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
A student claims that \(5-\sqrt{7}\) is rational because 5 is rational and \(\sqrt{7}\) is a number. What is the error in the student's reasoning?ExpertLevel 15What is the simplified form of (\sqrt{588}-\sqrt{300}+\sqrt{192})?ExpertLevel 15What is the value of (\frac{1}{\sqrt{10}+\sqrt{6}}+\frac{1}{\sqrt{10}-\sqrt{6}})?ExpertLevel 15Let \(a\) be a rational number and \(b\) be an irrational number. Which of the following expressions will always be irrational?ExpertLevel 15What is the value of (\frac{\sqrt{392}+\sqrt{200}}{\sqrt{2}})?ExpertLevel 15If (r=\sqrt{17}+\sqrt{11}) and (s=\sqrt{17}-\sqrt{11}), what is the value of (r^2-s^2)?ExpertLevel 15A student claims, “The square of every irrational number is irrational.” Which example disproves this claim?ExpertLevel 15If (P=\sqrt{500}+\sqrt{180}-\sqrt{320}), what is (P) equal to?ExpertLevel 15What is the value of ((\sqrt{125}+\sqrt{320})^2)?ExpertLevel 15In the number \(0.101001000100001\ldots\), the number of zeros between two successive 1s increases by one each time. A student says that it is rational because it contains only the digits 0 and 1. Which is the correct evaluation of the student's argument?ExpertLevel 15A student says, “The product of two irrational numbers is always irrational.” Which of the following examples proves that the statement is wrong?ExpertLevel 15Which is the simplified form of (\frac{\sqrt{31}+\sqrt{19}}{\sqrt{31}-\sqrt{19}})?ExpertLevel 15Which of the following examples shows that the product of two irrational numbers can be rational?ExpertLevel 15What is the simplified form of (\sqrt{726}-\sqrt{486}+\sqrt{150})?ExpertLevel 15Which is the rationalised form of (\frac{12}{\sqrt{43}-\sqrt{31}})?ExpertLevel 15Let r be a non-zero rational number and x be an irrational number. Which of the following statements is always true?ExpertLevel 15If a rectangle has length (\sqrt{41}+\sqrt{17}) and breadth (\sqrt{41}-\sqrt{17}), what will be its area?ExpertLevel 15In which option is the quotient of two irrational numbers rational?ExpertLevel 15A student claims that the sum of two irrational numbers is always irrational. Which of the following calculations gives a counterexample to the claim?ExpertLevel 19Which option is correct about \(\sqrt{3}\)?ExpertLevel 19