let \(p\) and \(q\) be coprime positive integers. In a proof by contradiction for the irrationality of \(\sqrt{3}\), if \(3\mid p^2\), which conclusion is necessary?HardLevel 16A student assumes \(\sqrt{3}=\frac{a}{b}\), where \(a,b\) are coprime positive integers. From \(a^2=3b^2\), the student concludes that 3 divides \(a\). Which rule justifies this conclusion?HardLevel 16If a proof assumes √3 rational and finally gets both p and q divisible by 3, with which initial condition is the contradiction?HardLevel 16For a positive integer \(n\), which statement correctly identifies when \(\sqrt{n}\) is irrational?HardLevel 17If (\sqrt{3}=\frac{u}{v}) is assumed in lowest form and (u^2=3v^2) is obtained, why is it correct to write (u=3k)?HardLevel 17Riya assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. After concluding from \(p^2=3q^2\) that \(p\) is divisible by 3, which is the correct next statement to obtain a contradiction?HardLevel 17If the square of a rational number is 3, what conclusion is obtained about its numerator and denominator in the proof that \(\sqrt{3}\) is irrational?HardLevel 17