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
If (t=\sqrt{17}+4), what is the value of (t+\frac{1}{t})?ExpertLevel 13Which of the following statements is necessarily true for every irrational number \(x\)?ExpertLevel 13What is the square of \(\sqrt{10+\sqrt{21}}\) equal to?ExpertLevel 13If a rectangle has length (\sqrt{14}+\sqrt{5}) and breadth (\sqrt{14}-\sqrt{5}), what will be its area?ExpertLevel 13If y = √8 + √18, what is the value of y/√2?MediumLevel 13Which of the following statements correctly identifies the decimal expansion of an irrational number?ExpertLevel 13A student claims, “The sum of two irrational numbers is always irrational.” Which of the following pairs is a counterexample to this claim?ExpertLevel 13Let \(x\) be an irrational number and \(r\) be a rational number. Which of the following expressions will always be irrational?ExpertLevel 13Which number definitely lies between (\sqrt{7}) and (\sqrt{8})?ExpertLevel 13A student says, “The product of two irrational numbers is always irrational.” Which example disproves this statement?ExpertLevel 13Let \(a\) and \(b\) be rational numbers, \(b\ne0\), and let \(m\) be a natural number that is not a perfect square. What type of number is \(a+b\sqrt{m}\)?ExpertLevel 13What is the simplified form of (\sqrt{300}-\sqrt{108}+\sqrt{75})?ExpertLevel 13Let \(x\) be an irrational number and \(r\) be a non-zero rational number. Which of the following expressions must be irrational?ExpertLevel 13What is the value of (\frac{\sqrt{72}+\sqrt{128}}{\sqrt{2}})?ExpertLevel 13If (r=\sqrt{7}+\sqrt{2}) and (s=\sqrt{7}-\sqrt{2}), what is the value of (r^2-s^2)?ExpertLevel 13If (P=\sqrt{242}+\sqrt{50}-\sqrt{98}), what is (P) equal to?ExpertLevel 13What is the value of ((\sqrt{28}+\sqrt{63})^2)?ExpertLevel 13Let \(x\) be an irrational number and \(r\) be a rational number. Which of the following expressions is always irrational?ExpertLevel 13If \(x\) is an irrational number, which of the following expressions will be irrational in every case?ExpertLevel 13What is the value of (√18 + √50)/√8?MediumLevel 13