Class 8 Mathematics Notes – Rational Numbers
(Odisha Board)
Definition:
A rational number is a number that can be expressed in the form p/q, where p and q are integers
and q ≠ 0. Examples: 1/2, -3/5, 7, 0 are all rational numbers.
Representation on Number Line:
Every rational number can be represented on the number line. For example, 1/2 lies between 0 and
1, and -3/4 lies between -1 and 0.
Properties of Rational Numbers:
• Closure Property: The sum, difference, and product of two rational numbers are also rational
numbers. Example: 1/2 + 1/3 = 5/6 (rational).
• Commutative Property: Rational numbers are commutative for addition and multiplication.
Example: 2/3 + 3/4 = 3/4 + 2/3.
• Associative Property: Rational numbers are associative for addition and multiplication.
Example: (1/2 + 1/3) + 1/4 = 1/2 + (1/3 + 1/4).
• Distributive Property: a × (b + c) = a×b + a×c for all rational numbers. Example: 2 × (1/3 +
1/6) = 2/3 + 1/3 = 1.
• Existence of Identity: Additive identity = 0 (a + 0 = a). Multiplicative identity = 1 (a × 1 = a).
• Existence of Inverse: For every rational number a, there exists an additive inverse (-a) and a
multiplicative inverse (1/a) if a ≠ 0.
Standard Form of a Rational Number:
A rational number is said to be in standard form when its denominator is positive and the numerator
and denominator have no common factor other than 1. Example: -3/4 is in standard form, but 6/-8 is
not.
Operations on Rational Numbers:
• Addition/Subtraction: To add or subtract rational numbers, make the denominators equal and
then add or subtract the numerators. Example: 2/3 + 1/6 = 4/6 + 1/6 = 5/6.
• Multiplication: Multiply the numerators and denominators. Example: (2/3) × (4/5) = 8/15.
• Division: Multiply the first rational number by the reciprocal of the second. Example: (2/3) ÷
(4/5) = (2/3) × (5/4) = 10/12 = 5/6.
Examples:
• Simplify: (3/4) + (-5/8) = (6/8 - 5/8) = 1/8.
(Odisha Board)
Definition:
A rational number is a number that can be expressed in the form p/q, where p and q are integers
and q ≠ 0. Examples: 1/2, -3/5, 7, 0 are all rational numbers.
Representation on Number Line:
Every rational number can be represented on the number line. For example, 1/2 lies between 0 and
1, and -3/4 lies between -1 and 0.
Properties of Rational Numbers:
• Closure Property: The sum, difference, and product of two rational numbers are also rational
numbers. Example: 1/2 + 1/3 = 5/6 (rational).
• Commutative Property: Rational numbers are commutative for addition and multiplication.
Example: 2/3 + 3/4 = 3/4 + 2/3.
• Associative Property: Rational numbers are associative for addition and multiplication.
Example: (1/2 + 1/3) + 1/4 = 1/2 + (1/3 + 1/4).
• Distributive Property: a × (b + c) = a×b + a×c for all rational numbers. Example: 2 × (1/3 +
1/6) = 2/3 + 1/3 = 1.
• Existence of Identity: Additive identity = 0 (a + 0 = a). Multiplicative identity = 1 (a × 1 = a).
• Existence of Inverse: For every rational number a, there exists an additive inverse (-a) and a
multiplicative inverse (1/a) if a ≠ 0.
Standard Form of a Rational Number:
A rational number is said to be in standard form when its denominator is positive and the numerator
and denominator have no common factor other than 1. Example: -3/4 is in standard form, but 6/-8 is
not.
Operations on Rational Numbers:
• Addition/Subtraction: To add or subtract rational numbers, make the denominators equal and
then add or subtract the numerators. Example: 2/3 + 1/6 = 4/6 + 1/6 = 5/6.
• Multiplication: Multiply the numerators and denominators. Example: (2/3) × (4/5) = 8/15.
• Division: Multiply the first rational number by the reciprocal of the second. Example: (2/3) ÷
(4/5) = (2/3) × (5/4) = 10/12 = 5/6.
Examples:
• Simplify: (3/4) + (-5/8) = (6/8 - 5/8) = 1/8.
