MATHEMATICS
CHAPTER – NUMBER SYSTEMS
NUMBER SYSTEMS
NUMBER SYSTEMS
You are already familiar with different kinds of numbers such as natural numbers, whole
numbers and integers. You also know the four fundamental operations of arithmetic:
Addition, Subtraction, Multiplication and Division
In this chapter, we revise these number systems and slowly extend our study to
rational numbers, irrational numbers and real numbers.
1. Natural Numbers
The counting numbers
1, 2, 3, 4, . . .
are called natural numbers.
The set of natural numbers is denoted by N.
N = {1, 2, 3, 4, . . .}
Examples:
• 5 is a natural number
• 12 is a natural number
• 0 is not a natural number
2. Whole Numbers
The number 0 together with all natural numbers are called whole numbers.
The set of whole numbers is denoted by W.
W = {0, 1, 2, 3, 4, . . .}
Thus,
W = N ∪ {0}
Examples:
1
, • 0, 6, 15 are whole numbers
• All natural numbers are whole numbers
3. Fundamental Operations on Whole Numbers
Whole numbers are used in the following four operations:
• Addition
• Subtraction
• Multiplication
• Division
4. Division Algorithm
If a and b are any two whole numbers such that b ̸= 0, then there exist unique whole
numbers q and r such that:
a=b×q+r where 0 ≤ r < b
This is called the Division Algorithm.
In words:
Dividend = Divisor × Quotient + Remainder
Example:
Divide 39 by 5.
39 = 5 × 7 + 4
Here,
• Dividend = 39
• Divisor = 5
• Quotient = 7
• Remainder = 4
5. Closure Property of Whole Numbers
A set is said to be closed under an operation if the result of the operation always belongs
to the same set.
2
, (a) Addition
Whole numbers are closed under addition.
12 + 8 = 20 ∈ W
(b) Multiplication
Whole numbers are closed under multiplication.
6 × 4 = 24 ∈ W
(c) Subtraction
Whole numbers are not closed under subtraction.
Example:
39 − 57 = −18 ∈/W
Since −18 is not a whole number, subtraction is not always possible in whole numbers.
6. Need for a New Number System
The system of whole numbers is not sufficient to solve all subtraction problems.
To find answers for such subtractions, we need to extend our number system. This
leads us to the system of integers.
7. Integers
The set of integers consists of positive numbers, negative numbers and zero. These
numbers are also called directed numbers because they have a direction on the number
line.
Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . .}
Examples:
• −7, 0, 9 are integers
• −15 lies to the left of 0 on the number line
• 12 lies to the right of 0 on the number line
3
CHAPTER – NUMBER SYSTEMS
NUMBER SYSTEMS
NUMBER SYSTEMS
You are already familiar with different kinds of numbers such as natural numbers, whole
numbers and integers. You also know the four fundamental operations of arithmetic:
Addition, Subtraction, Multiplication and Division
In this chapter, we revise these number systems and slowly extend our study to
rational numbers, irrational numbers and real numbers.
1. Natural Numbers
The counting numbers
1, 2, 3, 4, . . .
are called natural numbers.
The set of natural numbers is denoted by N.
N = {1, 2, 3, 4, . . .}
Examples:
• 5 is a natural number
• 12 is a natural number
• 0 is not a natural number
2. Whole Numbers
The number 0 together with all natural numbers are called whole numbers.
The set of whole numbers is denoted by W.
W = {0, 1, 2, 3, 4, . . .}
Thus,
W = N ∪ {0}
Examples:
1
, • 0, 6, 15 are whole numbers
• All natural numbers are whole numbers
3. Fundamental Operations on Whole Numbers
Whole numbers are used in the following four operations:
• Addition
• Subtraction
• Multiplication
• Division
4. Division Algorithm
If a and b are any two whole numbers such that b ̸= 0, then there exist unique whole
numbers q and r such that:
a=b×q+r where 0 ≤ r < b
This is called the Division Algorithm.
In words:
Dividend = Divisor × Quotient + Remainder
Example:
Divide 39 by 5.
39 = 5 × 7 + 4
Here,
• Dividend = 39
• Divisor = 5
• Quotient = 7
• Remainder = 4
5. Closure Property of Whole Numbers
A set is said to be closed under an operation if the result of the operation always belongs
to the same set.
2
, (a) Addition
Whole numbers are closed under addition.
12 + 8 = 20 ∈ W
(b) Multiplication
Whole numbers are closed under multiplication.
6 × 4 = 24 ∈ W
(c) Subtraction
Whole numbers are not closed under subtraction.
Example:
39 − 57 = −18 ∈/W
Since −18 is not a whole number, subtraction is not always possible in whole numbers.
6. Need for a New Number System
The system of whole numbers is not sufficient to solve all subtraction problems.
To find answers for such subtractions, we need to extend our number system. This
leads us to the system of integers.
7. Integers
The set of integers consists of positive numbers, negative numbers and zero. These
numbers are also called directed numbers because they have a direction on the number
line.
Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . .}
Examples:
• −7, 0, 9 are integers
• −15 lies to the left of 0 on the number line
• 12 lies to the right of 0 on the number line
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