Class IX Mathematics Number System Study Material

Chapter 1: Number System

Learning Objectives

  • Understand different types of numbers and their representations
  • Master operations with real numbers and their properties
  • Learn to work with exponents and roots
  • Apply concepts to solve practical problems

1. Representation of Numbers

a) Natural Numbers, Integers, and Rational Numbers on the Number Line

Natural numbers (ℕ): The counting numbers {1, 2, 3, 4, …} used for discrete quantities.
Whole numbers (𝕎): Natural numbers including zero {0, 1, 2, 3, …}.
Integers (ℤ): Whole numbers and their negatives {…, -3, -2, -1, 0, 1, 2, 3, …}.
Rational numbers (ℚ): Numbers expressible as p/q where p, q are integers and q ≠ 0.

Example 1: Represent -3/2 on the number line.
Solution:
  1. Divide the unit length between -2 and -1 into two equal parts
  2. The point midway between -2 and -1 represents -3/2

Practice Exercises

  1. Represent 5/3 on the number line.
  2. Find three rational numbers between 1/4 and 1/2.
  3. Which of these are integers: -3, 0, 5/2, √4, π?

b) Rational Numbers as Terminating and Non-Terminating Recurring Decimals

A rational number’s decimal representation is either:

  • Terminating: Finite decimal digits (denominator has only 2 and/or 5 as prime factors)
  • Non-terminating recurring: Infinite digits with repeating pattern (denominator has other prime factors)
Example 2: Convert 13/8 to decimal form.
Solution: 13 ÷ 8 = 1.625 (terminating)

Example 3: Convert 5/6 to decimal form.
Solution: 5 ÷ 6 = 0.8333… = 0.83 (repeating)

Practice Exercises

  1. Convert 7/16 to decimal form.
  2. Convert 2/11 to decimal form and identify the repeating pattern.
  3. Which of these fractions will have terminating decimals: 3/8, 7/12, 11/25?

2. Irrational Numbers

a) Examples such as √2, √3 and Their Representation

Irrational numbers cannot be expressed as simple fractions. Their decimal expansions are non-terminating and non-repeating.

Example 4: Prove that √3 is irrational.
Solution:
  1. Assume √3 is rational, so √3 = p/q (p, q coprime)
  2. Then 3 = p²/q² ⇒ p² = 3q² ⇒ p² is divisible by 3 ⇒ p is divisible by 3
  3. Let p = 3k ⇒ (3k)² = 3q² ⇒ 9k² = 3q² ⇒ q² = 3k² ⇒ q is divisible by 3
  4. Contradiction: p and q have common factor 3. Thus √3 is irrational.

Practice Exercises

  1. Represent √5 on the number line using geometric construction.
  2. Prove that 1/√2 is irrational.
  3. Identify which are irrational: √9, √10, 0.10110111011110…, π

b) Real Numbers and the Number Line

The real number system (ℝ) includes all rational and irrational numbers. Every real number corresponds to exactly one point on the number line.

Example 5: Find two irrational numbers between 2 and 3.
Solution:
  1. 2.10100100010000… (pattern increases zeros between 1s)
  2. √5 ≈ 2.236 (since 4 < 5 < 9 ⇒ 2 < √5 < 3)

Practice Exercises

  1. Find three irrational numbers between 1/2 and 2/3.
  2. Represent √2 + 1 on the number line.
  3. True or False: Between any two real numbers, there are infinitely many rational numbers.

3. Operations and Properties

a) Operations on Real Numbers

Real numbers follow these fundamental properties:

Property Addition Multiplication
Closure a + b ∈ ℝ a × b ∈ ℝ
Commutative a + b = b + a a × b = b × a
Associative (a + b) + c = a + (b + c) (a × b) × c = a × (b × c)
Distributive a × (b + c) = a × b + a × c
Example 6: Simplify (√3 + √2)(√3 – √2)
Solution: Using (a + b)(a – b) = a² – b²
(√3)² – (√2)² = 3 – 2 = 1

Practice Exercises

  1. Simplify (2 + √5)(3 – √5)
  2. Verify the associative property for addition with a = √2, b = π, c = √3
  3. Find the additive inverse of 3 – √7

b) Laws of Exponents for Real Numbers

For real numbers a, b and rational numbers m, n:

  1. am × an = am+n
  2. am ÷ an = am-n (a ≠ 0)
  3. (am)n = amn
  4. a-n = 1/an (a ≠ 0)
  5. a0 = 1 (a ≠ 0)
  6. (ab)n = anbn
  7. (a/b)n = an/bn (b ≠ 0)
Example 7: Simplify (53 × 5-2) ÷ (50 × 54)
Solution:
Numerator: 53+(-2) = 51
Denominator: 50+4 = 54
Result: 51-4 = 5-3 = 1/125

Practice Exercises

  1. Simplify (23)2 × 34 ÷ (62 × 32)
  2. Evaluate (1/3)-2 + (1/2)-3 + (1/4)-1
  3. If 2x = 16, find the value of x

c) nth Root of Real Numbers

The nth root of a is written as n√a or a1/n, where n is the index and a is the radicand.

Example 8: Simplify ∛(64) × √(25)
Solution:
∛64 = 4 (since 4³ = 64)
√25 = 5 (since 5² = 25)
Result: 4 × 5 = 20

Practice Exercises

  1. Evaluate ∜(81) + 5√(32)
  2. Simplify (√12 + √3)²
  3. Rationalize the denominator: 1/(√5 + √2)

Chapter Summary

  • Real numbers include both rational and irrational numbers
  • Every real number has a unique position on the number line
  • Real numbers follow fundamental operation properties
  • Exponent rules simplify complex expressions
  • Roots can be represented as fractional exponents

Self-Assessment Test

Part A: Multiple Choice (1 mark each)

  1. Which of these is irrational?
    1. √16
    2. √(9/4)
    3. √5
    4. 0.16
  2. The decimal expansion of 3/8 is:
    1. Terminating
    2. Non-terminating repeating
    3. Non-terminating non-repeating
    4. None of these

Part B: Short Answer (2 marks each)

  1. Find two irrational numbers between 0.5 and 0.6
  2. Simplify: (32 × 43) ÷ (62)

Part C: Long Answer (4 marks each)

  1. Prove that √5 is irrational
  2. Represent √7 on the number line geometrically

Additional Resources

  • Video Tutorials: Number System Basics
  • Interactive Number Line Tool
  • Practice Worksheets with Answer Keys
  • Historical Note: Discovery of Irrational Numbers

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