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
Which initial assumption is made to prove the irrationality of \(\sqrt{3}\) by the method of contradiction?MediumLevel 18In the proof that
sqrt3 is irrational, if
sqrt3 = p/q is assumed to be in lowest terms, why does a contradiction arise?MediumLevel 18It is known that \(\sqrt{3}\) is irrational. Which argument correctly completes a proof that \(\sqrt{12}\) is irrational?MediumLevel 18In the proof of (\sqrt{3}), after getting (q^2=3k^2), which conclusion about (q) is correct?MediumLevel 18Which of the following statements is correct in the contradiction proof that \(\sqrt{2}\) is irrational?MediumLevel 18If (\frac{a}{b}) is in lowest form, which situation is impossible?MediumLevel 18If p/q is in lowest form, which situation creates a contradiction in the proof of √3?MediumLevel 18Which option gives the correct order in the proof of irrationality of (\sqrt{2})?MediumLevel 18Which option gives the correct order in the proof of irrationality of (\sqrt{3})?MediumLevel 18A student believes that the statement “If the square of an integer is divisible by 3, then the integer is also divisible by 3” is false. Which statement correctly removes this misconception?MediumLevel 18In the proof of (\sqrt{3}), why is it incomplete to write directly from (p^2=3q^2) that (q) is divisible by (3)?MediumLevel 18A student says that since 5 is a rational number, \(5+\sqrt{3}\) must also be rational. Which statement correctly identifies the error in the argument?MediumLevel 18Which statement is used to move from p² to p in the proof of √3?MediumLevel 18Suppose a student writes \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime, while proving that \(\sqrt{3}\) is irrational. From \(p^2=3q^2\), which conclusion about \(p\) is correct?MediumLevel 18Why are p and q chosen coprime in √3 = p/q?MediumLevel 18If (p) and (q) are coprime, which situation is impossible in the proof of (\sqrt{3})?MediumLevel 18If \(\sqrt{3}\) is assumed to be \(\frac{p}{q}\) in lowest terms, what conclusion follows from \(p^2=3q^2\)?MediumLevel 18In the proof of (\sqrt{3}), if (p) is not divisible by (3), what conflict occurs with (p^2=3q^2)?MediumLevel 18Why is the proof of \(\sqrt{2}\) valid without decimal expansion?MediumLevel 18While proving \(\sqrt{3}\) irrational by contradiction, suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. What follows immediately from \(p^2=3q^2\)?MediumLevel 18