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
A 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 16A student claims that if \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, then \(p^2=3q^2\) proves only that \(p\) is divisible by 3. What is the correct improvement to the argument?MediumLevel 17What is the main reason for assuming (\sqrt{3}=\frac{p}{q}) in lowest form?MediumLevel 17In contradiction method, which initial assumption is taken for (\sqrt{2})?MediumLevel 17In the proof of (\sqrt{3}), why can we write (p=3k) from (p^2=3q^2)?MediumLevel 17While proving the irrationality of \(\sqrt{3}\) by contradiction, if assuming \(\sqrt{3}=\frac{p}{q}\) in lowest terms gives \(p^2=3q^2\), which conclusion creates the contradiction?MediumLevel 17In the proof of (\sqrt{3}), after getting (q^2=3k^2), what is the next conclusion toward final contradiction?MediumLevel 17A student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. If \(p^2=3q^2\) is obtained, what is the correct next conclusion in the proof?MediumLevel 17Which statement is a wrong conclusion in the proof of \(\sqrt{3}\)?MediumLevel 17Which of the following numbers has an irrational positive square root?MediumLevel 17In the proof of (\sqrt{3}), (p^2=3q^2) gives (p=3k) and then (q^2=3k^2). What will be the final conclusion?MediumLevel 17Which option shows the correct logical chain in the proof of (√2)?MediumLevel 17Which option shows the correct logical chain in the proof of (√3)?MediumLevel 17In a proof that √3 is irrational, suppose √3 = p/q, where p and q are coprime. Squaring gives p² = 3q², and on writing p = 3k, we get q² = 3k². Which conclusion is needed to complete the contradiction?MediumLevel 17A student says, “3 is not a perfect square, so \(\sqrt{3}\) is irrational.” Which step is needed to complete this argument?MediumLevel 17