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
How can the proof of (\sqrt{2}) be understood from the viewpoint of infinite descent?ExpertLevel 16How can the proof of (\sqrt{3}) be expressed in the language of infinite descent?ExpertLevel 16Which statement gives the deeper reason using prime exponents in perfect squares for the proof of (\sqrt{2})?ExpertLevel 16Why must \(p/q\) be taken in lowest terms in a proof by contradiction that \(\sqrt{2}\) is irrational?ExpertLevel 16A student assumes that \(\sqrt{3}=\frac{m}{n}\), where \(m\) and \(n\) are coprime. During the proof, if both \(m\) and \(n\) are shown to be divisible by 3, what is the error in the student's assumption?ExpertLevel 16A student claims that the diagonal of a square of side 1 unit is a rational number. Which argument correctly disproves this claim?ExpertLevel 16For a square with side length 1 cm, Reena claims that its diagonal must also be rational because the side is rational. Which statement correctly identifies the error in Reena’s conclusion?ExpertLevel 16Which statement alone does not break the rational assumption in the proof of (\sqrt{3})?ExpertLevel 16A student assumes \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime, and obtains \(p^2=3q^2\). Which next step is logically valid for reaching a contradiction?ExpertLevel 16A student assumes \(\sqrt{2}=\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. From \(a^2=2b^2\), the student concludes that both \(a\) and \(b\) are even. Which statement explains why this creates a contradiction?ExpertLevel 16Which option correctly states the different roles of (b\neq0) and (\gcd(a,b)=1) in the proof of (\sqrt{2})?ExpertLevel 16Which option correctly distinguishes (q\neq0) and (\gcd(p,q)=1) in the proof of (\sqrt{3})?ExpertLevel 16Which property is used decisively in the proof by contradiction that \(\sqrt{3}\) is irrational?ExpertLevel 16While proving the irrationality of \(\sqrt{2}\) by contradiction, which is the correct initial assumption for treating \(\sqrt{2}\) as rational?ExpertLevel 16A student assumes \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime, and obtains \(p^2=3q^2\). The student shows only that \(3\mid p\) and declares a contradiction. Which step correctly completes the proof?ExpertLevel 16If (a) is assumed odd in (a^2=2b^2), what contradiction appears?ExpertLevel 16If p is not divisible by 3 in p² = 3q², what contradiction appears?HardLevel 16Choose the correct order in the proof of √2.HardLevel 16Choose the correct order in the proof of √3.HardLevel 16In the proof by contradiction for the irrationality of \(\sqrt{2}\) and \(\sqrt{3}\), which prime-number property for an integer \(n\) is used decisively?ExpertLevel 16