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 (\sqrt{3}), if (u) is not divisible by (3), what conflict occurs from (u^2=3v^2)?If \(p\) and \(q\) are coprime integers and \(p^2=3q^2\), which conclusion necessarily follows?In a proof by contradiction, assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. If \(3\mid p^2\) is obtained, which conclusion must follow next?Suppose
\(\sqrt{3}=\frac{p}{q}\) is written in lowest terms. If
\(3\mid p^2\), which of the following conclusion is valid?Why does the proof of (\sqrt{3}) become weak without writing (\gcd(u,v)=1)?Suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion produces the contradiction in the proof that \(\sqrt{3}\) is irrational?If (u^2=3v^2) and (u=3t), what is the combined conclusion about (u) and (v)?If
\(\sqrt{3}=p/q\) is assumed with
\(p\) and
\(q\) coprime, which condition contradicts this assumption in the proof of irrationality?What is the correct argument involving the prime number 3 in the proof of √3?If
\(\sqrt{3}=\frac{p}{q}\) is assumed with
\(p\) and
\(q\) coprime, and the proof shows that both
\(p\) and
\(q\) are divisible by 3, what does this establish?In the proof of √3, what kind of step is it to write directly that v is divisible by 3 from u² = 3v²?A student claims that \(3\sqrt{2}\) is rational because 3 is a rational number. What is the correct evaluation of this claim?While assuming (\sqrt{3}) rational, which form is most suitable for proof?If
\(\sqrt{3}=\frac{p}{q}\) is assumed in lowest terms and
\(p^2=3q^2\) is obtained, what is the valid basis for concluding that
\(3\mid p\)?Suppose \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. On squaring, we get \(a^2=3b^2\). Which conclusion is justified at this stage?If assuming \(\sqrt{3}=\frac{p}{q}\) in lowest terms leads to \(p^2=3q^2\), which conclusion proves a contradiction in this assumption?In the standard proof, assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. Which number-theoretic fact is needed to obtain a contradiction from \(p^2=3q^2\)?Which statement is sufficient to reject the rational assumption for √3?Which idea about the exponents of prime factors explains the irrationality of √2?What is the deeper prime-factor reason in the proof of (\sqrt{3})?