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
In the proof of √2, why is it a contradiction when both a and b are even?MediumLevel 16Which statement is correct about \(\sqrt{2}\) and \(\sqrt{3}\)?MediumLevel 16Which order is correct in the proof of irrationality of (\sqrt{2})?MediumLevel 16Suppose \\(\sqrt{3}=\frac{p}{q}\\), where p and q are coprime positive integers. In the proof that \\(\sqrt{3}\\) is irrational, which fact gives the contradiction?MediumLevel 16If (x^2) is even then (x) is even. This statement is mainly used in which proof?MediumLevel 16If (x^2) is divisible by (3), then (x) is divisible by (3). This statement is central in which proof?MediumLevel 16Why is a/b taken in lowest form in the proof of √2?MediumLevel 16Reema says that \(\sqrt{3}\) is rational because 3 is a rational number. What is her error?MediumLevel 16A student says that \(4+\sqrt{3}\) is a rational number because 4 is rational. Which statement correctly explains the error?MediumLevel 16If (a^2=2b^2) and (a=2r), why will (b) be even?MediumLevel 16If (p^2=3q^2) and (p=3k), why will (q) be divisible by (3)?MediumLevel 16Which option correctly states the main difference between the proofs of (\sqrt{2}) and (\sqrt{3})?MediumLevel 16If a student writes (\sqrt{2}=\frac{a}{b}) and assumes (a) and (b) even from the start, what is the mistake?MediumLevel 16If a student assumes both (p) and (q) divisible by (3) immediately after writing (\sqrt{3}=\frac{p}{q}), what is the mistake?MediumLevel 16In which of the following options are both numbers irrational?MediumLevel 16Riya assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. On squaring, she gets \(p^2=3q^2\). Which argument correctly proves a contradiction in this assumption?MediumLevel 16In the proof of √3, after both p and q become divisible by 3, what is the final conclusion?MediumLevel 16Rima says that \(5+\sqrt{2}\) is a rational number because 5 is rational. Which is the correct evaluation of her statement?MediumLevel 16Which option shows the correct algebraic chain in the proof of (\sqrt{3})?MediumLevel 16If (\sqrt{2}) were rational, what condition should hold for (\frac{a}{b})?MediumLevel 16