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
Why is the equation a² = 3b² important in the proof of √3?EasyLevel 19If p/q is in lowest form, what is true about p and q?EasyLevel 19Rima claims that \(2+\sqrt{3}\) is a rational number. Which is the most appropriate argument to prove her claim wrong?EasyLevel 19A student claims that \(\sqrt{2}\) is rational because \(1.414=\frac{1414}{1000}\). What is the main error in the argument?EasyLevel 19What does assuming (\sqrt{2}) as (\frac{p}{q}) mean?EasyLevel 19What does assuming √3 = a/b mean?EasyLevel 19Suppose a is an integer and 3 divides \(a^2\). Which of the following conclusions is valid in the proof that \(\sqrt{3}\) is irrational?EasyLevel 19A student assumes that
",EasyLevel 19A student says that if \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, then when \(p\) is divisible by 3, \(q\) will also be divisible by 3. What conclusion does this statement lead to?EasyLevel 19Which of the following correctly describes the nature of the number \(\sqrt{2}\)?EasyLevel 19Which option shows the correct order in the proof of (\sqrt{2})?EasyLevel 19Which option shows the correct order in the proof of (\sqrt{3})?EasyLevel 19Why are (p) and (q) taken as integers in the proof of (\sqrt{2})?EasyLevel 19Why is (b\neq0) necessary in the proof of (\sqrt{3})?EasyLevel 19Assuming that \(\sqrt{3}\) is irrational, which of the following numbers must be irrational?EasyLevel 19In the proof of (\sqrt{3}), what type of conclusion is obtained twice?EasyLevel 19What is common in the proofs of irrationality of (\sqrt{2}) and (\sqrt{3})?EasyLevel 19In the proof that \(\sqrt{3}\) is irrational, if \(p^2\) is divisible by 3, which conclusion about \(p\) is correct?EasyLevel 20While proving (\sqrt{3}) irrational which statement is assumed at the start?EasyLevel 20If \(\sqrt{3}=\frac{m}{n}\) is assumed, where \(m\) and \(n\) are coprime integers, which conclusion produces the contradiction?EasyLevel 20