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 contradiction proof for the irrationality of \(\sqrt{3}\), assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. Why is the condition that \(p\) and \(q\) are coprime necessary?EasyLevel 20Rima says that if the decimal expansion of a number is infinite but non-repeating, then the number is rational. What is the error in Rima's statement?EasyLevel 20Which conclusion is correct in the proof that √2 is irrational?MediumLevel 20A student says that if \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, then from \(p^2=2q^2\) only \(p\) is even and nothing can be said about \(q\). What is the student's error?EasyLevel 20If the square of a number is not even then what type is the number?EasyLevel 20If a number is not divisible by (3), then its square is surely not divisible by what?EasyLevel 20Which of the following statements is the correct basis for proving that \(\sqrt{3}\) is irrational?EasyLevel 20A student says that \(\sqrt{3}\) is rational because 1.732 is a terminating decimal. What is the error in the student's reasoning?EasyLevel 20A student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. On squaring both sides, \(3q^2=p^2\) is obtained. Which conclusion proves the assumption wrong?EasyLevel 20Which of the following statements correctly describes the main idea used in proving that \(\sqrt{2}\) is irrational?EasyLevel 20A student says that if \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q\ne0\), then \(p^2=3q^2\) proves only that \(p\) is divisible by 3. What is the correct next conclusion in this argument?EasyLevel 20Why is it necessary to take (\frac{p}{q}) in lowest form in the proof of (\sqrt{3})?EasyLevel 20A student says that \(\sqrt{3}\) is rational because its value is approximately 1.73. Which comment about this statement is correct?EasyLevel 20Which of the following numbers has an irrational square root?EasyLevel 20In both proofs in what form is the number first written?EasyLevel 20A student writes: “3 is not a perfect square, so \(\sqrt{3}\) is irrational.” What is the most appropriate evaluation of this statement in a question asking to prove the irrationality of \(\sqrt{3}\)?EasyLevel 20In the proof of √2, when both a and b are even, which conclusion should not be taken?MediumLevel 20If a number can be written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are coprime integers and \(q\neq 0\), what is it called?EasyLevel 20What is the main aim of proving irrationality of (\sqrt{2}) and (\sqrt{3})?EasyLevel 20In the proof of \(\sqrt{2}\), what is the first conclusion after getting \(a^2=2b^2\)?EasyLevel 20