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 correct difference between the roles of (q\neq0) and (\gcd(p,q)=1) in the proof of (\sqrt{3})?Which option gives the correct order of the proof of \(\sqrt{2}\)?While proving \(\sqrt{3}\) irrational by contradiction, we assume \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. Which conclusion at the end of the proof establishes the contradiction?If \(r\) is a non-zero rational number, which statement about \(r\sqrt{3}\) is correct?If a student does not take (\frac{p}{q}) in lowest form in the proof of (\sqrt{3}), which conclusion becomes weak?Suppose \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. Which contradiction follows from this assumption?Why is it wrong to assume (p,q) divisible by (3) from the beginning in the proof of (\sqrt{3})?A student says, “For a non-zero rational number \(a\), \(a\sqrt{3}\) can be rational.” Which argument correctly explains the error in this statement?Which assumption is rejected by the contradiction obtained after assuming (\sqrt{3}) rational?A student claims that \(5+\sqrt{3}\) may be rational because 5 is rational. Which is the correct refutation of this claim?In a proof by contradiction, assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion from \(p^2=3q^2\) leads to the contradiction?In a proof by contradiction that \(\sqrt{3}\) is irrational, it is assumed to be \(\frac{p}{q}\), where \(p\) and \(q\) are coprime. Which conclusion contradicts the initial assumption?What is the main weakness in proving (\sqrt{3}) irrational using an approximate decimal?When proving the irrationality of \(\sqrt{3}\) by the contradiction method, which assumption is made at the start?If \(\sqrt{3}=\frac{p}{q}\) is assumed, where p and q are integers, why must \(\frac{p}{q}\) be taken in lowest terms in the proof by contradiction of its irrationality?In the proof by contradiction for the irrationality of \(\sqrt{3}\), assume \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. Which conclusion from \(p^2=3q^2\) is essential for the proof?In the proof by contradiction that \(\sqrt{2}\) is irrational, assume \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion contradicts this assumption?A student says, “\(\sqrt{3}\) is rational because it can be written as \(1.732\).” What is the main flaw in this reasoning?A student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. On squaring, he gets \(p^2=2q^2\). He says that \(q\) is even because the right-hand side is even. What should be the first correct conclusion in this argument?A student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q\ne0\), and concludes from \(p^2=2q^2\) that both \(p\) and \(q\) are even. If the student did not assume \(p\) and \(q\) to be coprime, why is this conclusion not automatically a contradiction?