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 \(\sqrt{2}=\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. On proceeding with the proof, the student finds that both \(a\) and \(b\) are even. What decisive conclusion follows?While proving the irrationality of \(\sqrt{3}\) by contradiction, if it is shown that 3 divides both \(a\) and \(b\), which conclusion creates the contradiction in the proof?If n² is divisible by 3 for an integer n, which statement about n must be true?Which statement creates the actual contradiction with the coprime condition in the proof of (\sqrt{2})?In the proof of (\sqrt{3}), which conclusion alone is not sufficient but is the first step toward final contradiction?If √2 = a/b is in lowest form, what does proving that both a and b are even show?A student says, “If the area of a square is rational, then its side must also be rational.” Which of the following examples disproves the statement?Suppose \\(\sqrt{3}=\frac{p}{q}\\), where \\(p\\) and \\(q\\) are coprime positive integers. Which conclusion from \\(p^2=3q^2\\) gives the contradiction needed to prove that \\(\sqrt{3}\\) is irrational?In the proof by contradiction, assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Why does this assumption lead to a contradiction?Which statement is correct in the proof of the irrationality of \(\sqrt{3}\), based on \(a^2=3b^2\), where \(a\) and \(b\) are coprime integers?In the proof of (\sqrt{3}), if (p) is not divisible by (3), why does (p^2=3q^2) create conflict?Which option gives the most correct full logical chain in the proof of \(\sqrt{2}\)?While proving the irrationality of \(\sqrt{3}\) by contradiction, which situation contradicts the assumption that \(p/q\) is in lowest terms?In the proof of √2, why is assuming a and b coprime more than a formality?A 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?If (a^2=2b^2) and (a=2r), what combined conclusion about (a) and (b) follows from (b^2=2r^2)?If (p^2=3q^2) and (p=3k), what combined conclusion about (p) and (q) follows from (q^2=3k^2)?Which option shows the correct contrapositive-style use needed in proving √2 irrational?Which option correctly uses the prime factor idea in the proof of (\sqrt{3})?While 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?