Video Lecture 1: What is a Rational Number? — The p/q Representation and Algebraic Axioms

# Video Lecture 1: What is a Rational Number? — The p/q Representation and Algebraic Axioms
**Speaker:** Prof. Arvind Swaminathan | **Duration:** ~05:50 mins

### Introduction: The Evolution of Numbers
From natural counting numbers ($\mathbb{N} = \{1, 2, 3, \dots\}$) to whole numbers ($\mathbb{W} = \{0, 1, 2, \dots\}$), to integers ($\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$), humans constantly expanded the number world to solve equations. When dividing $2$ by $3$, integers fail! This led to **Rational Numbers** ($\mathbb{Q}$).

### Definition:
A number that can be written in the form **$\frac{p}{q}$**, where $p$ and $q$ are integers and **$q \neq 0$**, is called a **Rational Number**.

### Key Algebraic Properties:
1. **Closure Property:**
- Rational numbers are closed under **Addition**, **Subtraction**, and **Multiplication** (e.g., sum, difference, or product of two rationals is always rational).
- They are **NOT closed under Division** because division by zero is undefined! (If 0 is excluded, non-zero rationals are closed under division).
2. **Commutative Property ($a + b = b + a$ and $a \times b = b \times a$):**
- Holds for Addition and Multiplication.
- Fails for Subtraction ($a - b \neq b - a$) and Division ($a \div b \neq b \div a$).