✍️ Class 9 Mathematics – Chapter 1: Number Systems

📘 Topic: Rational Numbers

🔷 Definition: Rational Numbers

A number is called a rational number if it can be written in the form:

$$r = \frac{p}{q}$$

Where:

• p and q are integers

• q ≠ 0

📌 The word ‘rational’ comes from ‘ratio’, and the symbol Q for rational numbers comes from ‘quotient’.

❗ Why q ≠ 0?

If the denominator q is zero, the expression becomes undefined in mathematics. For example, $\frac{3}{0}$ is not defined.

✳️ Co-prime Form of Rational Numbers

When we represent a rational number $\frac{p}{q}$, we usually write it in simplest form, where p and q have no common factors other than 1. That means they are co-prime.

🔄 Equivalent Rational Numbers

Two rational numbers are equivalent if they represent the same value, even if they have different numerators and denominators.

✴️ For example:

$$\frac{2}{3} = \frac{4}{6} = \frac{6}{9}$$

✅ True/False Statements – Reasoning Based

1. Are the following statements true or false?

(i) Every whole number is a natural number.
❌ False.
→ 0 is a whole number but not a natural number.
Natural numbers start from 1, but whole numbers start from 0.

(ii) Every integer is a rational number.
✅ True.
→ Every integer a can be written as $\frac{a}{1}$, which is a rational number.

(iii) Every rational number is an integer.
❌ False.
→ $\frac{3}{4}$ is a rational number but not an integer.

🚀 How to Find Rational Numbers Between Two Given Numbers

Let’s say we want to find rational numbers between two numbers, like between 1 and 2.

🌟 General Step-by-Step Procedure:

Step 1: Write both numbers with the same denominator.

1 = $\frac{1}{1}$ and 2 = $\frac{2}{1}$

Multiply numerator and denominator of both with 10:

$$1 = \frac{10}{10},\quad 2 = \frac{20}{10}$$

Step 2: Find numbers between the numerators:

Between 10 and 20, we have: 11, 12, 13, 14, 15...

So, rational numbers between 1 and 2 are:

$$\frac{11}{10}, \frac{12}{10}, \frac{13}{10}, \frac{14}{10}, \frac{15}{10}$$

Step 3: Simplify if needed.

📚 Exercise 1.1 – Solved Answers

Q1: Is zero a rational number?

✅ Yes, 0 is a rational number.

It can be written as:

$$0 = \frac{0}{1},\ \frac{0}{2},\ \frac{0}{-5} \quad \text{(where } q \ne 0 \text{)}$$

Q2: Find six rational numbers between 3 and 4.

$$3 = \frac{30}{10},\quad 4 = \frac{40}{10}$$

Numbers between 30 and 40: 31, 32, 33, 34, 35, 36

So, six rational numbers:

$$\frac{31}{10},\frac{32}{10},\frac{33}{10},\frac{34}{10},\frac{35}{10},\frac{36}{\mathrm{10}}$$

Q3: Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$

Multiply numerator and denominator by 10:

$$\frac{3}{5} = \frac{30}{50},\quad \frac{4}{5} = \frac{40}{50}$$

Between 30 and 40: 31, 32, 33, 34, 35

So five rational numbers:

$$\frac{31}{50}, \frac{32}{50}, \frac{33}{50}, \frac{34}{50}, \frac{35}{50}$$

Q4: State whether the following statements are true or false. Give reasons.

(i) Every natural number is a whole number.
✅ True
→ Natural numbers start from 1, and all natural numbers are included in whole numbers.

(ii) Every integer is a whole number.
❌ False
→ Integers include negative numbers like -1, -2, etc., which are not whole numbers.

(iii) Every rational number is a whole number.
❌ False
→ Rational numbers like $\frac{3}{5}$ are not whole numbers.

📌 Summary Chart

Set	Examples	Contains
Natural Numbers (N)	1, 2, 3, 4, ...	Counting numbers
Whole Numbers (W)	0, 1, 2, 3, ...	Natural + 0
Integers (Z)	..., -3, -2, -1, 0, 1, 2, ...	Positive and negative whole numbers
Rational Numbers (Q)	$\frac{3}{4}, -\frac{7}{2}, 0, 5$	Numbers written as $\frac{p}{q}$, q ≠ 0