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 problem occurs if (\gcd(m,n)=1) is not written in the proof of (\sqrt{2})?HardLevel 18Ravi claims that \(7+\sqrt{2}\) is a rational number because 7 is rational. What is the error in Ravi’s reasoning?HardLevel 18If \(m=2k\) and \(n^2=2k^2\), what is the combined conclusion in the proof of \(\sqrt{2}\)?HardLevel 18While proving the irrationality of \(\sqrt{3}\) by contradiction, if \(\sqrt{3}=\frac{p}{q}\) is assumed, which condition on \(p\) and \(q\) is necessary?HardLevel 18A student claims that if the square of an integer is divisible by 3, then the integer itself is divisible by 3. What is the correct evaluation of this claim?HardLevel 18What is the basis of 3 ∣ a² ⇒ 3 ∣ a in the proof of √3?HardLevel 18A student says that \(\sqrt{2}+\sqrt{3}\) is irrational because the sum of two irrational numbers is always irrational. What is the correct evaluation of this statement?HardLevel 18Why is it incomplete to write (b) divisible by (3) directly from (a^2=3b^2) in the proof of (\sqrt{3})?HardLevel 18In the proof by contradiction that \(\sqrt{3}\) is irrational, if \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers, which conclusion necessarily follows from \(3q^2=p^2\)?HardLevel 18While proving the irrationality of \(\sqrt{3}\) by contradiction, Riya shows that in the fraction \(a/b\) written in lowest terms, both \(a\) and \(b\) are divisible by 3. Why is this conclusion impossible?HardLevel 18While proving the irrationality of \(\sqrt{3}\) by contradiction, if \(\sqrt{3}=\frac{p}{q}\) is assumed in lowest terms and \(p^2=3q^2\) is obtained, which conclusion is necessary to reach the contradiction?HardLevel 18If \(\sqrt{3}=p/q \), where \(p \) and \(q \) are coprime positive integers, which conclusion is required to establish the contradiction?HardLevel 18In a proof that \(\sqrt{3}\) is irrational, if \(\frac{p}{q}\) is assumed to be in lowest terms and \(p^2=3q^2\) is obtained, which conclusion establishes the contradiction?HardLevel 18Which idea about exponents of prime factors deeply explains the irrationality of (\sqrt{2})?HardLevel 18Which idea about exponents of prime factors deeply explains the irrationality of (\sqrt{3})?HardLevel 18If (m,n) are coprime and both are proved even, what is the correct contradiction about (\gcd(m,n))?HardLevel 18If a and b are coprime and both are proved divisible by 3, what is the correct contradiction about gcd(a,b)?HardLevel 18What is the correct difference between the roles of (n\neq0) and (\gcd(m,n)=1) in the proof of (\sqrt{2})?HardLevel 18What is the correct difference between the roles of (b\neq0) and (\gcd(a,b)=1) in the proof of (\sqrt{3})?HardLevel 18Rima says, “If \(\sqrt{12}\) were rational, then \(\sqrt{3}=\frac{\sqrt{12}}{2}\) would also be rational, which contradicts the irrationality of \(\sqrt{3}\).” What is Rima’s conclusion?HardLevel 18