Chapter 1: Rational Numbers

A comprehensive study of rational numbers, their arithmetic operations, fundamental properties, additive & multiplicative identities, and insertion of rational numbers.

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15-min visual explanation of all properties

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Formula sheet & practice questions

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1. What is a Rational Number?

Any number that can be expressed in the form of p / q, where p and q are integers and q ≠ 0, is called a Rational Number.

Mathematical Representation

ℚ = { p / q : p, q ∈ ℤ and q ≠ 0 }

Examples: 2/3, -5/7, 0 (since 0 = 0/1), and -8 (since -8 = -8/1) are all rational numbers.

2. Fundamental Properties of Rational Numbers

Just like whole numbers and integers, rational numbers follow specific algebraic laws under the four fundamental arithmetic operations (Addition, Subtraction, Multiplication, Division):

A. Closure Property

Rational numbers are closed under Addition, Subtraction, and Multiplication.

Addition: If a and b are rational numbers, then a + b is also a rational number.
Example: 2/5 + 1/5 = 3/5 (Rational).
Subtraction: a - b is always a rational number.
Multiplication: a × b is always a rational number.
Division: Rational numbers are not closed under division because division by zero (e.g. 5/0) is undefined. However, excluding zero, they are closed.

B. Commutative Property

Changing the order of operands does not change the result:

Commutativity Rules

Addition: a + b = b + a
Multiplication: a × b = b × a

Note: Subtraction and Division are NOT commutative for rational numbers! (a - b ≠ b - a).

C. Associative Property

Grouping of operands does not alter the final result for addition and multiplication:

Associativity Rules

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

D. Distributive Property of Multiplication over Addition

This property allows us to simplify complex rational multiplications:

Distributive Law

a × (b + c) = (a × b) + (a × c)

3. Identities & Inverses

The Role of Zero (Additive Identity): Adding 0 to any rational number leaves it unchanged: a + 0 = 0 + a = a. Zero is the additive identity.
The Role of 1 (Multiplicative Identity): Multiplying any rational number by 1 leaves it unchanged: a × 1 = 1 × a = a.
Negative of a Number (Additive Inverse): For any rational number a/b, its additive inverse is -a/b, such that a/b + (-a/b) = 0.
Reciprocal (Multiplicative Inverse): For a rational number c/d (c ≠ 0), its reciprocal is d/c, such that c/d × d/c = 1. Note that 0 has no reciprocal!

4. Solved Exemplar Questions (NCERT Style)

Exemplar Problem 1 Distributive Property Application

Question: Simplify using appropriate properties:
(-2/3) × (3/5) + (5/2) - (3/5) × (1/6)

Rearrange using Commutativity: Group terms containing the common factor (3/5):
= (-2/3) × (3/5) - (3/5) × (1/6) + (5/2)

Apply Distributive Property: Take (3/5) as common:
= (3/5) × [ (-2/3) - (1/6) ] + (5/2)

Calculate inside brackets: LCM of 3 and 6 is 6:
(-4/6 - 1/6) = -5/6
= (3/5) × (-5/6) + (5/2)
= -1/2 + 5/2 = 4/2 = 2.

Exemplar Problem 2 Finding Rational Numbers Between Two Numbers

Question: Find 5 rational numbers between -2/5 and 1/2.

Make Denominators Equal: LCM of 5 and 2 is 10.
-2/5 = (-2 × 4) / (5 × 4) = -8/20
1/2 = (1 × 10) / (2 × 10) = 10/20

Pick Any 5 Numbers in between: Integers between -8 and 10 are -7, -6, -5, -4, -3, -2, -1, 0, 1...
Five rational numbers are: -7/20, -5/20, -1/20, 1/20, 3/20.

5. Quick Formula & Revision Matrix

Concept	Mathematical Formula	Exam Reminder
Additive Inverse	a/b → -a/b	Sum always equals 0
Multiplicative Inverse	a/b → b/a (a, b ≠ 0)	Product always equals 1; 0 has no reciprocal
Distributive Law	a(b + c) = ab + ac	Used to take common factors and simplify
Mean Method	(a + b) / 2	Gives a rational number strictly between a and b

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