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 a proof of irrationality, if 3 is prime and 3 divides the square p² of an integer p, which conclusion is correct?ExpertLevel 17A student believes that for some non-zero rational number \(q\), \(q\sqrt{3}\) can be rational. Which argument correctly refutes this belief?ExpertLevel 65A student claims that
\(\sqrt{2}+\sqrt{3}\) is a rational number. Which argument correctly identifies the error in this claim?ExpertLevel 17Which assumption is required at the beginning of a proof by contradiction that \(\sqrt{3}\) is irrational?ExpertLevel 65If (\sqrt{3}=\frac{u}{v}) is in lowest form, why is getting (3\mid u) and (3\mid v) a decisive contradiction?ExpertLevel 65Which of the following integers has a rational square root?ExpertLevel 17In the proof of (\sqrt{3}), which statement is a necessary middle step and not the final contradiction?ExpertLevel 65In a proof by contradiction, a student assumes \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime. After obtaining \(p^2=3q^2\), which statement correctly justifies the conclusion \(3\mid p\)?ExpertLevel 17A student claims that if \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, then \(p^2=2q^2\) implies only that \(p\) is even; nothing can be concluded about \(q\). What is the error in the student's reasoning?ExpertLevel 65A student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. During the proof, the student obtains \(p^2=3q^2\). What is the correct conclusion needed to establish a contradiction?ExpertLevel 17Why is writing (p=3q) from (p^2=3q^2) unacceptable in the proof of (\sqrt{3})?ExpertLevel 17A student assumes that \(\sqrt{3}\) is rational and writes it as \(\frac{p}{q}\) in lowest terms, where \(p\) and \(q\) are coprime. Which conclusion correctly follows from \(p^2=3q^2\)?ExpertLevel 17Suppose \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion creates the contradiction in the proof by contradiction?ExpertLevel 17What is the idea of prime factors of a perfect square in the proof of √2?ExpertLevel 17What is the idea of prime factors of a perfect square in the proof of √3?ExpertLevel 17In the proof of √2, writing y ≠ 0 is necessary, but why does it not give the final contradiction?ExpertLevel 17In the proof of (\sqrt{3}), what is the role of (n\neq0), and where does the final contradiction come from?ExpertLevel 17If a contradiction is obtained after assuming that √2 is rational, according to logic which conclusion is correct?ExpertLevel 17A student assumes \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime, to prove that \(\sqrt{3}\) is irrational. After obtaining \(3q^2=p^2\), the student says, “\(p\) is divisible by 3, so a contradiction has been reached.” Which statement correctly identifies the gap in the argument?ExpertLevel 65Why is the coprime fraction argument better than decimal approximation in the proof of (\sqrt{2})?ExpertLevel 17