Properties of whole number operations.
One day your mother asked you and your brother to decorate your room.
So, both of you go to a stationery shop to buy some colored wallpapers.
You bought
$4$
blue patterns and your brother bought
$3$
red patterns.
Your mother wants all the paper pieces on the walls. So she asks you about the total number of sheets you both bought.
Your brother added all the sheets and got the total number of sheets as
$7$
.
Friends, here you can observe that he added both the numbers which were whole numbers and the result was also a whole number.
Your brother then tells you that these are very interesting properties of whole numbers.
Letâ€™s discuss the properties of whole numbers.
We define whole numbers as-
Now, letâ€™s try to add two whole numbers
$2$
and
$3$
.
Here, from the addition of these whole numbers, we get another whole number.
We can observe that when we add two whole numbers we get another whole number.
Therefore, when we add two whole numbers the sum is always a whole number and this is the closure property for the addition of whole numbers.
Now, letâ€™s multiply two whole numbers and find the result.
When
$2$
is multiplied with
$4$
, we get
$8$
, which is also a whole number.
Here, we can observe that when we multiply two whole numbers, we get another whole number as the result.
So, we can say that whole numbers are closed under multiplication.
Now, letâ€™s pick two whole numbers from the number line.
Here, we add these numbers in different orders and find the result of the addition.
We can observe that we get the same result from both orders.
So, we can say that the addition is commutative for whole numbers and this property is known as commutativity for addition.
The general form of commutative property is shown above.
Similarly, we can say that whole numbers are commutative under multiplication.
The general form of commutative property under multiplication is given here.
Suppose, we pick numbers 3, 4, and 5 from the number line.
Letâ€™s consider two cases where we add
$3$
and
$4$
first and then add
$5$
to the sum.
And in the other case we add
$4$
and
$5$
first and then add
$3$
to the sum.
Here, we can observe that in both cases we get the same result, that is,
$12$
.
This property of whole numbers is known as associative property for addition.
The general form of associative property is shown above.
Now, letâ€™s multiply the
$3$
numbers that we have picked from the number line.
Performing multiplication by the methods above will give us the same result.
So, this is called the associative property for multiplication of whole numbers.
Letâ€™s discuss few more properties of whole numbers.
The distributive property allows us to get rid of brackets by multiplying each element inside the bracket with the one outside.
The general form of distributive property under addition is shown here.
Letâ€™s take three whole numbers 2, 3, and 4 to verify distributivity under addition.
First, we solve the left-hand side as shown above.
Then, we solve the right-hand side and observe that both sides are equal.
Revision
Whenever we add or multiply two whole numbers, the result is always a whole number and, it is the closure property of whole numbers.
The commutative property of whole numbers is shown here.
The associative property of whole numbers is shown here.
The distributive property of whole numbers is shown here.
The End