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Real Numbers Quiz - CBSE Class 9 Mathematics

Free practice quiz: Real Numbers (CBSE Class 9 Mathematics). 17 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This CBSE Class 9 Mathematics practice quiz has 17 multiple-choice questions on Real Numbers, and 17 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

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Rational and Irrational Numbers

  • A rational number can be written as p/q, where p and q are integers and q is not zero.
  • Rational decimals either terminate or repeat forever. Irrational decimals never end and never repeat, like π or √11.
  • Rationals and irrationals together form the real numbers. Square roots of negative numbers are not real.
Rational form
\[\frac{p}{q},\; p,q \in \mathbb{Z},\; q \neq 0\]
One repeating digit d
\[0.\overline{d} = \frac{d}{9}\]
Real numbers
\[\mathbb{R} = \mathbb{Q} \cup \text{Irrationals}\]

Common mistake: Thinking every square root is irrational. The square root of a perfect square, like √36, is rational.

Mixing Rational and Irrational Numbers

  • Rational plus irrational is always irrational. A non-zero rational times an irrational is also irrational.
  • Two irrationals can give a rational or an irrational answer, so check each case.
  • To prove √p is irrational for a prime p, assume √p = a/b in lowest terms and reach a contradiction.
Sum rule (r rational, s irrational)
\[r + s \text{ is irrational}\]
Product rule (r \neq 0)
\[r \times s \text{ is irrational}\]
Proof step
\[\sqrt{p}=\frac{a}{b} \Rightarrow p\,b^{2}=a^{2}\]

Common mistake: Assuming the product of two irrationals is always irrational. For example, √11 × √11 = 11, which is rational.

Working With Square Roots

  • Simplify a root by taking out perfect-square factors. Only like surds, with the same root, can be added.
  • If x² = k with k > 0, then x has two values, +√k and −√k.
Product of roots
\[\sqrt{a}\,\sqrt{b} = \sqrt{ab}\]
Taking out a square
\[\sqrt{a^{2}b} = a\sqrt{b},\; a \ge 0\]
Adding like surds
\[m\sqrt{a} + n\sqrt{a} = (m+n)\sqrt{a}\]
Square of a sum
\[(\sqrt{a}+\sqrt{b})^{2} = a + b + 2\sqrt{ab}\]

Common mistake: Writing √(a + b) = √a + √b. Roots split over multiplication, not over addition.

Exponent Laws and Roots

  • A negative power means the reciprocal. A fractional power means a root.
  • A cube root of a negative number is real and negative, because a negative cubed stays negative.
Negative exponent
\[a^{-n} = \frac{1}{a^{n}}\]
Unit fraction exponent
\[a^{\frac{1}{n}} = \sqrt[n]{a}\]
General fraction exponent
\[a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^{m}\]
Cube root of negative
\[\sqrt[3]{-a} = -\sqrt[3]{a}\]

Common mistake: Thinking a negative exponent makes the answer negative. In fact, a^(-n) is the reciprocal 1/aⁿ, which is still positive when a > 0.

Exam tips

Beta: made with AI and checked by a second AI. A teacher has not reviewed it yet.

Sample questions

  1. What is the least integer greater than the square root of 15?

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    Answer: B. 4

    The square root of 15 is approximately 3.87, so the least integer greater than this is 4.

  2. The number √81 can be classified as:

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    Answer: A. Rational

    √81 = 9, which is a whole number and thus a rational number.

  3. What is the product of the square roots of 16 and 25?

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    Answer: A. 20

    The square roots of 16 and 25 are 4 and 5, respectively. Their product is 4 x 5 = 20.

  4. Simplify the expression √50 + √32.

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    Answer: D. 9√2

    √50 = 5√2 and √32 = 4√2. Therefore, √50 + √32 = 5√2 + 4√2 = 9√2.

  5. Which number is not a real number?

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    Answer: C. i

    i represents the imaginary unit, which is not a real number.

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More questions in this quiz

Answers and explanations are in the full quiz.

  1. Which of the following numbers is irrational?
  2. What is the decimal representation of 1/3?
  3. What is the square root of 49?
  4. Is the sum of a rational and irrational number rational or irrational?
  5. If x is a real number such that x^2 = 4, what are the possible values of x?
  6. Convert the repeating decimal 0.666... to a fraction.
  7. Which of the following is an example of an irrational number?
  8. If a and b are rational numbers and √2 is an irrational number, is a√2 + b rational or irrational?
  9. What is (√3 + √5)^2?
  10. What is the cube root of -125?
  11. Prove that √7 is irrational.
  12. What is the value of (81)^(-1/2)?

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