NCERT Class 8 Maths Chapter 1 Explanation

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NCERT Class 8 Maths Chapter 1: Rational Numbers - Explanation

1. What are Rational Numbers?

  • A rational number is a number that can be expressed in the form p/q, where:
    • p and q are integers.
    • q ≠ 0 (denominator cannot be zero).

Examples:

5, 5/1, 7/2

  • (All these are rational numbers.)

2. Properties of Rational Numbers

Rational numbers follow several mathematical properties:

(i) Closure Property

  • Rational numbers are closed under addition, subtraction, and multiplication.
    • Example: (which is also a rational number).
    • 1/2 + 3/4 = 1
    • 2/3 - 1/3 = 1/3
    • 2/3 x 3/4 = 6/12 = 1/6
  • Not closed under division (as division by zero is not defined).
  • 5 ÷ 3 = 2.666...

(ii) Commutative Property

  • Addition and multiplication of rational numbers are commutative.
    • 2 + 3 = 3 + 2
    • 2 x 3 = 3 x 2

(iii) Associative Property

  • Addition and multiplication are associative.
    • (2 + 3) + 4 = 2 + (3 + 4)
    • (2 x 3) x 4 = 2 x (3 x 4)

(iv) Distributive Property

  • Multiplication distributes over addition and subtraction.
    • a x (b + c) = a x b + a x c

(v) Identity Elements

  • Additive Identity: 0 (as a + 0 = a).
  • Multiplicative Identity: 1 (as a x 1 = a).

(vi) Inverse of Rational Numbers

  • Additive Inverse: has an inverse (as a + (-a) = 0 ).
  • Multiplicative Inverse: has an inverse (as a x (1/a) =  1), provided  a not equal to 0

3. Representation of Rational Numbers on the Number Line

  • A rational number can be plotted on a number line.
  • Example: 3/4  is between 0 and 1.
  • To plot: Divide the section between 0 and 1 into 4 equal parts and mark the third part.

4. Standard Form of a Rational Number

A rational number is said to be in standard form when:

  • The denominator is positive.
  • The numerator and denominator have no common factors except 1.

Example:
Convert -12/18  into standard form:

  • Find the GCD (Greatest Common Divisor) of 12 and 18 → 6.
  • Divide numerator and denominator by 6: -12/18 = -2/3
    (which is in standard form).

5. Comparison of Rational Numbers

To compare rational numbers:

  • Convert them into like denominators.
  • The one with the greater numerator is larger.

Example: Compare and : 3/4 and 5/6

  • LCM of 4 and 6 = 12.
  • Convert fractions:
    .
  • Since 10 > 9, 5/6 > 3/4

6. Operations on Rational Numbers

(i) Addition

  • Find a common denominator and add numerators.
    • Example: 1/3 + 2/5 = 5/25 + 6/15 = 11/15

(ii) Subtraction

  • Similar to addition but subtract numerators.
  • Example: 7/3 - 2/3 = 5/3

(iii) Multiplication

  • Multiply numerators and denominators.
    • Example: 2/3 x 4/5 = 8/15

(iv) Division

  • Multiply by the reciprocal.
    • Example: 2/3 ÷ 4/5 = 2/3 x 5/4 = 10/12 = 5/6

7. Recurring and Terminating Decimals

A rational number is either:

  • Terminating Decimal (e.g. 1/2 = 0.5 ).
  • Recurring Decimal (e.g. 1/3 = 0.333..., ).

Summary

  • Rational numbers are in the form p/q (where q ≠ 0).
  • They follow closure, commutative, associative, and distributive properties.
  • Zero is the additive identity, and one is the multiplicative identity.
  • Operations include addition, subtraction, multiplication, and division.
  • Rational numbers can be compared and represented on the number line.
  • A rational number in standard form has no common factors between numerator and denominator.


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