# Rational Numbers Class 7 Notes: Chapter 9

## Introduction to Rational Numbers

### Introduction: Rational Numbers

• A rational number is defined as a number that can be expressed in the form $\frac{p}{q}$, 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 $\frac{p}{q}$. 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.
• $\frac{1}{3}=\frac{1}{3}\times \frac{2}{2}=\frac{2}{6}$

∴$\frac{2}{6}$ and $\frac{1}{3}$ are equivalent fractions.

• $\frac{15}{25}=\frac{15\div5}{25\div5}=\frac{3}{5}$ ∴$\frac{15}{25}$ and $\frac{3}{5}$ are 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.
• Example: Reduce $\frac{-4}{16}$.Here, the H.C.F. of 4 and 16 is 4. $\Rightarrow \frac{-4}{16}=\frac{\frac{-4}{4}}{\frac{16}{4}}=\frac{-4}{16}$ $\frac{-1}{4}$ is the standard form of $\frac{-4}{16}$.

### LCM

• The least common multiple (LCM) of two numbers is the smallest number (≠0) that is a multiple of both.
• Example: LCM of 3 and 4 can be calculated as shown below:
Multiples of 3: 0, 3, 6, 9, 12,15
Multiples of 4: 0, 4, 8, 12, 16
LCM of 3 and 4 is 12.

## Rational Numbers Between 2 Rational Numbers

### Rational Numbers between Two Rational Numbers

• There are unlimited number(infinite number) of rational numbers between any two rational numbers.
• Example: List some of the rational numbers between −35 and −13.
Solution: L.C.M. of 5 and 3 is 15.
⇒ The given equations can be written as $\frac{-9}{15}$ and $\frac{-5}{15}$ .
⇒ −615,−715,−815 are the rational numbers between −35 and −13.

Note: These are only few of the rational numbers between −35 and −13. There are infinte number of rational numbers between them. Following the same procedure, many more rational numbers can be inserted between them.

## Properties of Rational Numbers

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### Multiplication and Division of Rational Numbers

Multiplication of Rational Numbers