### Introduction: Rational Numbers

- A
**rational number**is defined as a number that can be expressed in the form p/q, where p and q are integers and q≠0. - In our daily lives, we use some quantities which are not whole numbers but can be expressed in the form of p/q. Hence we need rational numbers.

### Equivalent Rational Numbers

By **multiplying or dividing the numerator and denominator** of a rational number by a **same** **non zero integer**, we obtain another rational number equivalent to the given rational number.These are called **equivalent fractions**.

### Rational Numbers in Standard Form

A rational number is said to be in the **standard form** if its denominator is a positive integer and the **numerator and denominator have no common factor other than 1.**

**Operations on Rational Numbers**

**Addition**

Addition of two rational numbers with same denominators: Two rational numbers with the same denominators can be added by adding their numerators, keeping the denominator same.

Addition of two rational numbers with different denominators: As in the case of fractions, we first find the LCM of the two denominators. Then we find the rational numbers equivalent to the given rational numbers with this LCM as the denominator. Now, we add the two rational numbers as in (A).

**Subtraction**

While subtracting two rational numbers, we add the additive inverse of the rational number to be subtracted to the other rational number.

**Multiplication**

**Multiplication of a rational number by a positive integer:**

While multiplying a rational number by a positive integer, we multiply the numerator by that integer, keeping the denominator unchanged.

**Multiplication of rational number by a negative integer:**

While multiplying a rational number by a negative integer, we multiply the numerator by that integer, keeping the denominator unchanged.

**Multiplication of two rational numbers (none of which is an integer):**

Based on the above observations,

So, as done in fractions we multiply two rational numbers as follows:

- Step 1. Multiply the numerators of the two rational numbers.
- Step 2. Multiply the denominators of the two rational numbers.