Addition of Integers
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.
Example:
Add
the following integers.
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.
Additive Identity
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
Additive Inverse
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
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.