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 of the following numbers cannot be written as a ratio \(p/q\) of two integers, where \(q\ne0\)?MediumLevel 17Ravi says that \(\sqrt{2}\) is irrational because its decimal expansion is infinite. What is the flaw in his argument?MediumLevel 17In the proof of \(\sqrt{2}\), if both \(m\) and \(n\) are even, what can be said about their highest common factor?MediumLevel 17If a prime number has an odd exponent in the prime factorisation of a number, what can be concluded about the square root of that number?MediumLevel 17In the proof of (√2), if (m) is assumed odd, what problem arises from (m²=2n²)?MediumLevel 17In the proof of (\sqrt{3}), if (p) is not divisible by (3), what problem arises from (p^2=3q^2)?MediumLevel 17Which option gives the correct final sentence in the proof of √2?MediumLevel 17Which option gives the correct final sentence in the proof of (\sqrt{3})?MediumLevel 17Which method is suitable for the proofs of (\sqrt{2}) and (\sqrt{3}) and why?MediumLevel 17A student says, “\(\sqrt{3}=1.732\), so \(\sqrt{3}\) is a rational number.” What is the main error in this statement?MediumLevel 17A student says, “The square of \(\sqrt{2}\) is \(2\), and \(2\) is rational; therefore, \(\sqrt{2}\) is also rational.” What is the main error in this reasoning?MediumLevel 17In the proof of (\sqrt{2}), after (m^2=2n^2), what contradiction is prepared by writing (m=2r)?MediumLevel 17A student assumes that \(\sqrt{2}=\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. From \(a^2=2b^2\), the student stops after concluding that \(a\) is even. Which conclusion is necessary to complete the proof?MediumLevel 17What is the final contradiction in the proof by contradiction that \(\sqrt{3}\) is irrational?MediumLevel 17Suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion produces a contradiction in the proof that \(\sqrt{3}\) is irrational?MediumLevel 17What is the correct combined conclusion of the proofs of both (\sqrt{2}) and (\sqrt{3})?MediumLevel 17If \(\sqrt{3}=\frac{p}{q}\) is assumed to be in lowest terms, which statement produces the contradiction in the proof of its irrationality?MediumLevel 18In the proof that \(\sqrt{3}\) is irrational, assume \(\sqrt{3}=\frac{p}{q}\) in lowest terms. If \(p\) is shown to be divisible by 3, which conclusion about \(q\) is needed to obtain a contradiction?MediumLevel 18Which of the following decimal expansions indicates an irrational number?MediumLevel 18Which statement correctly describes the contradiction obtained while proving that \(\sqrt{2}\) is irrational?MediumLevel 18