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 the product of two irrational numbers is always irrational. Which of the following examples disproves the claim?ExpertLevel 13What is the rationalised form of (\frac{5}{\sqrt{17}+\sqrt{8}})?ExpertLevel 13If (y=\sqrt{14}-\sqrt{7}), which is the simplified value of (y^2)?ExpertLevel 13If a rectangle has length √18 + √2 and breadth √18 − √2, what will be its area?MediumLevel 13If u = 3√2 + √98 − √50, what is the value of u/√2?MediumLevel 13What is the value of (\frac{\sqrt{45}+\sqrt{80}}{\sqrt{5}})?ExpertLevel 13If \(x\) is an irrational number, which of the following expressions is not necessarily irrational?ExpertLevel 13Suppose \(x\) is an irrational number and \(q\) is a non-zero rational number. Which statement about \(qx\) is always true?ExpertLevel 13What is the square of \(\sqrt{12+\sqrt{35}}\) equal to?ExpertLevel 13If \(r\) is an irrational number and \(q\) is a rational number, which of the following statements is always true?ExpertLevel 13What is the rationalised form of (\frac{7}{\sqrt{19}-\sqrt{12}})?ExpertLevel 13If (P=\sqrt{363}-\sqrt{147}+\sqrt{75}), what is (P) equal to?ExpertLevel 13If (t=\sqrt{19}+3), what is the value of (t+\frac{10}{t})?ExpertLevel 13What is the value of ((\sqrt{32}+\sqrt{50})^2)?ExpertLevel 13Riya states, “The product of two irrational numbers is always irrational.” Which of the following examples disproves her statement?ExpertLevel 13What is the value of (\frac{1}{\sqrt{12}+\sqrt{5}}+\frac{1}{\sqrt{12}-\sqrt{5}})?ExpertLevel 13If (a=\sqrt{18}+\sqrt{8}) and (b=\sqrt{18}-\sqrt{8}), what is the value of (a^2-b^2)?ExpertLevel 13What is the simplified value of (\frac{\sqrt{243}-\sqrt{108}}{\sqrt{3}})?ExpertLevel 13A student claims that \(3+\sqrt{2}\) is a rational number. Which argument proves the student's claim wrong?ExpertLevel 14Which is the simplified form of (\frac{\sqrt{7}+\sqrt{2}}{\sqrt{7}-\sqrt{2}})?ExpertLevel 14