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
\(\sqrt{3}=\frac{p}{q}\) is assumed in lowest terms and
\(p^2=3q^2\) is obtained, what is the valid basis for concluding that
\(3\mid p\)?HardLevel 17Suppose \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. On squaring, we get \(a^2=3b^2\). Which conclusion is justified at this stage?HardLevel 17If assuming \(\sqrt{3}=\frac{p}{q}\) in lowest terms leads to \(p^2=3q^2\), which conclusion proves a contradiction in this assumption?HardLevel 17In the standard proof, assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. Which number-theoretic fact is needed to obtain a contradiction from \(p^2=3q^2\)?HardLevel 17Which statement is sufficient to reject the rational assumption for √3?HardLevel 17Which idea about the exponents of prime factors explains the irrationality of √2?HardLevel 17What is the deeper prime-factor reason in the proof of (\sqrt{3})?HardLevel 17If (x,y) are coprime and both are proved even, what is the correct contradiction about (\gcd(x,y))?HardLevel 17If u and v are coprime and both are proved divisible by 3, what contradiction about gcd(u,v) follows?HardLevel 17What is the correct difference between the roles of (y\neq0) and (\gcd(x,y)=1) in the proof of (\sqrt{2})?HardLevel 17What is the correct difference between the roles of (v\neq0) and (\gcd(u,v)=1) in the proof of (\sqrt{3})?HardLevel 17Which skipped step would make the proof of \(\sqrt{2}\) incomplete?HardLevel 17In a proof by contradiction that \(\sqrt{2}\) is irrational, suppose \(\sqrt{2}=\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. Which final condition contradicts this assumption?HardLevel 17If a proof writes (\sqrt{2}=\frac{x}{y}) but does not state lowest form, what is the biggest weakness?HardLevel 17While proving the irrationality of \(\sqrt{3}\) by contradiction, what conclusion about \(p\) is drawn from \(p^2=3q^2\)?HardLevel 17If both \(x,y\) are even, why is \(\frac{x}{y}\) considered reducible?HardLevel 17In the contradiction proof that √3 is irrational, \(p^2=3q^2\) gives \(3\mid p^2\). Which fact is used to conclude that \(3\mid p\)?HardLevel 17Which option best states the role of (2) in the proof of (\sqrt{2})?HardLevel 17Which statement is an essential basis of the proof by contradiction that \(\sqrt{2}\) is irrational?HardLevel 17If a student stops after proving only (x) even in the proof of (\sqrt{2}), what is the main error?HardLevel 17