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
If (r=3t) and (r^2=3s^2), what follows next?EasyLevel 21A student writes that \(\sqrt{3}=1.732\), so \(\sqrt{3}\) is rational. What is the correct correction to this statement?EasyLevel 21In the proof of (\sqrt{3}), what conclusion about (s) follows from (s^2=3t^2)?EasyLevel 21A student claims that \(\sqrt{3}=\frac{6}{10}\) because a decimal number close to 3 can be written. What is the error in this claim?EasyLevel 21While proving the irrationality of \(\sqrt{2}\), a student assumes \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. On obtaining \(p^2=2q^2\), the student immediately says that \(q\) is even. How should the teacher correct the error?EasyLevel 21Reema says, “The decimal expansion of \(\sqrt{3}\) is infinite, so it is irrational.” What is the flaw in her reasoning?EasyLevel 21In the proof by contradiction that \(\sqrt{3}\) is irrational, if \(\sqrt{3}=\frac{p}{q}\) where \(p\) and \(q\) are coprime, which statement follows from \(p^2=3q^2\) and creates the contradiction?EasyLevel 21If \(\sqrt{2}\) is assumed to be \(\frac{p}{q}\) in lowest terms, which conclusion creates the contradiction in proving that it is irrational?EasyLevel 21A student says, “If n is an integer, then \(\sqrt{n}\) will also be an integer.” Which example is most suitable to disprove this statement?EasyLevel 21Aman assumes that ext{\(\sqrt{3}=\frac{p}{q}\)}, where ext{\(p\)} and ext{\(q\)} are coprime. On squaring, he gets ext{\(p^2=3q^2\)}. Aman says, “ ext{\(3\mid p^2\)} does not necessarily imply ext{\(3\mid p\)}.” What is the correct evaluation of Aman’s statement?EasyLevel 21What does assuming (\sqrt{2}) as (\frac{m}{n}) mean?EasyLevel 21What does assuming (\sqrt{3}) as (\frac{r}{s}) mean?EasyLevel 21A student claims that √12 is irrational because √12 = 2√3. Which statement is needed to make this argument valid?EasyLevel 21What is the purpose of taking r/s in lowest form in the proof that √3 is irrational?MediumLevel 21Which is the correct short order of the proof that √2 is irrational?MediumLevel 21Which is the correct short order of the proof of (\sqrt{3})?EasyLevel 21A student says that \(\sqrt{3}\) is rational because \(1.7^2=2.89\), which is very close to 3. What is the error in the student's reasoning?EasyLevel 21A student says, “1.732 is a terminating decimal and is very close to \(\sqrt{3}\), so \(\sqrt{3}\) is rational.” What is the main error in the argument?EasyLevel 21A student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. The student obtains \(p^2=3q^2\). Which conclusion from this step establishes the contradiction?EasyLevel 21In the proof of (\sqrt{3}), when both numbers are divisible by (3), which common factor is obtained?EasyLevel 21