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Subjects

Mathematics

Real Numbers

वास्तविक संख्याएँ

Real Numbers is a Class 9 Mathematics topic in the Number Systems chapter. Students learn how rational and irrational numbers together form the real number system, represent them on the number line, and distinguish between their decimal expansions. The topic develops understanding of terminating and non-terminating decimals, recurring and non-recurring forms, and the key properties of real numbers under addition, subtraction, multiplication, and division. It also builds a foundation for working confidently with numbers in algebra and geometry.

Practice questions

What is the simplified form of ( (2+\sqrt{3})^3 )?HardLevel 1What is the simplified form of ( (3-\sqrt{2})^3 )?HardLevel 1Which of the following conditions about the decimal expansion of a real number definitely identifies it as irrational?HardLevel 1If \(15<\sqrt{m}<16\), what is the correct range for \(m\)?HardLevel 1When is ( \sqrt{a}+\sqrt{b}=\sqrt{a+b} ) generally not correct?HardLevel 1What is the value of ( \sqrt{a}\times\sqrt{a} ) where (a\geq0)?HardLevel 1A student claims that the sum of two irrational numbers is always irrational. Which of the following examples disproves the claim?HardLevel 1Which product made from ( \sqrt{2} ), ( \sqrt{3} ), and ( \sqrt{5} ) is rational?HardLevel 1What is the value of 1/(2 + √3) + 1/(2 − √3)?HardLevel 5What is the simplified surd form of ( \sqrt{1125} )?HardLevel 2If \(r\) is a rational number and \(x\) is an irrational number, which of the following will always be irrational?HardLevel 2What is the simplified form of (5\sqrt{20}-2\sqrt{125}+\sqrt{45})?HardLevel 2Choose the correct simplified form of (3\sqrt{75}+4\sqrt{48}-2\sqrt{27}).HardLevel 2A student says that \(0.101001000100001\ldots\) is a rational number because it contains only the digits 0 and 1. Which is the correct evaluation of the student's statement?HardLevel 2What is the expansion of ( (4\sqrt{3}+3\sqrt{2})^2 )?HardLevel 2Let \(x=\sqrt{2}+\sqrt{3}\). Which classification of \(x\) and \(x^2\) is correct?HardLevel 2A student says that the number \(0.101001000100001\ldots\) is rational because the digit 1 occurs repeatedly. The number of zeros before each successive 1 keeps increasing. Which is the correct evaluation of the student's statement?HardLevel 2What is the rationalised form of ( \frac{1}{4\sqrt{3}+3\sqrt{5}} )?HardLevel 2What is the rationalised form of ( \frac{5}{\sqrt{13}-\sqrt{8}} )?HardLevel 2What is the value of ((4+√7)/(4−√7)+(4−√7)/(4+√7))?HardLevel 2