Properties of Whole Numbers (Additional)
When we look into various operations on numbers closely, we notice several properties of whole numbers. These properties help us to understand the numbers better. Moreover, they make calculations under certain operations very simple.
Do This
Let each one of you in the class take any two whole numbers and add them. Is the result always a whole number? Your additions may be like this:
| Whole Number | + | Whole Number | = | Result |
|---|---|---|---|---|
| 7 | + | 8 | = | |
| 5 | + | 5 | = | |
| 0 | + | 15 | = |
Is the sum always a whole number?
Did you find a pair of whole numbers whose sum is not a whole number?
Hence, we say that sum of any two whole numbers is a whole number i.e. the collection of whole numbers is closed under addition. This property is known as the closure property for addition of whole numbers.
Are the whole numbers closed under multiplication too? How will you check it?
Your multiplications may be like this :
| Whole Number | x | Whole Number | = | Result |
|---|---|---|---|---|
| 7 | x | 8 | = | |
| 5 | x | 5 | = | |
| 0 | x | 15 | = |
The multiplication of two whole numbers is also found to be a whole number again. We say that the system of whole numbers is closed under multiplication.
Closure property: Whole numbers are closed under addition and also under multiplication.When we say that whole numbers are "closed" under addition and multiplication, it means that if you add or multiply any two whole numbers, the answer will always be another whole number.
Think, discuss and write
- The whole numbers are not closed under subtraction. Why? Your subtractions may be like this :
| Whole Number | - | Whole Number | = | Result |
|---|---|---|---|---|
| 6 | - | 2 | = | |
| 7 | - | 8 | = | |
| 5 | - | 4 | = | |
| 3 | - | 9 | = |
- Are the whole numbers closed under division?
.
Observe this table :
| Whole Number | ÷ | Whole Number | = | Result |
|---|---|---|---|---|
| 8 | ÷ | 4 | = | |
| 5 | ÷ | 7 | = | |
| 12 | ÷ | 3 | = | |
| 6 | ÷ | 5 | = |
Division by zero
Division by a number means subtracting that number repeatedly.
Let us find 8 ÷ 2. Now, subtract 2 again and again from 8.
8 - 2 =
6 - 2 =
4 - 2 =
2 - 2 =
After how many moves did we reach 0?
So, we write 8 ÷ 2 = 4.
Using this, find 24 ÷ 8 =
Let us now try 2 ÷ 0.
2 - 0 =
2 - 0 =
2 - 0 =
and so on ...........
In every move we get 2 again! Will this ever stop?
We say, 2 ÷ 0 is not defined.
Check it for 5 ÷ 0, 16 ÷ 0.
This remains true for any whole number.
Thus, division of a whole number by 0 is not defined.
What is 2 + 3?
Try it out on the given below numberline.
In both the cases we reach 5.
What about 3 + 2?
So, 3 + 2 is same as 2 + 3.
Similarly, 5 + 3 is same as 3 + 5. (Try it on the above numberline)
Is this true when any two whole numbers are added?
You will not get any pair of whole numbers for which the sum is different when the order of addition is changed.
You can add two whole numbers in any order.
We say that addition is commutative for whole numbers. This property is known as commutativity for addition.
Let's take a multiplication example.
You have a small party at home. You want to arrange 6 rows of chairs with 8 chairs in each row for the visitors. The number of chairs you will need is 6 × 8. You find that the room is not wide enough to accommodate rows of 8 chairs. You decide to have 8 rows of chairs with 6 chairs in each row. How many chairs do you require now?
Will you require more number of chairs?
Is there a commutative property of multiplication?
Multiply numbers 4 and 5 in different orders. You will observe that 4 × 5 = 5 × 4. Is it true for the numbers 3 and 6; 5 and 7 also?
You can multiply two whole numbers in any order
We say multiplication is commutative for whole numbers. Thus, addition and multiplication are commutative for whole numbers.
Associativity of addition and multiplication
Observe the following equations:
(a) (2 + 3) + 4 =
(b) 2 + (3 + 4) = 2 +
In (a) above, you can add 2 and 3 first and then add 4 to the sum and in (b) you can add 3 and 4 first and then add 2 to the sum.
Are the results same?
