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 both proofs in what form is the number first written?EasyLevel 20A student writes: “3 is not a perfect square, so \(\sqrt{3}\) is irrational.” What is the most appropriate evaluation of this statement in a question asking to prove the irrationality of \(\sqrt{3}\)?EasyLevel 20In the proof of √2, when both a and b are even, which conclusion should not be taken?MediumLevel 20If a number can be written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are coprime integers and \(q\neq 0\), what is it called?EasyLevel 20What is the main aim of proving irrationality of (\sqrt{2}) and (\sqrt{3})?EasyLevel 20In the proof of \(\sqrt{2}\), what is the first conclusion after getting \(a^2=2b^2\)?EasyLevel 20While proving that \(\sqrt{2}\) is irrational by contradiction, assume \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. After obtaining \(p^2=2q^2\) and showing that \(p\) is even, which conclusion must be established to get a contradiction?EasyLevel 20In proving irrationality of (\sqrt{2}), with which assumption does contradiction method begin?EasyLevel 21While proving (\sqrt{3}) irrational, what is the first rational assumption?EasyLevel 21When a rational number is written in lowest form as p/q, what is true about p and q?EasyLevel 21Riya says, “Since 2 is an integer,
\(\sqrt{2}\) must also be a rational number.” What is Riya’s error?EasyLevel 21A student says, “\(\sqrt{3}=1.732\), so \(\sqrt{3}\) is rational.” What is the main error in the student's reasoning?EasyLevel 21After squaring \(\sqrt{3}=\frac{r}{s}\), which equation is correct?EasyLevel 21If \(\sqrt{3}\) is assumed to be rational and written as \(\frac{p}{q}\), which condition is necessary for \(p\) and \(q\)?EasyLevel 21If (r^2=3s^2), what is true about (r^2)?EasyLevel 21If the square of an integer (x) is even, what type is (x)?EasyLevel 21If the square of an integer (x) is divisible by (3), then (x) is divisible by what?EasyLevel 21While proving the irrationality of \(\sqrt{3}\) by contradiction, suppose \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime. Which conclusion produces a contradiction to this assumption?EasyLevel 21Which statement creates a contradiction in the proof that \(\sqrt{2}\) is irrational?EasyLevel 21If (m=2k) and (m^2=2n^2), which relation follows?EasyLevel 21