Class 8 Mathematics Chapter 1: Rational Numbers 

These notes of Rational Numbers  cover the entire NCERT Class 8 Chapter 1 (Rational Numbers) syllabus and are sufficient for exams, homework, and revision. Practice all examples and exercise questions from NCERT for complete mastery!

1. Introduction to Rational Numbers

• Numbers that can be written in the form p/q where p and q are integers and q ≠ 0 are called Rational Numbers.
• Denominator can never be zero because division by zero is undefined.
• Symbol for rational numbers is ℚ.
• Examples: 3/5, -7/2, 0/1 (= 0), 8/1 (= 8), -9/-4 (= 9/4), etc.
• Every integer is a rational number because any integer ‘a’ can be written as a/1.
• Decimal numbers that are terminating or non-terminating repeating are also rational numbers.

2. Properties of Rational Numbers

A. Closure Property

• Addition: a, b ∈ ℚ ⇒ a + b ∈ ℚ (closed)
• Subtraction: a, b ∈ ℚ ⇒ a – b ∈ ℚ (closed)
• Multiplication: a, b ∈ ℚ ⇒ a × b ∈ ℚ (closed)
• Division: a, b ∈ ℚ, b ≠ 0 ⇒ a ÷ b ∈ ℚ (closed, except division by 0)

B. Commutative Property

• Addition and multiplication are commutative:
• a + b = b + a
• a × b = b × a Subtraction and division are NOT commutative.

C. Associative Property

• Addition and multiplication are associative:
  • (a + b) + c = a + (b + c)
  • (a × b) × c = a × (b × c)
• Subtraction and division are NOT associative.

D. Distributive Property

Multiplication is distributive over addition and subtraction:

• a × (b + c) = a × b + a × c
• a × (b – c) = a × b – a × c

E. Identity Property

• Additive identity: 0 (a + 0 = a = 0 + a)
• Multiplicative identity: 1 (a × 1 = a = 1 × a)

F. Additive Inverse

For any rational number a, the additive inverse is –a (a + (–a) = 0)

G. Multiplicative Inverse (Reciprocal)

For any non-zero rational number a, reciprocal is 1/a (a × 1/a = 1)

3. Representation of Rational Numbers on Number Line

• Positive rational numbers lie to the right of 0.
• Negative rational numbers lie to the left of 0.
• To represent p/q:
  1. Divide each unit into q equal parts.
  2. Move |p| parts towards right (if positive) or left (if negative).

Example: Represent 5/3 and –7/4 on number line.

4. Rational Numbers Between Two Rational Numbers

There are infinitely many rational numbers between any two rational numbers.

Method 1: Using average

• (a + b)/2 gives one rational number between a and b.
• Keep repeating to get more.

Method 2: Converting to equivalent fractions with same denominator

Example: Rational numbers between 3/5 and 4/5 → 3/5 = 6/10, 4/5 = 8/10 → 7/10 lies between them.

Method 3: Writing with larger denominator

Between 1/2 and 3/4 LCM of 2 and 4 = 4 1/2 = 2/4, 3/4 = 3/4 Multiply numerator and denominator by 10 → 20/40 and 30/40 So numbers: 21/40, 22/40, …, 29/40

5. Standard Form of Rational Number

A rational number p/q is in standard (lowest) form if:

• p and q have no common factor other than 1 (i.e., HCF(p, q) = 1)
• q is always taken positive.

Example: 12/18 → divide by 6 → 2/3 (standard form)

6. Comparison of Rational Numbers

Case 1: Same denominator

The number with greater numerator is greater. Example: 5/7 > 3/7

Case 2: Different denominators

• Cross multiply method: For a/b and c/d, If a×d > b×c → a/b > c/d If a×d < b×c → a/b < c/d
• Or convert both into equivalent fractions with same denominator (LCM of denominators) and then compare numerators.

Negative rational numbers

A negative rational number is always less than a positive one. Among negative numbers, the one with greater absolute value is smaller. Example: –5/2 < –3/2 (because 5/2 > 3/2)

7. Operations on Rational Numbers (Step-by-Step)

Addition

• Same denominator: Add numerators, keep denominator same. Example: 2/7 + 3/7 = 5/7
• Different denominator: Take LCM, convert, then add. Example: 1/6 + 3/4 = (2/12) + (9/12) = 11/12

Subtraction

Similar to addition, but subtract numerators. Example: 5/8 – 3/8 = 1/8 Example: 7/7 – 1/6 = (42 – 7)/42 = 35/42 = 5/6

Multiplication

Multiply numerators together and denominators together. Simplify if possible. Example: (-3/4) × (8/15) = (-24/60) = –2/5

Division

Multiply by reciprocal of second number. Example: (–5/9) ÷ (10/21) = (–5/9) × (21/10) = (–5 × 21)/(9 × 10) = –7/6

8. Important Identities Used

(a + b)² = a² + 2ab + b² (a – b)² = a² – 2ab + b² a² – b² = (a + b)(a – b)

These are useful when simplifying expressions involving rational numbers.

9. Word Problems (Typical Types)

1. Sum of numbers: Find two rational numbers, etc.
2. Fraction of quantity: Ram has 5/2 litres milk. He uses 3/4 litre. How much left?
3. Repeated addition as multiplication: If a car covers 2/3 km in 1 minute, how much in 15 minutes? → 15 × 2/3 = 10 km
4. Division: A rope of 25/2 m is cut into pieces of 5/6 m each. How many pieces? → (25/2) ÷ (5/6) = 15 pieces

10. Summary Table of Properties

Key Points to Remember

• 0 is a rational number (0/1).
• Every natural, whole, integer is rational.
• Rational numbers are dense on number line.
• Negative of negative rational number is positive.
• Reciprocal of negative number is negative.
• Division by zero is undefined.

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Chapter 2: Linear Equations in One Variable

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