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
Reema claims that for any rational number \(r\), \(r+\sqrt{7}\) will be irrational. Which of the following arguments correctly evaluates her claim?ExpertLevel 14What is the simplified form of (\sqrt{242}-\sqrt{128}+\sqrt{72})?ExpertLevel 14Rina says in class that the sum of two irrational numbers is always irrational. Which of the following examples disproves her statement?ExpertLevel 14Let \(r\) be a rational number. Which statement about \(r+\sqrt{2}\) is always true?ExpertLevel 14A student claims that \(5+\sqrt{7}\) is rational because it is the sum of two real numbers. Which argument proves the claim wrong?ExpertLevel 14If \(r\) is a non-zero rational number and \(x\) is an irrational number, which of the following statements is always true?ExpertLevel 14If (P=\sqrt{363}+\sqrt{147}-\sqrt{75}), what is (P) equal to?ExpertLevel 14What is the rationalised form of (\frac{9}{\sqrt{26}+\sqrt{17}})?ExpertLevel 14If \(p\) is a non-zero rational number and \(q\) is an irrational number, which of the following expressions must always be irrational?ExpertLevel 15Which is the simplified form of (\frac{\sqrt{11}+\sqrt{6}}{\sqrt{11}-\sqrt{6}})?ExpertLevel 15A student says, “The sum of two irrational numbers is always irrational.” Which of the following expressions is a counterexample to this statement?ExpertLevel 15What is the simplified form of (\sqrt{605}-\sqrt{320}+\sqrt{125})?ExpertLevel 15Reena concludes, “If the square of a real number is rational, then the number itself must be rational.” Which of the following examples disproves her conclusion?ExpertLevel 15If (y=\sqrt{20}+\sqrt{45}+\sqrt{80}), what is the value of (y^2)?ExpertLevel 15Which is the rationalised form of (\frac{7}{\sqrt{30}-\sqrt{23}})?ExpertLevel 15If s = √242 + √128, what is the value of s/√2?MediumLevel 15What is the value of (\sqrt{15+\sqrt{26}}\times\sqrt{15+\sqrt{26}})?ExpertLevel 15Which of the following decimal expansions definitely identifies the number as irrational?ExpertLevel 15Which is the simplified form of (\frac{\sqrt{17}+\sqrt{5}}{\sqrt{17}-\sqrt{5}})?ExpertLevel 15If \(x\) is an irrational number, which of the following expressions will always be rational?ExpertLevel 15