The properties of addition whole numbers are as follows:

Closure property:

If a and b are two whole numbers, then a + b is also a whole number. In other words, the sum of any two whole numbers is a whole number or, whole numbers are closed for addition.

Verification: In order to verify this property, let us take any two whole numbers and add them. We find that the sum is always a whole numbers as shown below

7 + 3 = 10 (10 is also a whole number)

0 + 8 = 8 (8 is also a whole number)

29 + 37 = 66 (66 is also a whole number)

Commutative Property:

If a and b are any two whole numbers, then a + b = b + a.

In other words, the sum of two whole numbers remains the same even if the order of whole numbers (called addends) is changed.

Verification: In order to verify this property, let us consider some pairs of whole numbers and add them in two different orders. We find that the sum remains the same as shown below:

9 + 3 = 3 + 9

13 + 25 = 25 + 13

0 + 32 = 32 + 0

If a is any whole number, then

a + 0 = a = 0 + a

In other words, the sum of any whole number and zero is the number itself. That is, zero is the only whole number that does not change the value (identity) of the number it is added to.

The whole number 0 (zero) is called the additive identity or the identity element for addition of whole numbers.

Verification: In order to verify this property, we take any whole number and add it to zero. We find that the sum is the whole number itself as shown below:

5 + 0 = 5 = 0 + 5

27 + 0 = 27 = 0 + 27

137 + 0 = 137 = 0 + 137

Note:

Zero is called the additive identity because it maintains or does not change the identity (value) of the numbers during the operation of addition.

Associativity:

If a, b, c are any three whole numbers, then

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

In other words, the addition of whole numbers is associative.

Verification: In order to verify this property, we take three whole numbers, say a, b, c and find the values of the expression (a + b) + c and a + (b + c). We find that the values of these expression remain same, as shown below;

(i) (2 + 5) + 7 = 2 + (5 + 7)

then, 7 + 7 = 2 + 12

14 = 14

(ii) (5 + 10) + 13 = 5 + (10 + 13)

then, 15 + 13 = 5 + 23

28 = 28

(iii) (9 + 0) + 11 = 9 + (0 + 11)

then, 9 + 11 = 9 + 11

20 = 20

Let us consider any three whole numbers a, b, c.

We have, (a + b) + c

= (b + a) + c [By using commutativity of addition we have a + b = b + a]

= b + (a + c) [By using associativity of addition]

= b + (c + a) [By using commutativity of addition]

= (b + c) + a [By using associativity of addition]

= (c + b) + a [By using commutativity of addition]

Property of Opposites:

For any real number a, there is a unique real number –a such that

a + (–a) = 0 and (–a) + a = 0

The sum of the real number (a) and its opposite real number (-a) is zero then they are known as the additive inverses of each other.

Verification:

5 + (-5) = 0 and (-5) + 5 = 0

or, 5 - 5 = 0 and -5 + 5 = 0

Here 5 is real number and (-5) is it's opposite real number. Sum of 5 and (-5) is zero.

Therefore, (-5) is additive inverses of 5

or, 5 is additive inverses of (-5).

Property of Opposite of a Sum:

If a and b are any two whole numbers,then

–(a + b) = (–a) + (–b)

The opposite of the sum of whole numbers is equal to the sum of the opposites whole numbers.

Verification:

-(3 + 4) = (-3) + (-4)

or, -(7) = -3 -4

or, -7 = -7

Here the opposite of the sum of 3 and 4 is equal to -7.

The opposites of 3 and 4 are (-3) and (-4) respectively.

The sum of the opposites (-3) and (-4) is equal to -7.

Property of Successor of a Sum:

If a is any whole number, then

a + 1 = (a + 1), which is a successor of "a".

If we add 1 with the sum of a number, we will have successor of the number.

Verification:

2420 + 1 = 2421

2421 is the successor of 2420.

Similarly, 1 + 2542 = 2543

2543 is the successor of 2542.

The Number Zero

Properties of Whole Numbers

Successor and Predecessor

Representation of Whole Numbers on Number Line

Properties of Subtraction

Properties of Multiplication

Properties of Division

Division as The Inverse of Multiplication