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
What is the combined conclusion of the proofs of (\sqrt{2}) and (\sqrt{3})?MediumLevel 18If the rational assumption for (\sqrt{2}) were correct, what should be true about (\frac{m}{n})?MediumLevel 18Before proving the irrationality of \(\sqrt{3}\) by contradiction, in what form must \(\frac{p}{q}\) be assumed?MediumLevel 18In the proof that
\(\sqrt{3}\) is irrational, suppose
\(\sqrt{3}=p/q\) is in lowest terms and squaring gives
\(p^2=3q^2\). What correctly fixes a student's error that
\(3\mid p^2\) does not imply
\(3\mid p\)?MediumLevel 18Which initial assumption is made to prove the irrationality of \(\sqrt{2}\) by the method of contradiction?MediumLevel 18A student argues 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. Identify the student's error.MediumLevel 18Which statement gives both correct reason and conclusion in the proof of (\sqrt{3})?MediumLevel 18Riya says that \(\sqrt{3}\) is rational because its decimal form is \(1.732\). Which statement correctly identifies the error in her claim?MediumLevel 18In the proof of √3, what does p = 3k show?MediumLevel 18While proving the irrationality of \(\sqrt{3}\) by contradiction, suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. What is the first conclusion obtained from \(3q^2=p^2\)?MediumLevel 18A student says that if \(\sqrt{2}=\frac{p}{q}\), then both \(p\) and \(q\) may be even. Why is this statement incorrect in the proof that \(\sqrt{2}\) is irrational?MediumLevel 18If assuming √2 is rational gives a contradiction, which assumption is proved false?MediumLevel 18A student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. On squaring, the student gets \(p^2=2q^2\). Which conclusion proves that this assumption is invalid?MediumLevel 18Which option correctly describes the role of (m) and (n) in the proof of (\sqrt{2})?MediumLevel 18Which option correctly describes the role of (p) and (q) in the proof of (\sqrt{3})?MediumLevel 18Which option gives the correct middle objective in the proof of (\sqrt{2})?MediumLevel 18Which option gives the correct middle objective in the proof of (\sqrt{3})?MediumLevel 18If a student writes (a=2b) from (a^2=2b^2) in the proof of (\sqrt{2}), what is the mistake?MediumLevel 18If a student writes (p=3q) directly from (p^2=3q^2) in the proof of (\sqrt{3}), what is the mistake?MediumLevel 18Which of the following statements is essential for proving the irrationality of \(\sqrt{3}\) by the contradiction method?MediumLevel 18