CBSE Class 8 Maths – Chapter 1: Rational Numbers

CBSE Class 8 Maths – Chapter 1: Rational Numbers (Notes)

Clear, quick notes with definitions, properties, examples and summary. Perfect for last-minute revision and tests.

Contents

1. What are Rational Numbers?
2. Properties of Rational Numbers
3. Representation on Number Line
4. Standard Form
5. Reciprocal
6. Numbers Between Two Rational Numbers
7. Quick Summary
8. Practice Questions
9. Answer Key

1) What are Rational Numbers?

A rational number is any number that can be written in the form p/q, where p and q are integers and q ≠ 0.

Examples

³₅, −⁷₉, 0 = ⁰₁, 2 = ²₁

Note

Every integer is a rational number (denominator can be 1).

2) Properties of Rational Numbers

Closure

• Addition: a/b + c/d is rational.
• Subtraction: a/b − c/d is rational.
• Multiplication: a/b × c/d is rational.
• Division: (a/b) ÷ (c/d) is rational if c/d ≠ 0.

Commutativity

• Addition: a/b + c/d = c/d + a/b
• Multiplication: a/b × c/d = c/d × a/b

Associativity

• Addition: (x + y) + z = x + (y + z)
• Multiplication: (xy)z = x(yz)

Distributive Law

a/b × ( c/d + e/f ) = (a/b × c/d) + (a/b × e/f)

Identity Elements

• Additive identity: 0 (adding 0 keeps the number same).
• Multiplicative identity: 1 (multiplying by 1 keeps the number same).

3) Representation on Number Line

Rational numbers can be marked on a number line. For example, ²₃ lies between 0 and 1.

Tip: To place ^m_n, divide the interval from 0 to 1 into n equal parts and count m steps from 0 (move left if negative).

4) Standard Form of a Rational Number

• Denominator should be positive.
• Numerator and denominator should have no common factor other than 1 (i.e., in lowest terms).

Example: −4/−6 = ²₃ (standard form).

Remember: If both numerator and denominator are negative, the number is positive.

5) Reciprocal

Reciprocal of a/b is b/a, provided a ≠ 0.

Example: Reciprocal of ³₄ is ⁴₃.

6) Rational Numbers Between Two Rational Numbers

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

Example: Between ¹₃ and ¹₂, one number is ⁵₁₂.

Quick method: For two rationals A and B, the mediant (p1+p2q1+q2) lies between them when both are positive and have positive denominators.

Quick Summary

• Rational numbers include integers, fractions and negatives of fractions.
• Closed under +, −, × and ÷ (division by 0 is not allowed).
• Follow commutative, associative and distributive laws.
• Can be shown on a number line and written in standard form.
• Infinitely many rationals exist between any two rationals.

Practice Questions

1. Write each in standard form:
   1. −¹²₁₈
   2. ⁻⁸₋₁₄
2. Find the reciprocal (if it exists):
   1. ⁷₉
   2. 0
3. Insert one rational number between:
   1. ²₅ and ³₅
   2. −¹₂ and ¹₃
4. Simplify using distributive law:

   (³₄) × ( ⁵₆ + ¹₃ )

Answer Key (Quick Check)

1. (a) −²₃    (b) ⁴₇
2. (a) ⁹₇    (b) Not defined (reciprocal of 0 doesn’t exist)
3. (a) ¹₂ (example)
   (b) −¹₂ + ¹₃ → One option is −¹₃ (using mediant or common denominator)

4. (³₄ × ⁵₆) + (³₄ × ¹₃) = ¹⁵₂₄ + ³₁₂ = ⁵₈ + ¹₄ = ⁷₈.

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