Topic 1.4 : Integers Topic 1.4 : पूर्णांक

1. Meaning of Integers

The collection of positive whole numbers, negative whole numbers and zero is called the set of integers.

Z = {..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...}

The set of integers is generally represented by the symbol Z.

2. Types of Integers

Positive Integers

Integers greater than zero are called positive integers.

Examples: 1, 2, 3, 4, 5, ...

Negative Integers

Integers less than zero are called negative integers.

Examples: −1, −2, −3, −4, −5, ...

Zero

Zero is an integer but is neither positive nor negative.

3. Natural Numbers, Whole Numbers and Integers

According to the convention followed in this study material:

Natural Numbers:
N = {1, 2, 3, 4, ...}

Whole Numbers:
W = {0, 1, 2, 3, 4, ...}

Integers:
Z = {..., −3, −2, −1, 0, 1, 2, 3, ...}

4. Integers on the Number Line

Integers can be represented on a number line:

← −5   −4   −3   −2   −1   0   1   2   3   4   5 →

• Positive integers lie to the right of zero.
• Negative integers lie to the left of zero.
• Moving towards the right increases the value.
• Moving towards the left decreases the value.

5. Comparison of Integers

• Every positive integer is greater than zero.
• Zero is greater than every negative integer.
• Every positive integer is greater than every negative integer.
• Among negative integers, the one nearer to zero is greater.

Example:
−3 > −8 because −3 lies to the right of −8.

Similarly: −7 > −15

6. Successor and Predecessor

For any integer n:

Successor = n + 1
Predecessor = n − 1

Example: For −7:

• Successor of −7 = −6
• Predecessor of −7 = −8

7. Opposite Integers and Additive Inverse

Two integers having the same distance from zero but lying on opposite sides of zero are called opposite integers.

Examples:

• 5 and −5
• 12 and −12
• 100 and −100

The additive inverse of an integer a is −a because:

a + (−a) = 0

For example, the additive inverse of −17 is 17 because:

−17 + 17 = 0

8. Absolute Value of an Integer

The absolute value of an integer is its distance from zero on the number line, irrespective of direction.

It is represented using vertical bars.

|5| = 5
|−5| = 5
|0| = 0

Therefore:

|a| ≥ 0

9. Addition of Integers

Case 1: Same Signs

Add the absolute values and retain the common sign.

(+7) + (+5) = +12

(−7) + (−5) = −12

Case 2: Different Signs

Subtract the smaller absolute value from the larger absolute value and use the sign of the number having the larger absolute value.

(+12) + (−7) = +5

(−15) + (+6) = −9

10. Subtraction of Integers

Subtraction can be converted into addition by changing the sign of the integer being subtracted.

a − b = a + (−b)

Example 1:
8 − (−5)
= 8 + 5
= 13

Example 2:
−6 − 4
= −6 + (−4)
= −10

11. Multiplication and Division: Sign Rules

The following sign rules apply to both multiplication and division:

The same sign rule applies to division.

12. Sign of the Product of Several Integers

When several non-zero integers are multiplied:

• If the number of negative factors is even, the product is positive.
• If the number of negative factors is odd, the product is negative.

Example:
(−2)(−3)(−4)(−5)

There are four negative factors. Since 4 is even, the product is positive.

Product = 120

13. Properties of Integers

Closure Property

Integers are closed under:

• Addition
• Subtraction
• Multiplication

Integers are not closed under division.

Example:
5 ÷ 2 = 2.5, which is not an integer.

Commutative Property

Addition and multiplication are commutative:

a + b = b + a
a × b = b × a

Subtraction and division are not commutative in general.

Associative Property

Addition and multiplication are associative:

(a + b) + c = a + (b + c)
(a × b) × c = a × (b × c)

Subtraction and division are not associative in general.

Distributive Property

Multiplication is distributive over addition and subtraction:

a(b + c) = ab + ac

a(b − c) = ab − ac

14. Identity Elements

Additive Identity = 0

a + 0 = a

Multiplicative Identity = 1

a × 1 = a

15. Important Properties of Zero

• a + 0 = a
• a − 0 = a
• 0 − a = −a
• a × 0 = 0
• 0 ÷ a = 0, provided a ≠ 0
• a ÷ 0 is undefined

16. Distance Between Two Integers

The distance between integers a and b on the number line is:

|a − b|

Example: Find the distance between −7 and 5.

Distance = |−7 − 5|
= |−12|
= 12

17. Solved Examples

Example 1

Arrange −8, 3, 0, −2 and 7 in ascending order.

Solution:
−8 < −2 < 0 < 3 < 7

Example 2

Evaluate: −15 + 8 − (−6)

Solution:
−15 + 8 + 6
= −15 + 14
= −1

Example 3

Find the additive inverse of −125.

Solution:
The additive inverse of −125 is 125, because:
−125 + 125 = 0

Example 4

Evaluate: (−8) × (−5) × (−2)

Solution:
There are three negative factors.
An odd number of negative factors gives a negative product.

8 × 5 × 2 = 80

Therefore, the answer is −80.

Example 5

Which is greater: −19 or −11?

Solution:
−11 lies to the right of −19 on the number line.

Therefore, −11 > −19.

Example 6

Find the distance between −15 and −4.

Solution:
Distance = |−15 − (−4)|
= |−15 + 4|
= |−11|
= 11

18. Common Exam Traps

19. Quick Revision

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Previous Year Questions (PYQs)

Practice verified previous-year questions based on integers and related number-system concepts. Try to solve each question before revealing the answer and explanation.

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Practice MCQs

Practice these exam-oriented questions to strengthen your understanding of integers, number-line comparison, absolute value, additive inverse, sign rules and properties of integers.