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
Where \(q\) is any rational number, which of the following expressions is guaranteed to be irrational?ExpertLevel 14If \(x\) is an irrational number and \(r\) is a rational number, which of the following expressions will be irrational in every case?ExpertLevel 14Let \(r\) be an irrational number and \(q\) be a non-zero rational number. Which of the following statements is always true?ExpertLevel 14Which is the rationalised form of (\frac{5}{\sqrt{14}-3})?ExpertLevel 14If s = √108 + √300, what is the value of s/√3?MediumLevel 14What is the value of (\sqrt{8+\sqrt{11}}\times\sqrt{8+\sqrt{11}})?ExpertLevel 14Nidhi says that the number \(0.101001000100001\ldots\) is rational because its decimal expansion contains only 0 and 1. Which statement correctly explains Nidhi's error?ExpertLevel 14Which is the simplified form of (\frac{\sqrt{13}+\sqrt{3}}{\sqrt{13}-\sqrt{3}})?ExpertLevel 14Which of the following decimal expansions identifies a real number as irrational?ExpertLevel 14Which of the following numbers is rational despite containing square-root signs?ExpertLevel 14If (t=\sqrt{23}+4), what is the value of (t+\frac{7}{t})?ExpertLevel 14What is the value of (\frac{\sqrt{98}-\sqrt{50}}{\sqrt{2}})?ExpertLevel 14Let \(x\) be an irrational number and \(q\) be a non-zero rational number. Which of the following statements is always true?ExpertLevel 14What is the simplified form of (\sqrt{539}-\sqrt{275}+\sqrt{99})?ExpertLevel 14What is the rationalised form of (\frac{6}{\sqrt{19}-\sqrt{10}})?ExpertLevel 14A student claims that every non-terminating decimal is irrational. Which of the following examples disproves the claim?ExpertLevel 14What is the square of \(\sqrt{13+\sqrt{42}}\) equal to?ExpertLevel 14If a rectangle has length (\sqrt{22}+\sqrt{7}) and breadth (\sqrt{22}-\sqrt{7}), what will be its area?ExpertLevel 14Let \(x\) be an irrational number. Which of the following statements is always true?ExpertLevel 14If y = √27 + √75, what is the value of y/√3?MediumLevel 14