Properties Of Addition And Subtraction Of Integers
Closure under Addition
We have learnt about whole numbers and integers. We have also learnt about addition and subtraction of integers.
Let us see whether this property is true for integers or not.
What do you observe?
Is the sum of two integers always an integer?
Did you find a pair of integers whose sum is not an integer?
Since addition of integers gives integers, we say integers are closed under addition.
In general, for any two integers a and b, a + b
Closure under Subtraction
What happens when we subtract an integer from another integer?
Let us see some examples.
What do you observe?
Is there any pair of integers whose difference is not an integer?
Can we say integers are closed under subtraction?
Yes, we can see that integers are closed under subtraction.
Thus,if a and b are two integers then a – b is also an integer.
Do the whole numbers satisfy this property?
Commutative Property
We know that 3 + 5 = 5 + 3 = 8, that is, the whole numbers can be added in any order. In other words, addition is
Can we say the same for integers also? We have:
5 + (– 6) = –1 and (– 6) + 5 = –1
So,
5 + (– 6) = (– 6) +
What about (– 8) + (– 9) and (– 9) + (– 8)?
Let us see whether they are same.
- (– 8) + (– 9) =
- Next, (– 9) + (– 8) =
- In both the cases, we get the result equal to
. - Thus, LHS
RHS
Let's try some more.
let us see one more example:
(– 23) + 32 and 32 + (– 23) ?
- (– 23) + 32 =
- Next, 32 + (– 23) =
- In both the cases, we get the result equal to
. - Thus, LHS
RHS
Can you think of any pair of integers for which the sums are different when the order is changed?
Thus addition is
In general, for any two integers a and b, we can say
a + b =
We know that subtraction is not commutative for whole numbers. Is it commutative for integers?
Consider the integers 5 and (–3).
Is 5 – (–3) the same as (–3) –5?
- 5 – (–3) =
- Next, (–3) –5 =
- We see that the result is
. - Thus, LHS
to RHS
We conclude that subtraction is
Associative Property
Observe the following examples:
Consider the integers –3, –2 and –5.
Look at (–5) + [(–3) + (–2)] and [(–5) + (–3)] + (–2).
In the first expression, we have the sum of (–3) and (–2) and in the second the sum of (–5) and (–3). Do we get the same results i.e.
(–5) + [(–3) + (–2)] = [(–5) + (–3)] + (–2) OR Are the results different?
- (–5) + [(–3) + (–2)] = (-5) +
= - Next, [(–5) + (–3)] + (–2) =
+ (-2) = - In both the cases, we get the result equal to
. - LHS
RHS
Similarly, check if:
(–3) + [1 + (–7)] = [(–3) + 1] + (–7)
- ( –3) + [1 + (–7)] = –3 +
= - Next, [(–3) + 1] + (–7) =
+ -7 = - In both the cases, we get the result equal to
. - LHS
RHS
Addition is
In general for any integers a, b and c, we can say
a + (b + c) = (a + b) + c
Additive Identity
When we add zero to any whole number, we get the same whole number. Zero is an additive identity for whole numbers. Is it an additive identity again for integers also?
Observe the following and fill in the blanks:
| (i) (– 8) + 0 = – 8 | (ii) 0 + (– 8) = – 8 |
| (iii) (–23) + 0 = | (iv) 0 + (–37) = –37 |
| (v) 0 + (–59) = | (vi) 0 + |
| (vii) – 61 + | (viii) -21 + 0 = |
The above examples show that zero is an additive identity for integers. You can verify it by adding zero to any other five integers.
In general, for any integer a
a + 0 = a = 0 + a
1. Write a pair of integers
(a) whose sum gives a negative integer:
first number : -5 ; Second number:
(b) whose sum gives zero :
first number: -9 ; Second number:
(c) whose sum gives an integer smaller than both the integers :
first number: 5 ; Second number:
(d) whose sum gives an integer smaller than only one of the integers:
first number: 3 ; Second number:
(e) whose sum is greater than both the integers :
first number: 2 ; Second number:
2. Write a pair of integers
(a) whose difference gives a negative integer :
first number: 3 ; Second number:
(b) whose difference gives zero :
first number: 4 ; Second number:
(c) whose difference gives an integer smaller than both the integers :
first number: 7 ; Second number:
(d) whose difference gives an integer greater than only one of the integers :
first number: 5 ; Second number:
(e) whose difference gives an integer greater than both the integers :
first number: -1 ; Second number:
Note: These are the example based on integers for sum and difference you can try it for some more examples.
EXAMPLE 1 : Write down a pair of integers whose.
(a) sum is –3
(b) difference is –5
(c) difference is 2
(d) sum is 0
Solution:
(a) (-1) + (-2) = -3 or (-5) + 2 =
(b) (-9) - (- 4) = -5 or (-2) - 3 =
(c) (-7) - (-9) = 2 or 1 - (-1) =
(d) (-10) + 10 = 0 or 5 + (-5) =