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
While proving the irrationality of \(\sqrt{3}\) by contradiction, if \(\sqrt{3}=\frac{p}{q}\) is assumed to be in lowest terms, which conclusion about \(p\) and \(q\) produces the contradiction?HardLevel 18If the square of an integer is divisible by 3, which conclusion about the integer must be true?HardLevel 18If ext{\(\sqrt{3}\)} is assumed to be ext{\(p/q\)} , where ext{\(p\)} and ext{\(q\)} are coprime integers, which fact produces the contradiction in the proof?HardLevel 18A student assumes that \(\sqrt{2}=\frac{m}{n}\), where \(m\) and \(n\) are coprime. Which conclusion from \(m^2=2n^2\) proves that this assumption is contradictory?HardLevel 18A student claims that \(\sqrt{12}\) is rational because 12 is not a perfect square. Which is the correct simplified form of \(\sqrt{12}\) that identifies the error in the claim?HardLevel 18If
b^2=3a^2 is obtained when
b is in lowest form, which property is used to prove that
b is divisible by 3?HardLevel 18In a proof that \(\sqrt{3}\) is irrational, a student assumes \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime, and obtains \(p^2=3q^2\). The student directly writes that \(q\) is divisible by 3. Which statement is needed to make the reasoning valid?HardLevel 18If (\sqrt{3}=\frac{a}{b}) with (\gcd(a,b)=1), but both (a,b) are proved divisible by (3), which contradiction is correct?HardLevel 18Which option shows an invalid shortcut in the proof of √2?HardLevel 18Which option shows a wrong shortcut in the proof of √3?HardLevel 18If \\(\sqrt{3}\\) is written as \\(a/b\\), where \\(a\\) and \\(b\\) are coprime integers, which conclusion produces the contradiction in the proof of its irrationality?HardLevel 18In the proof of (\sqrt{3}), if (a) is not divisible by (3), what inconsistency arises from (a^2=3b^2)?HardLevel 18Which option gives the correct complete logical chain for the proof of \(\sqrt{2}\)?HardLevel 18Which option gives the correct complete logical chain for the proof of \(\sqrt{3}\)?HardLevel 18What problem occurs if (\gcd(m,n)=1) is not written in the proof of (\sqrt{2})?HardLevel 18Ravi claims that \(7+\sqrt{2}\) is a rational number because 7 is rational. What is the error in Ravi’s reasoning?HardLevel 18If \(m=2k\) and \(n^2=2k^2\), what is the combined conclusion in the proof of \(\sqrt{2}\)?HardLevel 18While proving the irrationality of \(\sqrt{3}\) by contradiction, if \(\sqrt{3}=\frac{p}{q}\) is assumed, which condition on \(p\) and \(q\) is necessary?HardLevel 18A student claims that if the square of an integer is divisible by 3, then the integer itself is divisible by 3. What is the correct evaluation of this claim?HardLevel 18What is the basis of 3 ∣ a² ⇒ 3 ∣ a in the proof of √3?HardLevel 18