# Properties Of Addition And Subtraction Of Integers

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## Closure under Addition

We have learnt about whole numbers and integers in Class VI. 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 an integer.**

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## Closure under Subtraction

What happens when we subtract an integer from another integer? Can we say that their difference is also an integer?

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?

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## 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) + 5**

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What about (– 8) + (– 9) and (– 9) + (– 8)?

- (– 8) + (– 9) =
- Next, (– 9) + (– 8) =
- In both the cases, we get the result equal to –17.
- Thus, LHS = RHS

Let's try some more.

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Or (– 23) + 32 and 32 + (– 23) ?

- (– 23) + 32 =
- Next, 32 + (– 23) =
- In both the cases, we get the result equal to 9.
- 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 for integers.*

In general, for any two integers a and b, we can say

a + b = b + a

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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 not the same.
- Thus, LHS not equal to RHS

*We conclude that subtraction is for integers.*

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## 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 –10.
- LHS = RHS

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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 –9.
- LHS = RHS

Addition is

In general for any integers a, b and c, we can say

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

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## 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