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 assumption is taken first to prove the irrationality of (\sqrt{2})?If (\sqrt{2}=\frac{p}{q}) is assumed then which condition is necessary for (p) and (q)?Which of the following numbers is irrational and can be proved irrational using a contradiction based on divisibility by 3?From (p^2=2q^2) what conclusion is obtained about (p^2)?If (p^2) is even then what is the correct conclusion about (p)?A student sees \(\sqrt{3}\) displayed as 1.732 on a calculator and claims that \(\sqrt{3}\) is rational because the decimal ends. Which statement correctly identifies the error?If (p=2r) and (p^2=2q^2), which conclusion follows next?From (q^2=2r^2) what conclusion follows about (q)?What is the final contradiction in the proof that √2 is irrational?Which method is used in the proof of the irrationality of √2?While proving the irrationality of \(\sqrt{3}\) by contradiction, which property is used to conclude \(3\mid p\) from \(3\mid p^2\)?In the proof by contradiction that \(\sqrt{2}\) is irrational, if \(\sqrt{2}=\frac{p}{q}\) is assumed where \(p,q\) are coprime, which conclusion creates a contradiction with the initial assumption?From (a^2=3b^2), which conclusion is obtained about (a^2)?If (a^2) is divisible by (3), what is the correct conclusion about (a)?Rima says that \(\sqrt{2}\) is rational because its decimal form begins with 1.414.... What is the correct error in her reasoning?If (a=3k) and (a^2=3b^2), what conclusion follows next?From (b^2=3k^2), what conclusion follows about (b)?Suppose \(\sqrt{2}=\frac{m}{n}\), where \(m\) and \(n\) are integers. Which condition on \(m\) and \(n\) is necessary at the start of a proof by contradiction that \(\sqrt{2}\) is irrational?What kind of beginning is used in the proofs of both √2 and √3?Amit claims that \(4\sqrt{3}\) is a rational number. Which argument correctly shows the error in his claim?