We also have, (5 + 7) + 3 = 12 + 3 =
So, (5 + 7) + 3 = 5 + (7 + 3)
This is associativity of addition for whole numbers.
Let's check if it holds for multiplication as well. Take the whole numbers: 2,3 and 4. Now, find the product of these three numbers. The answer is
How did you multiply them? What was the arrangement of the numbers? As it turns out:
(2 × 3) × 4 = 2 × (3 × 4) (Irrespective of the order)
Similarly, you can see that (3 × 5) × 4 = 3 × (5 × 4) Try this for (5 × 6) × 2 and 5 × (6 × 2); (3 × 6) × 4 and 3 × (6 × 4). This is associative property for multiplication of whole numbers.
Think on and find:
Which is easier and why?
(a) (6 × 5) × 3 or 6 × (5 × 3) :
(b) (9 × 4) × 25 or 9 × (4 × 25):
Solution: We observe that it is easier to mentally calculate the products once we encounter a multiple of 5. Even better if a multiple of 10 is involved as all we need to do is to count the number of zeroes and add it to the original number. For example, in above problem (a):
30 × 3 = 90 where we only need to know the value of 3 × 3 and then add the total number of zeroes involved in the multiplication.
Try These
Find : (i) 7 + 18 + 13 (ii) 16 + 12 + 4
(i) 7 + 18 + 13 = (7 + 13) + 18 =
(ii) 16 + 12 + 4 = (16 + 4) + 12 =
Try These
Find : (i) 25 × 8358 × 4 (ii) 625 × 3759 × 8
(i) 25 × 8358 × 4 = (25 × 4) × 8358 =
625 × 3759 × 8 = (625 × 8) × 3759 =
Think, discuss and write
Is (16 ÷ 4) ÷ 2 = 16 ÷ (4 ÷ 2)?
Is there an associative property for division?
What about (28 ÷ 14) ÷ 2 ? and 28 ÷ (14 ÷ 2) ?
Distributivity of multiplication over addition
What is the total number of squares?
We have a total of ten squares. Now, let's split it into two as shown above. We get a 3 x 2 and a 2 x 2 grid.
The number of squares in 3 x 2 grid is
Is it (3 × 2) + (2 × 2)?
Does it mean that 2 × 5 = (2 × 3) + (2 × 2)?
But, 2 × 5 = 2 × (3 + 2)
This shows that: 2 × (3 + 2) = (2 × 3) + (2 × 2) Similarly, you will find that 2 × (3 + 5) = (2 × 3) + (2 × 5)
This is known as distributivity of multiplication over addition.
Observe the following multiplication and discuss whether we use here the idea of distributivity of multiplication over addition.
425 × 136 =
[2550 ← 425 × 6 (multiplication by 6 ones)] +
[12750 ← 425 × 30 (multiplication by 3 tens)] +
[42500 ← 425 × 100 (multiplication by 1 hundred)] +
= 57800 ← 425 × (6 + 30 + 100 )
Try These
Find the following using distributive property:
(i) 15 × 68 = 15 × (60 + 8) =
(ii) 17 × 23 = 17 × (20 + 3) =
(iii) 69 × 78 + 22 × 69 = 69 × (78 + 22) = 68 ×
Identity (for addition and multiplication)
How is the collection of whole numbers different from the collection of natural numbers? It is just the presence of 'zero' in the collection of whole numbers.
This number 'zero' has a special role in addition. The following table will help you guess the role.
When you add zero to any whole number what is the result?
It is the
Zero is called an identity for addition of whole numbers or additive identity for whole numbers.
Zero has a special role in multiplication too. Any number when multiplied by zero becomes
For example, observe the pattern :
5 × 6 = 30
5 × 5 = 25 Observe how the products
5 × 4 = 20 Do you see a pattern?
5 × 3 = 15 Can you guess the last step?
5 × 2 =
5 × 1 =
5 × 0 =
You came across an additive identity for whole numbers. A number remains unchanged when added to zero.
Similar is the case for a multiplicative identity for whole numbers. Observe this table.
| Whole Number | Identity | Product |
|---|---|---|
| 7 | × 1 | = |
| 5 | × 1 | = |
| 1 | × 12 | = |
| 1 | × 100 | = |
| 1 | × n | = |
You are right. 1 is the identity for multiplication of whole numbers or multiplicative identity for whole numbers.