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
Rina is proving that \(\sqrt{12}\) is irrational. Which of the following arguments contains no error?ExpertLevel 16A square has an area of \(2\text{ cm}^2\). A student says, “Since the area is rational, the side of the square must also be rational.” What is the correct correction to this statement?ExpertLevel 16What problem appears when a² = 2b² is viewed through prime factors in the proof of √2?HardLevel 16Why must \(\sqrt{3}=\frac{p}{q}\) be assumed to be in lowest terms while proving the irrationality of \(\sqrt{3}\) by contradiction?ExpertLevel 16If a student tries to prove irrationality of √2 by writing its decimal value, what is the correct evaluation?MediumLevel 16A student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, and on squaring obtains \(p^2=2q^2\). Which reasoning correctly proves irrationality?ExpertLevel 16Which divisibility rule is crucial in the proof by contradiction that \(\sqrt{3}\) is irrational?ExpertLevel 16How would the decimal expansion of \(\sqrt{2}\) be classified?ExpertLevel 16Why must \(p/q\) be taken in lowest terms in the standard proof by contradiction that \(\sqrt{3}\) is irrational?ExpertLevel 16In a proof by contradiction that √2 is irrational, a student assumes √2 = a/b, where a and b are coprime positive integers. From 2b² = a², the student concludes that a is even. Which is the correct basis for this conclusion?ExpertLevel 16Suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which contradiction follows from this assumption?ExpertLevel 16Suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which property is used to infer \(3\mid p\) from \(3\mid p^2\) in the proof?ExpertLevel 16If \(\sqrt{3}=\frac{p}{q}\) is assumed to be in lowest terms, where \(p\) and \(q\) are coprime integers, which statement produces the contradiction in the proof of its irrationality?ExpertLevel 16While proving the irrationality of \(\sqrt{2}\) by contradiction, assume that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion makes this assumption impossible?ExpertLevel 16If both (a,b) are assumed even from the beginning while proving (\sqrt{2}), what is the mistake?ExpertLevel 16If (p,q) are assumed divisible by (3) from the start in the proof of (\sqrt{3}), what is the mistake?ExpertLevel 16What is the most important exam caution in the proofs of √2 and √3?MediumLevel 16A student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. After obtaining \(p^2=3q^2\), the student says, “\(p\) is divisible by 3, but \(q\) need not be divisible by 3.” Which statement correctly identifies the error?ExpertLevel 16In irrationality of (\sqrt{3}), which assumption is rejected by the contradiction?ExpertLevel 16In the proof of (\sqrt{2}), if (a,b) are assumed coprime and later (a=2r) and (b=2s) are obtained, which conclusion is most precise?ExpertLevel 16