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 c/d is not taken in lowest form in the proof of √2, which conclusion will not remain decisive?ExpertLevel 18A student claims that if \(3p^2=q^2\), then \(p\) must be divisible by 3. What is the correct evaluation of the claim?ExpertLevel 18A student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. After obtaining \(3q^2=p^2\), which of the following conclusion is correct?ExpertLevel 18Why is it wrong to assume h and k are divisible by 3 from the beginning in the proof of √3?ExpertLevel 18Which of the following statements is essential in a proof by contradiction that \(\sqrt{3}\) is irrational?ExpertLevel 18In a proof by contradiction, suppose \(\sqrt{2}=\frac{h}{k}\), where \(h\) and \(k\) are coprime integers. Which conclusion contradicts this assumption?ExpertLevel 18Which of the following number-theoretic facts is used centrally in proving that \(\sqrt{3}\) is irrational?ExpertLevel 18In the proof of (\sqrt{2}), if (\frac{c}{d}) can be changed into (\frac{c/2}{d/2}), what becomes clear?ExpertLevel 18A student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. On squaring, \(p^2=2q^2\) is obtained. Which conclusion follows correctly?ExpertLevel 18If (\gcd(c,d)=1), after proving (2\mid c) in the proof of (\sqrt{2}), which conclusion should not be drawn immediately?ExpertLevel 18In the proof of √3, after proving 3 divides h, what kind of step is writing h = 3r?ExpertLevel 18In the proof of (\sqrt{2}), if (c=2u) and (d=2v) are obtained, which form is possible for (\frac{c}{d})?ExpertLevel 18In the proof of (\sqrt{3}), if (h=3r) and (k=3s) are proved, what happens to the lowest fraction condition?ExpertLevel 18A student claims that \(\sqrt{3}\) is rational and writes it as \(\frac{p}{q}\) in lowest terms, where \(p,q\) are coprime. If \(p^2=3q^2\), what is the error in this claim?ExpertLevel 18A student claims that \(\sqrt{2}+\sqrt{3}\) is a rational number. Which argument correctly identifies the error in the claim?ExpertLevel 18If both (c) and (d) become even in the proof of (\sqrt{2}), which option correctly indicates infinite descent?ExpertLevel 18If a student assumes that \(\sqrt{2}+\sqrt{3}\) is a rational number \(r\), which conclusion correctly proves a contradiction in this claim?ExpertLevel 18A student says that \(\sqrt{3}\) is irrational because its decimal expansion \(1.732\ldots\) continues endlessly. What is the main error in this argument?ExpertLevel 18In the proof of √3, 3 | h² implies 3 | h. Which broader principle does this illustrate?HardLevel 18In the proof by contradiction that \(\sqrt{3}\) is irrational, which property is used to conclude \(3\mid p\) from \(3\mid p^2\)?ExpertLevel 18