Mathematics
Irrational numbers
अपरिमेय संख्याएँ
In this Class 9 Mathematics topic from the Number Systems chapter, students learn that irrational numbers cannot be written in the form p/q, where p and q are integers and q is not zero. They explore familiar examples such as √2 and π, understand their non-terminating, non-repeating decimal expansions, and distinguish them from rational numbers. The topic also develops skills for representing irrational numbers on the number line and understanding their place within the real number system.
Practice questions
Let \(p\) be a non-zero rational number and \(x\) be an irrational number. Which of the following statements is always true?ExpertLevel 15If (t=\sqrt{31}+5), what is the value of (t+\frac{6}{t})?ExpertLevel 15What is the value of (\frac{\sqrt{300}-\sqrt{108}}{\sqrt{3}})?ExpertLevel 15If \(x\) is an irrational number and \(r\) is a non-zero rational number, what type of number must \(rx\) be?ExpertLevel 15What is the simplified form of (\sqrt{847}-\sqrt{363}+\sqrt{147})?ExpertLevel 15What is the rationalised form of (\frac{8}{\sqrt{27}-\sqrt{11}})?ExpertLevel 15Kavya says, “The product of any two irrational numbers is always irrational.” Which of the following examples proves that her statement is incorrect?ExpertLevel 15What is the square of \(\sqrt{16+\sqrt{45}}\) equal to?ExpertLevel 15If a rectangle has length (\sqrt{30}+\sqrt{11}) and breadth (\sqrt{30}-\sqrt{11}), what will be its area?ExpertLevel 15A student claims that the sum of any two irrational numbers is always irrational. Which of the following examples disproves this claim?ExpertLevel 15If y = √75 + √147, what is the value of y/√3?MediumLevel 15Which is the simplified form of (\frac{\sqrt{10}-\sqrt{6}}{\sqrt{10}+\sqrt{6}})?ExpertLevel 15Riya says, “The square of every irrational number is also irrational.” Which of the following examples proves her statement wrong?ExpertLevel 15What is the value of \(\sqrt{11}\left(\sqrt{539}-\sqrt{275}\right)\)?ExpertLevel 15A student calls the number 0.101001000100001… rational because it contains only the digits 0 and 1. Why is the student’s conclusion incorrect?ExpertLevel 15What is the rationalised form of (\frac{5}{\sqrt{26}+\sqrt{17}})?ExpertLevel 15If p and q are irrational numbers and \(p+q\) is rational, which of the following expressions must be irrational?ExpertLevel 15Which number definitely lies between (\sqrt{26}) and (\sqrt{30})?ExpertLevel 15A student claims that every non-terminating decimal expansion is irrational. Which of the following numbers is a counterexample to the claim?ExpertLevel 15Reena states: “\(\sqrt{12}+\sqrt{27}\) is irrational because the sum of two irrational numbers is always irrational.” What is the correct evaluation of her statement?ExpertLevel 15