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
In the proof of √3, after which sequence is n proved divisible by 3?ExpertLevel 65If \(p\) and \(q\) are coprime integers and \(p^2=3q^2\), which conclusion is essential in the proof that \(\sqrt{3}\) is irrational?ExpertLevel 17Why must the fraction be taken in lowest terms when assuming \(\sqrt{2}=\frac{p}{q}\) in the proof that \(\sqrt{2}\) is irrational?ExpertLevel 65Using exponents of prime factors in perfect squares, which idea is correct in the proof of √2?ExpertLevel 65Using exponents of prime factors in perfect squares, which idea is correct in the proof of √3?ExpertLevel 65While proving the irrationality of \(\sqrt{3}\) by contradiction, assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. Which conclusion from \(p^2=3q^2\) is necessary to reach a contradiction?ExpertLevel 65If (m) is not divisible by (3) and (m^2=3n^2), what inconsistency is obtained?ExpertLevel 65What is the correct difference between the roles of (s\neq0) and (\gcd(r,s)=1) in the proof of (\sqrt{2})?ExpertLevel 65What is the correct difference between the roles of (n\neq0) and (\gcd(m,n)=1) in the proof of (\sqrt{3})?ExpertLevel 65While proving the irrationality of \(\sqrt{3}\) by contradiction, if \(\frac{p}{q}\) is in lowest terms, which immediate conclusion follows from \(3\mid p^2\)?ExpertLevel 65If a student does not take (\frac{m}{n}) in lowest form in the proof of (\sqrt{3}), which conclusion becomes weak?ExpertLevel 65If \(\sqrt{3}=\frac{p}{q}\) is assumed in lowest terms and \(p^2=3q^2\) is obtained, which conclusion establishes the contradiction in the proof of irrationality?ExpertLevel 65Why is it wrong to assume (m,n) divisible by (3) from the beginning in the proof of (\sqrt{3})?ExpertLevel 65A student says that if \(\sqrt{3}=\frac{p}{q}\), then \(p^2=3q^2\) only implies that \(p\) is divisible by 3; nothing can be concluded about \(q\). What is the student's error?ExpertLevel 17In a proof of irrationality, if 3 is prime and 3 divides the square p² of an integer p, which conclusion is correct?ExpertLevel 17A student believes that for some non-zero rational number \(q\), \(q\sqrt{3}\) can be rational. Which argument correctly refutes this belief?ExpertLevel 65A student claims that
\(\sqrt{2}+\sqrt{3}\) is a rational number. Which argument correctly identifies the error in this claim?ExpertLevel 17Which assumption is required at the beginning of a proof by contradiction that \(\sqrt{3}\) is irrational?ExpertLevel 65If (\sqrt{3}=\frac{u}{v}) is in lowest form, why is getting (3\mid u) and (3\mid v) a decisive contradiction?ExpertLevel 65Which of the following integers has a rational square root?ExpertLevel 17