The World of Numbers – Class 9 Mathematics notes, important questions and MCQs

Chapter overview

Real numbers include rational numbers (p/q) and irrational numbers like √2 and π. Every real number is a point on the number line, and decimal expansions tell them apart.

Types of numbers

• Natural N = 1, 2, 3, …; whole W = 0, 1, 2, …; integers Z = …, −2, −1, 0, 1, 2, ….
• Rational: p/q with integers p, q and q ≠ 0.
• Irrational: cannot be written as p/q (√2, √3, π).

Decimal expansions

• Rational numbers have terminating (1/4 = 0.25) or non-terminating repeating (1/3 = 0.333…) decimals.
• Irrational numbers have non-terminating, non-repeating decimals.
• 0.333… = 1/3 and 0.999… = 1.

Laws of exponents and surds

• aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1.
• a^(1/n) = ⁿ√a; a^(m/n) = (ⁿ√a)ᵐ.
• Rationalise 1/√a by multiplying with √a/√a; 1/(√a + √b) by (√a − √b).

Key terms

Rational number

a number of the form p/q, q ≠ 0

Irrational number

a number with a non-terminating, non-repeating decimal

Rationalising

removing a root from the denominator

Formulas

• (√a + √b)(√a − √b) = a − b
• aᵐ × aⁿ = aᵐ⁺ⁿ

Important questions with answers

Quick revision

• Every integer is rational.
• Repeating decimal → rational.
• Real numbers = rational + irrational.

Exam tips

• Between any two rational numbers there are infinitely many rational numbers.
• π = 22/7 is only an approximation; π is irrational.

Practice MCQs

1. Which is irrational?

   1. 0.75
   2. √9
   3. √3
   4. 2/5
   Show answer

   Answer: (c) √3. 3 is not a perfect square.

2. 1/8 as a decimal is

   1. 0.125
   2. 0.8
   3. 0.0125
   4. 0.18
   Show answer

   Answer: (a) 0.125. 1 ÷ 8 = 0.125.

3. Every integer is a

   1. natural number
   2. rational number
   3. whole number
   4. irrational number
   Show answer

   Answer: (b) rational number. n = n/1.

4. 0.666… written as p/q is

   1. 1/6
   2. 3/5
   3. 6/10
   4. 2/3
   Show answer

   Answer: (d) 2/3. x = 0.666…, 10x − x = 6 ⇒ x = 2/3.

5. The value of 8^(2/3) is

   1. 16
   2. 4
   3. 2
   4. 6
   Show answer

   Answer: (b) 4. (∛8)² = 2² = 4.

6. Rationalising 1/√5 gives

   1. √5
   2. 5/√5
   3. √5/5
   4. 1/5
   Show answer

   Answer: (c) √5/5. Multiply by √5/√5.

7. 1/(√3 + √2) equals

   1. √3 − √2
   2. √3 + √2
   3. 1
   4. (√3 − √2)/5
   Show answer

   Answer: (a) √3 − √2. Multiply by (√3 − √2): denominator 3 − 2 = 1.

8. 0.4777… (7 repeating) as a fraction is

   1. 47/99
   2. 47/90
   3. 43/90
   4. 43/99
   Show answer

   Answer: (c) 43/90. 100x − 10x = 47.7… − 4.7… = 43 ⇒ x = 43/90.