CBSE Class 9th : Mathematics : Chapter 1

1. The Number Line Hierarchy

Before diving into operations, it is essential to understand how numbers are categorized.

Natural Numbers ( $N$ ): Counting numbers starting from $1, 2, 3, \dots$
Whole Numbers ( $W$ ): Natural numbers including zero ( $0, 1, 2, 3, \dots$ ).
Integers ( $Z$ ): All positive and negative whole numbers (.... $\dots, -2, -1, 0, 1, 2, \dots$ ).
Rational Numbers ( $Q$ ): Any number that can be written in the form p/q where $p$ and $q$ are integers and "q does not equal to 0" .
Irrational Numbers: Numbers that cannot be written in $\frac{p}{q}$ form (e.g.√2).
Real Numbers ( $R$ ): The collection of both rational and irrational numbers. Every real number corresponds to a unique point on the number line.

How a number looks as a decimal tells you if it is rational or irrational:

Type of Decimal	Category	Example
Terminating	Rational	0.5, 2.125
Non-terminating Repeating	Rational	$0.333...$ (or $0.\bar{3}$ )
Non-terminating Non-repeating	Irrational	$1.41421...$ or 丌

To find $n$ rational numbers between $a$ and $b$ :

When you mix rational and irrational numbers, follow these rules:

The sum or difference of a rational and an irrational number is irrational.
The product or quotient of a non-zero rational number with an irrational number is irrational.
If we add, subtract, multiply, or divide two irrational numbers, the result may be rational or irrational.