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
Suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. After concluding from \(p^2=3q^2\) that \(p\) is divisible by 3, which conclusion completes the contradiction proving \(\sqrt{3}\) is irrational?MediumLevel 16In the proof that \(\sqrt{3}\) is irrational, \(a^2=3b^2\) gives \(3\mid a^2\). Which rule justifies the next step?MediumLevel 16A student claims that \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. Which correct conclusion follows from this claim?MediumLevel 16In the proof of √2, what is finally proved false?MediumLevel 16In the proof of (\sqrt{3}), which assumption is finally rejected?MediumLevel 16A student writes: If \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime, then \(a^2=3b^2\). The student concludes that 3 divides \(b\). What is the error in this conclusion?MediumLevel 16Which option gives the correct conclusion and reason for the proof of √3?MediumLevel 16If (a) and (b) are coprime, which situation is not possible?MediumLevel 16Why is \(\frac{p}{q}\) taken in lowest terms while proving the irrationality of \(\sqrt{2}\) by contradiction?MediumLevel 16If assuming \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime, leads to \(p^2=3q^2\), which conclusion creates the contradiction?MediumLevel 16A student claims that if the square of an integer is divisible by 2, then the integer itself is divisible by 2. How is this statement useful in proving the irrationality of \(\sqrt{2}\)?MediumLevel 16Which option gives the correct rational form used in the proof of (\sqrt{2})?MediumLevel 16Which option gives the correct rational form used in the proof of (\sqrt{3})?MediumLevel 16A student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. After obtaining \(p^2=3q^2\), which conclusion is correct?MediumLevel 16A student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. After obtaining \(p^2=2q^2\), which conclusion creates a contradiction in this assumption?MediumLevel 16Which option is the correct final statement for both (\sqrt{2}) and (\sqrt{3})?MediumLevel 16In the proof of √2, if after taking a = 2r we get b² = 2r², what does it prove next?MediumLevel 16In the proof of (\sqrt{3}), if after taking (p=3k) we get (q^2=3k^2), which conclusion does it lead to?MediumLevel 16A student has to prove that \(1+\sqrt{3}\) is irrational. Which of the following arguments is correct?MediumLevel 16Which option gives the correct final argument in the proof of (\sqrt{3})?MediumLevel 16