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 √2, if m/n is in lowest form, which situation is impossible?MediumLevel 21Riya says that since a calculator displays \(\sqrt{2}\) as 1.414, \(\sqrt{2}\) is a rational number. What is the error in her reasoning?EasyLevel 21Why is finding the decimal value not necessary in the proof of (\sqrt{2})?EasyLevel 21Which thing is unnecessary in the proof of (\sqrt{3})?EasyLevel 21In the proof of (\sqrt{2}), what is shown false by the rational assumption?EasyLevel 21In the proof of \(\sqrt{3}\), which conclusion is rejected by the rational assumption?EasyLevel 21Riya assumes that \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. If the proof shows that both \(a\) and \(b\) are divisible by 3, what conclusion follows?EasyLevel 21What is the main purpose of assuming \(\frac{p}{q}\) is in lowest terms in the contradiction proof that \(\sqrt{2}\) is irrational?EasyLevel 21While proving the irrationality of \(\sqrt{3}\) by contradiction, if \(\sqrt{3}=p/q\) where \(p\) and \(q\) are coprime, which conclusion is essential to the proof?EasyLevel 21A student claims that \(2\sqrt{2}\) is rational because 2 is a rational number. Which statement correctly explains the error?EasyLevel 21What is the main aim of the proofs of (\sqrt{2}) and (\sqrt{3})?EasyLevel 21Which option gives a correct statement for both (\sqrt{2}) and (\sqrt{3})?EasyLevel 21Which mistake should be avoided in the proof that √2 is irrational?MediumLevel 21Which mistake should be avoided in the proof of (\sqrt{3})?EasyLevel 21In the proof of irrationality of (\sqrt{2}), what type of numbers are (m) and (n) assumed to be?EasyLevel 21In the proof of irrationality of (\sqrt{3}), what type of numbers are (r) and (s) assumed to be?EasyLevel 21After assuming √2 = a/b in lowest rational form, we get a² = 2b². Which immediate conclusion is correct?MediumLevel 16Assuming (\sqrt{3}=\frac{p}{q}), we get (p^2=3q^2). What is the correct conclusion about (p)?MediumLevel 16A student claims that \(\sqrt{3}\) is rational because \(1.7^2=2.89\), which is very close to 3. What is the main error in the student's reasoning?MediumLevel 16A student says that \(2\sqrt{3}\) is rational because 2 is a rational number. What is the error in the student's statement?MediumLevel 16