Question

6) Write the properties of integers under addition.​

Answer 1

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--> The whole numbers and negative numbers together are called integers. The set of all integers is denoted by Z.

Z  =  {... - 2, - 1,0,1,2, ...}, is the set of all integers

Here, we are going to see the following four properties of addition of integers. 

(i)   Closure property

(ii)   Commutative property

(iii)  Associative property

(iv)  Additive identity

--> Closure Property of Addition of Integers

Observe the following examples:

(i)  19 + 23 = 42

(ii)  - 10 + 4 = - 6

(iii)  18 + (- 47) = - 29

In general, for any two integers a and b, a + b is an integer.

Therefore the set of integers is closed under addition.

--> Commutative Property of Addition of Integers

Two integers can be added in any order. In other words, addition is commutative for integers.

We have 8 + (- 3)  =  5 and (- 3) + 8  =  5

So, 8 + (- 3)  =  (- 3) + 8

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

a + b = b + a

Therefore addition of integers is commutative.

--> Associative Property of Addition of Integers

Observe the following example :

Consider the integers 5, – 4 and 7.

Look at

5 + [(– 4) + 7] = 5 + 3 = 8

and 

[5 + (– 4)] + 7 = 1 + 7 = 8

Therefore, 5 + [(– 4) + 7] = [5 + (– 4)] + 7

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

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

Therefore addition of integers is associative.

--> Additive Identity

When we add zero to any integer, we get the same integer.

Observe the example: 5 + 0 = 5.

In general, for any integer a, a + 0 = a.

Therefore, zero is the additive identity for integers.