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 statement correctly describes the method used to prove that \(\sqrt{2}\) is irrational?EasyLevel 20A student assumes
dfrac{p}{q}
is in lowest terms and obtains
p^2=3q^2
, where
p
and
q
are coprime. Which conclusion proves a contradiction in this assumption?EasyLevel 20A student claims that \(\sqrt{2}\) is rational because \(1.414\) can be written as \(\frac{1414}{1000}\). What is the main error in the student's reasoning?EasyLevel 20What is known about (a^2) from (a^2=2b^2)?EasyLevel 20A student says, “\(\sqrt{3}=\frac{6}{\sqrt{12}}\), so \(\sqrt{3}\) is rational.” What is the error in the argument?EasyLevel 20If (a^2) is even what is the correct conclusion about (a)?EasyLevel 20If (p^2) is divisible by (3) what is the correct conclusion about (p)?EasyLevel 20If \(\sqrt{3}=\frac{p}{q}\) is assumed, where \(p\) and \(q\) are coprime, which conclusion produces the contradiction?EasyLevel 20The diagonal of a square with side 1 unit is \(\sqrt{2}\) units long. A student wants to write it as a ratio of two integers. Which conclusion is correct?EasyLevel 20If (a=2k) and (a^2=2b^2) then which relation follows?EasyLevel 20If (p=3r) and (p^2=3q^2) then which relation is formed next?EasyLevel 20From (b^2=2k^2) what conclusion is obtained about (b)?EasyLevel 20From (q^2=3r^2) what conclusion is obtained about (q)?EasyLevel 20A student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. After obtaining \(p^2=3q^2\), which conclusion proves this assumption contradictory?EasyLevel 20If \(\sqrt{2}\) is assumed to be \(\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers, which conclusion creates the contradiction in the proof?EasyLevel 20Why is the initial assumption considered false in the contradiction method?EasyLevel 20A student says, “\(3\) is an integer, so \(\sqrt{3}\) is rational.” What is the main error in this reasoning?EasyLevel 20Which statement is correct in proving irrationality of (\sqrt{3})?EasyLevel 20A student claims that \(\sqrt{3}=\frac{26}{15}\). Which of the following checks immediately proves that this claim is false?EasyLevel 20When irrationality of \(\sqrt{3}\) is proved, which conclusion is correct?EasyLevel 20