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About Rational & Irrational Numbers — Class 9 ICSE

Distinguish between rational and irrational numbers and represent them on the number line. This topic is part of the ICSE Class 9 mathematics syllabus (chapter: Chapter 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 Rational & Irrational Numbers

  • Rational vs Irrational Numbers
  • Proving Irrationality
  • Representing Irrationals on the Number Line
  • Rationalising the Denominator
  • Operations with Surds and Review

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

Rational & Irrational Numbers — solved examples for Class 9 ICSE

Example 1easy

Which of the following is a rational number?
  1. A)sqrt(2)
  2. B)sqrt(3)
  3. C)3/4
  4. D)sqrt(5)

Step-by-step solution

  1. A rational number is any number that can be written as a fraction p/q, where p and q are integers and q is not zero.
  2. 3/4 is already in the form p/q, so it is rational.
  3. sqrt(2), sqrt(3), sqrt(5) are all irrational numbers as they cannot be expressed as fractions.

Answer: 3/4

Example 2medium

Rationalise the denominator: 1/sqrt(5)
  1. A)sqrt(5)/5
  2. B)5/sqrt(5)
  3. C)1/5
  4. D)sqrt(5)/25

Step-by-step solution

  1. To rationalise, multiply numerator and denominator by sqrt(5):
  2. Calculate:
    1/sqrt(5)xsqrt(5)/sqrt(5)=sqrt(5)/51/sqrt(5) x sqrt(5)/sqrt(5) = sqrt(5)/5

Answer: sqrt(5)/5

Example 3hard

Rationalise the denominator: 1/(sqrt(7) - sqrt(5))
  1. A)(sqrt(7) + sqrt(5))/2
  2. B)(sqrt(7) - sqrt(5))/2
  3. C)(sqrt(7) + sqrt(5))/12
  4. D)(sqrt(7) + sqrt(5))/7

Step-by-step solution

  1. Multiply by the conjugate:
  2. Numerator:
    sqrt(7)+sqrt(5)sqrt(7) + sqrt(5)
  3. Denominator:
    (sqrt(7))2−(sqrt(5))2=7−5=2(sqrt(7))^2 - (sqrt(5))^2 = 7 - 5 = 2
  4. Result:
    (sqrt(7)+sqrt(5))/2(sqrt(7) + sqrt(5))/2

Answer: (sqrt(7) + sqrt(5))/2

Practice questions on Rational & Irrational Numbers

  1. Q1.easy

    Which of the following is an irrational number?
    1. A)0.75
    2. B)22/7
    3. C)sqrt(7)
    4. D)0.333...
    Show answer

    Answer: sqrt(7)

    Hint: An irrational number has a non-terminating, non-repeating decimal expansion.

  2. Q2.easy

    The decimal expansion of 1/3 is:
    1. A)0.3
    2. B)0.33
    3. C)0.333... (non-terminating repeating)
    4. D)0.3 (terminating)
    Show answer

    Answer: 0.333... (non-terminating repeating)

    Hint: Divide 1 by 3 using long division.

  3. Q3.easy

    Which of these represents a terminating decimal?
    1. A)1/3
    2. B)5/6
    3. C)7/8
    4. D)2/9
    Show answer

    Answer: 7/8

    Hint: A fraction gives a terminating decimal when its denominator (in lowest terms) has only 2 and 5 as prime factors.

  4. Q4.medium

    Simplify: sqrt(50)
    1. A)5sqrt(2)
    2. B)2sqrt(5)
    3. C)10sqrt(5)
    4. D)25sqrt(2)
    Show answer

    Answer: 5sqrt(2)

    Hint: Factor 50 and look for perfect square factors.

  5. Q5.medium

    Express 0.4777... as a fraction in simplest form.
    1. A)43/90
    2. B)47/99
    3. C)47/90
    4. D)43/99
    Show answer

    Answer: 43/90

    Hint: Let x = 0.4777... Multiply by 10 and by 100 to eliminate the repeating part.

  6. Q6.medium

    Simplify: 3sqrt(2) + 5sqrt(2)
    1. A)8sqrt(2)
    2. B)15sqrt(2)
    3. C)8sqrt(4)
    4. D)15sqrt(4)
    Show answer

    Answer: 8sqrt(2)

    Hint: These are like terms -- add the coefficients.

  7. Q7.hard

    If x = (3 + sqrt(5))/2, find x^2 + 1/x^2.
    1. A)7
    2. B)5
    3. C)9
    4. D)11
    Show answer

    Answer: 7

    Hint: First find x + 1/x, then use (x + 1/x)^2 = x^2 + 2 + 1/x^2.

  8. Q8.hard

    Simplify: (sqrt(5) - 2)/(sqrt(5) + 2) - (sqrt(5) + 2)/(sqrt(5) - 2)
    1. A)-8sqrt(5)
    2. B)8sqrt(5)
    3. C)-4sqrt(5)
    4. D)0
    Show answer

    Answer: -8sqrt(5)

    Hint: Rationalise each fraction, then subtract.

  9. Q9.hard

    If a = 5 + 2sqrt(6) and b = 1/a, find the value of a^2 + b^2.
    1. A)98
    2. B)100
    3. C)96
    4. D)102
    Show answer

    Answer: 98

    Hint: Find b by rationalising 1/a. Then find a + b and use (a+b)^2 = a^2 + 2ab + b^2.

These are 9 of the 60 questions available for Rational & Irrational Numbers. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.