Whole Numbers

Counting Numbers or Natural Numbers

The numbers 1, 2, 3, are called natural numbers or counting numbers.

Whole Numbers

Let us add one more number i.e., zero (0), to the collection of natural numbers. Now the numbers are 0, 1, 2,… These numbers are called whole numbers.

We can say that whole numbers consist of zero and the natural numbers. Therefore, except zero all the whole nos. are natural numbers.

Facts of Whole numbers

Ø The smallest natural number is 1.

Ø The number 0 is the first and the smallest whole numbers.

Ø There are infinitely many or uncountable number of whole-numbers.

Ø All natural numbers are whole-numbers.

Ø All whole-numbers are not natural numbers. For example, 0 is a whole-number but it is not a natural number.

The first 50 whole numbers are

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,

21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,

41, 42, 43, 44, 45, 46, 47, 48, 49, 50

Some other Important terms to remember

SUCCESSOR

PREDECESSOR

F The successor of a whole number is the number obtained by adding 1 to it.

F Clearly, the successor of 1 is 2; successor of 2 is 3; successor of 3 is 4 and so on.

F The predecessor of a whole number is one less than the given number.

F Clearly, the predecessor of 1 is 0; predecessor of 2 is 1; predecessor of 3 is 2 and so on.

F The whole number 0 does not have any predecessor.

Example

1) Write the successor of

(a) 10701 (b) 100499 (c) 5099999 (d) 5670

Solution:

a) 10702
b) 100500
c) 5100000
d) 5671

2) Write the predecessor of

(a) 14 (b) 100000 (c) 8090 (d) 4321

Solution:

a) 12
b) 99999
c) 8089
d)4320

Number Line

A line on which numbers are marked at equal intervals to show simple numerical operations is called a number line.

In order to represent whole numbers on a number line, we draw a straight line and mark a point O on it.

From ‘0’ point, mark points 1, 2, 3, 4, 5, 6, 7, 8, 9, etc. on the line at equal intervals to the right of ‘0’.

https://educationwithfun.com/pluginfile.php/1545/mod_page/intro/Number%20Line.PNG

The arrow-head on the right-side on the number line shows that the numbers can continue up to infinity. With the help of number line we can compare two whole numbers (i.e. we can easily find out which number is greater or smaller).

On the number line we see that the number 6 is on the right of 3.

Hence 6 is greater than 3 (i.e. 6 > 3). Number 1 lies on the left of 3.

Therefore 1 is smaller than 3 (i.e. 1 < 3).

 Addition on the number line

https://educationwithfun.com/pluginfile.php/1545/mod_page/intro/Addition%20On%20Number%20Line.PNG

Here, 3 is added to the given number i.e. 2, so we will make 3 jumps to the right of 2.

i.e. 1st jump – from 2 to 3,

     2nd jump – 3 to 4

     And 3rd jump – 4 to 5

Therefore, the sum of 2 and 3 is 5

 Subtraction on the number line

https://educationwithfun.com/pluginfile.php/1545/mod_page/intro/Subtraction%20On%20Number%20Line.PNG

Here, 2 is subtracted from the given number i.e. 6, so we will make 2 jumps to the left of 6.

i.e. 1st jump – from 6 to 5,

     And 2nd jump – 5 to 4

Therefore, we get 6 – 2 = 4

 Multiplication on the number line

https://educationwithfun.com/pluginfile.php/1545/mod_page/intro/Multiplication%20On%20Number%20Line.PNG

Start from 0, move 2 units at a time to the right, make such 3 moves. And we will reach 6.

So, we say, 2 × 3 = 6.

Properties of Whole Numbers

Closure Property

Closure property on Addition for Whole Number

0+2=2
1+3=4
5+6=11

So Whole number are closed on Addition

Closure property on Multiplication for Whole Number

0×2=0
1×4=4
5×1=5

So Whole number are closed on Multiplication

Closure property on subtraction of Whole number

5−0=5
0−5=?
1−3=?
3−1=2

So Whole number are not closed on Subtraction

Closure property on Division of Whole number

=2
=?
=0
=? (Division by Zero is undefined)

So Whole Number are not closed on Division

Closure Property

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

Commutative property

Commutativity property on Addition for Whole Number

0+2=2+0=2

So Whole number are Commutative on Addition

Commutativity property on Multiplication for Whole Number

0×2=0 or 2×0=0

So Whole number are Commutative on Multiplication

Commutativity property on subtraction of Whole number

5−0=5 but 0−5=?

So Whole number are not Commutative on Subtraction

Commutativity property on Division of Whole number

=2 but =?

So Whole Number are not Commutative on Division

You can add two whole numbers in any order. You can multiply two whole numbers in
any order.

Commutative property

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

 Associative property

Associativity property on Addition for Whole Number

0+(2+3)=(0+2)+3=5
1+(2+3)=6=(1+2)+3

So Whole number are Associative on Addition

Associativity property on Multiplication for Whole Number

0×(2×3)=0 or (0×2)×3=0

So Whole number are Associative on Multiplication

Associativity property on subtraction of Whole number

10−(2−1)=9 but (10−2)−1=7

So Whole number are not Associative on Subtraction

Associativity property on Division of Whole number

16÷(4÷2)=8 but (16÷4)÷2 =2

So Whole Number are not Associative on Division

If a, b and c are any two whole numbers, then (a+b)+c = a+(b+c) and (a×b)×c = a×(b×c).

Distributive property

If a, b and c are any two whole numbers, then a(b+c)=a×b+a×c

Additive Identity

If a is any whole number, then a+0=a=0+a.

Example

2+0=2
0+3=3
5+0=5

Multiplicative Identity

If a is any whole number, then a×1=a=1×a.

Example

1×1=1
5×1=5
6×1=6

Multiplication by zero

If a is any whole number, then a×0=0=0×a.

Example

1×0=0
5×0=0
0×0=0

Division by zero

If a is any whole number, then a ÷ 0 is not defined.