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
In the proof of √2, why is assuming a and b coprime more than a formality?HardLevel 16A student says, “Since \((\sqrt{2})^2 = 2\) and 2 is rational, \(\sqrt{2}\) must also be rational.” What is the correct error in this argument?HardLevel 16If (a^2=2b^2) and (a=2r), what combined conclusion about (a) and (b) follows from (b^2=2r^2)?HardLevel 16If (p^2=3q^2) and (p=3k), what combined conclusion about (p) and (q) follows from (q^2=3k^2)?HardLevel 16Which option shows the correct contrapositive-style use needed in proving √2 irrational?HardLevel 16Which option correctly uses the prime factor idea in the proof of (\sqrt{3})?HardLevel 16While proving the irrationality of \(\sqrt{3}\) by contradiction, assume that \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. Which conclusion creates a contradiction to the initial assumption?HardLevel 16In the proof of (\sqrt{3}), if someone writes directly from (p^2=3q^2) that (q) is divisible by (3), which analysis is correct?HardLevel 16In the proof of √2, which option correctly distinguishes an ordinary rational form from the proof-ready form?HardLevel 16For a positive integer \(n\), which of the following statements is correct?HardLevel 16Which option gives the correct comparative analysis of the proofs of (\sqrt{2}) and (\sqrt{3})?HardLevel 16A student claims that \(5+\sqrt{3}\) is rational because 5 is rational. Which is the correct refutation of this claim?HardLevel 16In the proof by contradiction that \(\sqrt{3}\) is irrational, if \(p\) and \(q\) are coprime and \(p^2=3q^2\), which conclusion follows immediately?HardLevel 16Which statement is sufficient to reject the rational assumption in the proof of (\sqrt{2})?HardLevel 16Which statement is sufficient to reject the rational assumption in the proof of (√3)?HardLevel 16Which option states the correct hidden principle used in the proof of (\sqrt{2})?HardLevel 16Which option gives the deeper prime-factor reason for irrationality of (\sqrt{3})?HardLevel 16While proving the irrationality of \(\sqrt{3}\) by contradiction, suppose \(\sqrt{3}=\frac{p}{q}\) is in lowest terms. What conclusion about \(p\) follows from \(p^2=3q^2\)?HardLevel 16Which of the following conclusions about radicals is correct?HardLevel 16If
\(\sqrt{3}=\frac{p}{q}\) is assumed in lowest terms to prove irrationality, which statement establishes the contradiction?HardLevel 16