### How to add integers using a number line?

To add integers, we have the following cases:

Case 1: When both the integers are positive.
Let us add 4 and 7.

We first move 4 steps towards the right from 0 to reach 4 and then we move 7 steps to the right of 4. Then we reach 11.
So, 4 + 7 = 11

Case 2: When both the integers are negative.
Let us add 4 and 7.

Here first we take 4 steps towards the left of 0 to reach − 4 and from there, we move 7 steps further to the left to reach −11.
So, 4 + (7) = – 11

Case 3: When both the integers are of opposite signs.
a.      Let us add 4 and 7.

First move 4 steps towards the right of 0 to reach 4 and from there, move 7 steps towards the left to reach −3.
So, (+4) + (−7) = −3

b.     Let us add 4 and 7.

First we move 4 steps towards the left of 0 to reach −4 and from there, we move 7 steps towards the right to reach +3.
So, (−4) + 7 = +3

### To add two integers, use the following steps:

1. If both the integers are positive, then their absolute values are added together. Hence, the sum will be a positive integer.
2. If both the integers are negative, then their absolute values are added and the
sum will be negative.
3. If two integers have opposite signs, i.e., a positive and a negative integer, then their
difference is calculated and the difference takes the sign of the integer with the larger
absolute value.

a.      6 and 10
b.      –12 and –18
c.       14 and –25
d.       –16 and 35

Solution:
a.      Here, both the integers are positive.
6 + 10 = 16
b.      Here, both the integers are negative.
(– 12) + (– 18) = – 30
c.       Here, the integers are of different signs.
14 + (– 25) = – 11
d.      Here, the integers are of different signs.
(16) + 35 = 19

## Properties of Addition of Integers

### Closure Property

The sum of any two whole numbers is a whole number. This property is called the closure property of addition for whole numbers.
For any two integers a and b, a + b is an integer.
For example,
a.      4 + 5 = 9, 9 is an integer.
b.      (–6) + 11 = 5, 5 is an integer.
Thus, integers are closed under addition.

### Commutative Property

Let us find the sum of –7 and 5.
(–7) + 5 = –2
If we add 5 and (–7), we still get the sum as –2 or (–7) + 5 = 5 + (–7).
For any two integers a and b, a + b = b + a
For example,
a.      (–3) + 7 = 7 + (–3) = 4
b.      (–7) + (–3) = (–3) + (–7) = –10
So, we can say that addition is commutative for integers.

### Associative Property

Consider the integers (–12), +5 and (–8). We can find the sum of these three integers in the following ways.
a. [(–12) + 5] + (–8) = (–7) + (–8) = –15
b. (–12) + [5 + (–8)] = (–12) + (–3) = –15
In both the cases, we get the same result.
Thus, [(–12) + 5] + (–8) = (–12) + [5 + (–8)]
For any of three integers a, b and c, a + (b + c) = (a + b) + c

For example,
a.      5 + (3 + 2) = (5 + 3) + 2 = 10
b.      8 + [–2 + (–3)] = [8 + (–2)] + (–3) = 3
So, we can conclude that addition is associative for integers.

If we add zero (0) to any whole number, we get the same whole number. Zero (0) is known as the additive identity for whole numbers.

For example,
(–32) + 0 = (–32)
24 + 0 = 24

From this, we can conclude that zero (0) is the additive identity for integers.
For any integer a, we have a + 0 = 0 + a = a

If we add 25 with 25, we get zero (0) as the sum. Thus, 25 is known as the additive inverse of 25 and 25 is known as the additive inverse of 25.
For example,
(–42) + (42) = 0
21 + (–21) = 0
If a is an integer, then there exists an integer –a such that a + (–a) = 0

Example 2: Apply the properties and find the sum of the following.
a. –50, –150 and 200
b. –2, –65, –8 and 35

Solution:
a. (–50) + (–150) + 200
By associative property
= [(–50) + (–150)] + 200
= –200 + 200 = 0
b. (–2) + (–65) + (–8) + 35
(By associative property)
= [(–2) + (–8)] + [(–65) + 35]
= (–10) + (–30) = –40

## Subtraction of Integers

How to subtract integers using a number line?

Subtraction is the process opposite to that of addition.
On a number line, while subtracting a positive integer we move towards the left and while subtracting a negative integer we move towards the right.

To subtract integers, we have the following cases:

Case 1: When both the integers are positive.
Let us subtract 5 from 3.
On the number line, start from 3 and move 5 steps to the left of 3 and reach –2.

Thus, we get 3 – 5 = –2.

Case 2: When both the integers are negative.
Let us subtract –4 from –9.
On the number line, start from –9 and move 4 steps to the right of –9 and reach –5.

Thus, we get (–9) – (–4) = –9 + 4 = –5.

Case 3: When both the integers are of opposite signs.

a.      Let us subtract 3 from 2.
On the number line, start from –2 and move 3 steps to the left of –2 and reach –5.

Thus, we get (–2) – (3) = –2 – 3 = –5.

b.     Let us subtract 4 from 3.
On the number line, start from 3 and move 4 steps to the right of 3 and reach 7.

Thus, we get (3) – (–4) = 3 + 4 = 7.

From the above, we can say that subtracting a negative integer from an integer is adding
the additive inverse of the integer to the given number.

Example 1: Subtract 8 from –6.

Solution: The additive inverse of 8 is –8.
Thus, –6 – (8) = –6 + Additive inverse of 8 = –6 + (–8) = –14

Example 2: Subtract (–7) from (–12).

Solution: The additive inverse of –7 is +7.
Thus, (–12) – (–7) = (–12) + Additive inverse of (–7) = –12 + 7 = –5

## Properties of Subtraction of Integers

### Closure property

The difference between any two integers is always an integer i.e., if a and b are any two integers then their difference, a b or b a, will always be an integer.
For example,
a. 7 – 3 = 4, 4 is an integer.
b. (–4) – (–3) = –1, –1 is an integer.
Thus, subtraction is closure for integers.

### Commutative property

For any two integers a and b, a b is not equal to b a, i.e., a b b a
Thus, difference of integers is not commutative.
For example,
3 – (–4) (–4) – 3
As 3 – (–4) = 7 and (–4) – 3 = –7
Thus, 3 – (–4) (–4) – 3
Thus, subtraction is not commutative for integers.

### Associative property

For any three integers a, b and c, (a b) – c a – (b c)
Thus, subtraction of integers is not associative.
For example,
a. 5 – (2 – 6) (5 – 2) – 6
b. (–7) – (3 – 5) (–7 – 3) – 5
Thus, subtraction is not associative for integers.

## Successor of an Integer

Like whole numbers, every integer has a successor. 0 is the successor of –1, –1 is the
successor of –2, –2 is the successor of –3 and so on.
Thus, one added to an integer gives its successor.

## Predecessor of an Integer

Every integer has a predecessor. 1 is the predecessor of 2, 2 is the predecessor of 3, 3 is
the predecessor of 4 and so on.
Thus, one subtracted from an integer gives its predecessor.

Example: Find the successor and the predecessor of each of the following.
a. 15                          b. –11                 c. 8                    d. –6

Solution: We know the successor of an integer is one more than the given integer and the predecessor of an integer is one less than the given integer.

a. The successor of 15 is 15 + 1 = 16.
The predecessor of 15 is 15 – 1 = 14.
b. The successor of –11 is –11 + 1 = –10.
The predecessor of –11 is –11 – 1 = –12.
c. The successor of 8 is 8 + 1 = 9.
The predecessor of 8 is 8 – 1 = 7.
d. The successor of –6 is –6 + 1 = –5.
The predecessor of –6 is –6 – 1 = –7.