Table of Contents
Number System – Class 9 Mathematics
The Number System forms the foundation for algebra, geometry, and higher mathematics. In this blog, we will understand different types of numbers in a simple and clear manner.
1. What is a Number System?
A number system is a way of representing numbers and classifying them based on their properties. All numbers that we use in mathematics belong to one or more categories of the number system.
2. Types of Numbers
(a) Natural Numbers
Natural numbers are the numbers used for counting objects.
They are represented as:
\( \mathbb{N} = \{1, 2, 3, 4, 5, \dots\} \)
(b) Whole Numbers
Whole numbers include all natural numbers along with zero.
\( \mathbb{W} = \{0, 1, 2, 3, 4, \dots\} \)
(c) Integers
Integers include positive numbers, negative numbers, and zero.
\( \mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\} \)
(d) Rational Numbers
A number is called a rational number if it can be expressed in the form:
\[
\frac{p}{q}, \quad \text{where } p, q \in \mathbb{Z} \text{ and } q \neq 0
\]
Rational numbers include fractions and terminating or repeating decimals.
\( \frac{1}{2}, \frac{3}{5}, -4, 0.25, 0.\overline{3} \)
(e) Irrational Numbers
Irrational numbers cannot be written in the form \( \frac{p}{q} \).
Their decimal expansion is non-terminating and non-repeating.
\( \sqrt{2}, \sqrt{3}, \pi \)
(f) Real Numbers
All rational and irrational numbers together form the set of real numbers.
\[
\mathbb{R} = \text{Rational Numbers} \cup \text{Irrational Numbers}
\]
-2, 0, 5, \( \sqrt{7} \), \( \pi \)
3. Decimal Expansion of Rational Numbers
Rational numbers have either:
- Terminating decimals (e.g., \( \frac{1}{4} = 0.25 \))
- Non-terminating recurring decimals (e.g., \( \frac{1}{3} = 0.\overline{3} \))
4. Properties of Real Numbers
- Closure Property: For real numbers \( a \) and \( b \), \( a + b \in \mathbb{R} \)
- Commutative Property: \( a + b = b + a \)
- Associative Property: \( (a + b) + c = a + (b + c) \)
- Distributive Property: \( a(b + c) = ab + ac \)