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About Real Number System — Class 9 IB

Classify real numbers, work with surds, and understand the completeness of the number line. This topic is part of the IB Class 9 mathematics syllabus (chapter: Unit 1). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Real Number System

  • Classifying Real Numbers
  • Rational Numbers: Forms and Conversions
  • Irrational Numbers and Their Properties
  • Operations with Radicals
  • Properties of Real Numbers

Interactive lesson · about 20 minutes · checkpoint question after every unit

Real Number System — solved examples for Class 9 IB

Example 1easy

Which of the following is an irrational number?
  1. A)3/4
  2. B)0.5
  3. C)√2
  4. D)7

Step-by-step solution

  1. A rational number can be written as p/q where p and q are integers.
  2. √2 = 1.41421356... is a non-terminating, non-repeating decimal, so it is irrational.
    √2∉Q√2 ∉ ℚ

Answer: √2

Example 2medium

Simplify √72 completely.
  1. A)6√2
  2. B)4√3
  3. C)3√8
  4. D)2√18

Step-by-step solution

  1. 72 = 36 × 2
    √72=√(36×2)√72 = √(36 × 2)
  2. √36 = 6, so √72 = 6√2.
    √72=6√2√72 = 6√2

Answer: 6√2

Example 3hard

Rationalize the denominator: 1/(√5 − √3).
  1. A)(√5 + √3)/2
  2. B)(√5 − √3)/2
  3. C)(√5 + √3)/8
  4. D)(√5 + √3)

Step-by-step solution

  1. Multiply by conjugate: 1/(√5−√3) × (√5+√3)/(√5+√3)
    (√5+√3)/((√5)2−(√3)2)(√5+√3)/((√5)²−(√3)²)
  2. (√5+√3)/(5−3) = (√5+√3)/2.
    (√5+√3)/2(√5+√3)/2

Answer: (√5 + √3)/2

Practice questions on Real Number System

  1. Q1.easy

    Which set does the number -3 belong to?
    1. A)Natural numbers only
    2. B)Whole numbers only
    3. C)Integers
    4. D)Irrational numbers
    Show answer

    Answer: Integers

    Hint: Think about which number sets include negative whole numbers.

  2. Q2.easy

    Simplify √18.
    1. A)3√2
    2. B)2√3
    3. C)6√3
    4. D)9√2
    Show answer

    Answer: 3√2

    Hint: Factor 18 into a perfect square times another number.

  3. Q3.easy

    Which of the following is a rational number?
    1. A)π
    2. B)√5
    3. C)0.333...
    4. D)√7
    Show answer

    Answer: 0.333...

    Hint: A repeating decimal can always be expressed as a fraction.

  4. Q4.medium

    Rationalize the denominator: 1/√3.
    1. A)√3/3
    2. B)3/√3
    3. C)1/3
    4. D)√3
    Show answer

    Answer: √3/3

    Hint: Multiply both numerator and denominator by √3.

  5. Q5.medium

    Simplify 3√8 + 2√2.
    1. A)8√2
    2. B)5√10
    3. C)5√8
    4. D)6√2 + 2√2
    Show answer

    Answer: 8√2

    Hint: First simplify √8, then combine like surds.

  6. Q6.medium

    Express 0.̄36̄ (0.363636...) as a fraction.
    1. A)4/11
    2. B)36/100
    3. C)36/99
    4. D)9/25
    Show answer

    Answer: 4/11

    Hint: Let x = 0.363636... and multiply by 100 to shift the repeating block.

  7. Q7.hard

    Prove that √2 is irrational. Which proof technique is typically used?
    1. A)Direct proof
    2. B)Proof by contradiction
    3. C)Proof by induction
    4. D)Proof by exhaustion
    Show answer

    Answer: Proof by contradiction

    Hint: Assume √2 is rational and show this leads to a logical contradiction.

  8. Q8.hard

    Simplify: (√6 + √2)(√6 − √2).
    1. A)4
    2. B)8
    3. C)√12
    4. D)6 − 2
    Show answer

    Answer: 4

    Hint: This is a difference of squares: (a+b)(a−b) = a² − b².

  9. Q9.hard

    If x = (3 + 2√2) and y = (3 − 2√2), what is xy?
    1. A)1
    2. B)5
    3. C)9 − 8
    4. D)17
    Show answer

    Answer: 1

    Hint: Use the difference of squares: (a+b)(a−b) = a² − b².

These are 9 of the 60 questions available for Real Number System. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.