All numbers which we generally use for counting like 1, 2, 3, 4 etc. are known as ‘Natural numbers’, because these numbers come naturally when we start to count.
If number 1 is added to any natural number then we can get the next number. This next number is called its “successor”
For example,
12 is a natural number and when 1 is added to 12 we get 13 (12+1=13). So here 13 is the successor of 12. Similarly the successor of 13 is 14 (13+1=14) and so on.
If number 1 is subtracted from any natural number then we get the previous number. This previous number is called its “predecessor”
For example,
12 is a natural number and when 1 is subtracted from 12 we get 11 (12-1=11). So here 11 is the predecessor of 12. Similarly the predecessor of 13 is 12 (13-1=12) and so on.
The population in our village can be counted; even the population in our country can be counted. But, it is difficult to count the stars in the sky. If we are able to count the stars there is a number for that also. If 1 is added to that number, it will be a larger number than original number. Hence we can say that “Every natural number has a successor”. Also “There is no largest natural number”.
There is no predecessor for number 1 in Natural numbers, but we can add ‘0’ as the predecessor for 1. When 0 is added to the set of natural numbers, this becomes set of whole numbers. In other words, the Natural numbers along with ‘0’ forms the collection of ‘Whole Numbers’.
The whole numbers starts from zero and continues like 0, 1, 2, 3…. and so on. Just like natural numbers, there is no largest whole number.
Let us consider the following example,
Which number is at the farthest left when we mark 15, 6 and 9 on a number line?
From the number line, we can see that number 6 is on the farthest left.
Which number is at the farthest right when we mark 1005 and 9756 on a number line?
From the number line, we can see that number 9756 is on the farthest right.
Addition of whole numbers is shown in the above image on the number line. Let us consider the addition of 3 and 4 on a number line,
Subtraction of whole numbers is shown in the above image on the number line. Let us consider the subtraction of 6 and 8 on a number line,
Multiplication of whole numbers is shown in the above image on the number line. Let us consider the multiplication of 2 and 3 on a number line.
We generally perform four basic operations on the whole numbers. They are Addition, Subtraction, Multiplication and Division. Now let us study the properties of these operations on whole numbers.
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.
For Addition and Multiplication
Whole numbers are closed only under ADDITION and MULTIPLICATION.
Consider the following examples,
For Subtraction and Division
Whole numbers are not closed under SUBTRACTION and DIVISION.
Consider the following examples,
*Division by zero: Division of any whole number by zero is not defined.
For Addition and Multiplication
Let us consider the addition of 4+6 and 6+4 on a number line, we can find that the result is the same in both cases.
So, we can add two whole numbers in any order.
This property is known as commutative property of addition for whole numbers.
Let us consider the multiplication of 2×3 and 3×2, whatever may be the order of multiplication the result will be the same.
This is known as commutative property of multiplication for the whole numbers.
Thus we can say that ADDITION and MULTIPLICATION are commutative for whole numbers.
For subtraction and division:
Let us consider the subtraction 10-7 and 7-10, we can find that the result is not the same in both cases,
Again consider division of 15/5 and 5/15, here also the result is not the same in both cases.
Hence from the above cases we can say SUBTRACTION and DIVISION are not commutative for whole numbers.
For addition and multiplication
Let us take the addition (1+2) +3= 3+3 = 6
Now consider let us change the order of addition,
1+ (2+3) = 1 + 5 = 6
In both the cases the result is the same. This is called associative property of addition for the whole numbers.
Let us take the multiplication (1×2) ×3 = 2 × 3 = 6
Now let us change the order of multiplication
1× (2×3) =1×6 =6
Again in both the result is the same. This is called associative property of multiplication for the whole numbers.
Example: Find 10+15+20
This can be written as (10+15) + 20 =45 or 10 + (15 +20) =45
This represents that ADDITION and MULTIPLICATION are associative for whole numbers.
For division
Let us consider the division (16/4)/2=4/2=2
Now let us change the order of division
16 / (4/2) = 16/2=8
The result is not same in both the cases. Hence we can say that DIVISION is not associative for whole numbers.
For multiplication over addition
Let us consider the example of 4 × (5+8)
4 × (5+8) =4× (13) =52
Now take (4×5) + (4×8) = 20+32 =52.
We find in both the cases, the result is the same i.e. 4× (5+8) = (4×5) + (4×8) =52
Hence if we take any three whole numbers for example, a, b, c we have
a× (b + c) = (a × b) + (a × c).
This is known as distributive property of multiplication over addition.
Example: A hotel charges Rs.30 for lunch and Rs.10 for milk each day. How much money do you spend in 10 days on these things?
Solution: Add amount for milk for 10 days and amount for lunch for 10 days.
Cost of lunch = 10×30 = Rs.300
Cost of milk = 10×10 = Rs.100
Total cost = Rs. (300+100) = Rs.400
This example shows that 10× (30+10) = (10×30) + (10×10)
The difference between the collection of whole numbers and natural numbers is the presence of zero.
Let us add zero to whole number,
1+0=1
2+0=2
5+0=0
The same whole number comes again. Hence zero is called the additive identity of whole numbers.
Now let us consider the multiplication of whole numbers with zero. In multiplication any whole number multiplied by zero the result is zero.
1×0=0
2×0=0
5×0=0
Let us take the multiplication of whole numbers with one,
1×1=1
2×1=2
5×1=5
We can observe that whatever may be the whole number when multiplied with one, the result is the same whole number. Hence we can say that one is the multiplicative identity of whole numbers.
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