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
Why is divisibility reasoning necessary instead of approximate decimal in the proof of (\sqrt{3})?ExpertLevel 17What is the highest-level description of the similarity between the proofs of √2 and √3?ExpertLevel 17A student claims that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. From \(p^2=3q^2\), which conclusion about \(p\) is necessary?ExpertLevel 17A student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. On squaring, the student gets \(p^2=2q^2\). Which of the following is the valid next inference?ExpertLevel 17If \(p/q\) is in lowest terms and \(p^2=3q^2\), which conclusion establishes the contradiction in the proof that \(\sqrt{3}\) is irrational?ExpertLevel 17Which of the following statements is correctly used in the proof that
sqrt(3) is irrational in number systems?ExpertLevel 18A student assumes that \(3\sqrt{2}\) is a rational number. Which argument correctly shows a contradiction in this assumption?ExpertLevel 18If
ext{\(\sqrt{3}\)} is assumed to be
ext{\(\frac{p}{q}\)}, where
ext{\(p\)} and
ext{\(q\)} are coprime, which conclusion produces the contradiction in a proof by contradiction?ExpertLevel 18In the proof by contradiction for the irrationality of sqrt(3), before assuming sqrt(3) = p/q, which condition is essential for the fraction p/q?ExpertLevel 18In the contradiction proof of the irrationality of \(\sqrt{3}\), we assume \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime. If both \(p\) and \(q\) are finally found to be divisible by 3, which conclusion is correct?ExpertLevel 18A student claims that if the square of an integer is divisible by 2, then the integer itself is divisible by 2. Which argument correctly supports this claim?ExpertLevel 18In the contradiction proof of the irrationality of \(\sqrt{3}\), what does writing \(\frac{p}{q}\) in lowest terms mean?ExpertLevel 18A student assumes \(\sqrt{2}=p/q\), where \(p\) and \(q\) are coprime integers, to prove that \(\sqrt{2}\) is irrational. From \(p^2=2q^2\), the student writes \(p=2m\) and obtains \(q^2=2m^2\). The student says that since \(p\) and \(q\) are coprime, \(q\) must be odd. What is the correct correction to this statement?ExpertLevel 18Which option identifies an invalid shortcut in the proof of √2?HardLevel 18If \(\sqrt{3}=\frac{p}{q}\) is assumed to be in lowest terms and \(p^2=3q^2\) is obtained, which conclusion establishes the contradiction in this assumption?ExpertLevel 18If
\(\sqrt{3}=\frac{p}{q}\) is assumed to be in lowest terms and
\(p^2=3q^2\) is obtained, which conclusion about
\(p\) and
\(q\) contradicts this assumption?ExpertLevel 18If (h) is not divisible by (3) and (h^2=3k^2), what inconsistency appears?ExpertLevel 18Which of the following numbers can be proved irrational using its prime factorisation?ExpertLevel 18Suppose \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion follows from this assumption and contradicts the fraction being in lowest terms?ExpertLevel 18In proving the irrationality of
ext{\(\sqrt{3}\)}
, Ravi assumes that
ext{\(\sqrt{3}=p/q\)}
, where
ext{\(p\)}
and
ext{\(q\)}
are coprime. From
ext{\(p^2=3q^2\)}
, he says that only
ext{\(p\)}
is divisible by 3 and nothing can be concluded about
ext{\(q\)}
. Which fact corrects his error?ExpertLevel 18