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
If (a^2) is divisible by (3), what is the correct conclusion about (a)?EasyLevel 19Rima says that \(\sqrt{2}\) is rational because its decimal form begins with 1.414.... What is the correct error in her reasoning?EasyLevel 19If (a=3k) and (a^2=3b^2), what conclusion follows next?EasyLevel 19From (b^2=3k^2), what conclusion follows about (b)?EasyLevel 19Suppose \(\sqrt{2}=\frac{m}{n}\), 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?EasyLevel 19What kind of beginning is used in the proofs of both √2 and √3?EasyLevel 19Amit claims that \(4\sqrt{3}\) is a rational number. Which argument correctly shows the error in his claim?EasyLevel 19Why are a and b assumed to be coprime in √3 = a/b?EasyLevel 19If a number is even then what type will its square be?EasyLevel 19If a number is divisible by (3), what about its square?EasyLevel 19Which statement is used most in the proof of (\sqrt{2})?EasyLevel 19Which statement is mainly used in the proof of (\sqrt{3})?EasyLevel 19Which number-theoretic fact is used decisively while proving the irrationality of \(\sqrt{3}\) by contradiction?EasyLevel 19A student says, “If \(\sqrt{3}\) is irrational, then \(5\sqrt{3}\) is also irrational.” Which argument correctly proves the statement?EasyLevel 19A student says that \(5+\sqrt{3}\) may be a rational number. Which argument correctly identifies the error in this statement?EasyLevel 19Riya says that \(\sqrt{2}\) is irrational because its decimal expansion never terminates. What is the flaw in her reasoning?EasyLevel 19What is the main purpose of the final step in the contradiction method?EasyLevel 19A calculator displays \(\sqrt{3}\) as 1.732. Mohan concludes that \(\sqrt{3}\) is rational. What is Mohan’s error?EasyLevel 19Which statement about √3 is correct?EasyLevel 19A student assumes that \(\sqrt{3}\) can be written as \(\frac{a}{b}\) in lowest terms. During the proof, it is found that 3 divides both \(a\) and \(b\). Which conclusion about the student’s assumption is correct?EasyLevel 19