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
For a square with an area of 2 cm², Aarav says, “Since the area is a rational number, the perimeter of the square must also be rational.” Which is the correct evaluation of Aarav’s statement?ExpertLevel 17Which statement is correct about the decimal expansion of \(\sqrt{3}\)?ExpertLevel 17In the proof by contradiction for the irrationality of \(\sqrt{3}\), why are \(p\) and \(q\) chosen to be coprime when assuming \(\sqrt{3}=\frac{p}{q}\)?ExpertLevel 17A student claims that \(1+\sqrt{2}\) is rational because 1 is a rational number. Which argument correctly refutes the claim?ExpertLevel 65Assume that a/b is in lowest terms and obtain a² = 3b². Which conclusion about a is essential in the proof that √3 is irrational?ExpertLevel 17Why is only (p) being divisible by (3) not the final contradiction in the proof of (\sqrt{3})?ExpertLevel 65In the contradiction proof that \(\sqrt{3}\) is irrational, assume \(\sqrt{3}=\frac{p}{q}\), where \(\gcd(p,q)=1\). This gives \(p^2=3q^2\). Which principle is correctly applied from \(3\mid p^2\)?ExpertLevel 17For a positive integer \(n\), which condition identifies when \(\sqrt{n}\) is irrational?ExpertLevel 65While proving the irrationality of \(\sqrt{3}\), assume that \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime. Which conclusion obtained from \(p^2=3q^2\) contradicts this assumption?ExpertLevel 17A student assumes \(\sqrt{2}=\frac{m}{n}\), where \(m,n\) are integers, to prove that \(\sqrt{2}\) is irrational. Later, the student finds that both \(m\) and \(n\) are even and calls this a contradiction. Which essential condition is missing from the argument?ExpertLevel 17If someone writes (a=2b) from (a^2=2b^2), what is the correct correction?ExpertLevel 17If someone writes (p=3q) from (p^2=3q^2), what is the correct correction?ExpertLevel 17In the proof of √2, the idea of infinite descent is connected with which situation?ExpertLevel 65How can the proof of (\sqrt{3}) be understood in the language of infinite descent?ExpertLevel 17Which statement about exponents of prime factors in perfect squares connects to the proof of √2?ExpertLevel 65Which statement about exponents of prime factors in perfect squares is useful in the proof of √3?ExpertLevel 65In a proof by contradiction for the irrationality of \(\sqrt{3}\), which conclusion is necessary to proceed after obtaining \(a^2=3b^2\)?ExpertLevel 65If (p) is not divisible by (3) in the proof of (\sqrt{3}), what contradiction follows from (p^2=3q^2)?ExpertLevel 65Rima says that \(\sqrt{3}\) is irrational because its decimal expansion continues infinitely. What is the main flaw in her argument?ExpertLevel 65What is the main purpose of assuming \(\sqrt{2}=\frac{p}{q}\) in lowest terms in the standard proof by contradiction?ExpertLevel 65