Whole Numbers
Whole numbers are a part of real numbers which
includes positive integers with 0. Whole numbers do not include negative
numbers, fractions, decimals or rational numbers. We see many numbers around us
such as to represent money, to tell time, to measure temperature, to measure
length, weight and capacity. In all these numbers, the counting numbers along
with 0 are whole numbers.
What are Whole Numbers?
When 0 is taken together with natural numbers, it forms a group of whole numbers.
We know that the natural numbers are
the set of counting
numbers starting from 1, 2, 3, … . On the other hand, the whole numbers are
the numbers which includes 0 along with the counting numbers. We can say that
the positive integers along with 0 are called whole numbers. Or the set of non-negative
integers are known as the whole numbers. The main difference between the whole
numbers and the natural
numbers is the presence of 0 in whole numbers.
Whole Numbers Definition
Whole numbers are defined as the group of all natural
numbers along with 0 is called a group of whole numbers. It is written as: 0, 1,
2, 3, 4, 5, 6, 7, 8, 9, ………. ∞
Examples of Whole Numbers
Some examples of whole numbers are as
follows:
0, 67, 251, 8632, 26931, 792102, 4967303,
……….
Whole Numbers Symbol
Whole numbers are denoted by the
symbol, W.
Set of Whole Numbers
The set of whole numbers is the
collection of all the counting numbers along with 0. The set of whole numbers
is denoted by the capital letter W. It is written as:
W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
………. ∞}
The set of whole numbers can be
written in three forms.
1. Description form: In description
form, the set of whole numbers is written as:
W = {all counting numbers along with 0}
2. Roaster form: In roaster form, the
set of whole numbers is written as:
W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …….}
3. Set builder form: In set builder
form, the set of whole numbers is written as:
W = {x : x is a non-negative integer}
Smallest and Largest Whole Numbers
The whole numbers start from 0. So,
the smallest whole number is 0. What is the largest whole number? Can you find
it?
Suppose I write a very large whole
number, for example, 4663094810434086542145698869732190867284684762549577120374618459624
Can you say this is the largest whole
number? Of course, no. Because there are many whole numbers greater than this
number. Hence, if you write a very large whole number, there are many numbers
larger than that number.
Thus, no largest whole number exists.
Difference
Between Whole Numbers and Natural Numbers
The
whole numbers are the set of natural numbers including 0. They are written as:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …..
The natural numbers are the set of
counting numbers starting from 1. They are written as: 1, 2, 3, 4, 5, 6, 7, 8, 9, …..
Whole Numbers |
Natural Numbers |
The set of whole numbers is denoted by W and written as: W = {0, 1, 2, 3, 4,
5, 6, 7, 8, 9, …….} |
The set of natural numbers is denoted by N and written as: N = {1, 2, 3, 4, 5,
6, 7, 8, 9, ……} |
The smallest whole number is 0. |
The smallest natural number is 1. |
All the natural numbers are whole numbers. |
All the whole numbers are natural numbers, except 0. |
Whole Numbers
on Number Line
Let us draw a ray and mark a point at its starting point. Label
this point as 0. Make some more points on it at equal distances, on the right
side of 0. Label these points as 1, 2, 3, 4, ... .
This gives you a number line for whole
numbers. The number line extends endlessly in one direction.
On the number line for whole numbers, no number is marked to the left of 0. Every number on the number line is greater than any number on its left side.
Properties of Whole Numbers
Closure Property
The sum and the product of two or more
whole numbers are always whole numbers. But the difference and the quotient of
two whole numbers are not always whole numbers. In general, a + b = c and a × b
= c. Where c is also a whole number.
Closure property
of addition: 5 + 9 = 14; 3 + 8
= 11; 50 + 35 = 85; 80 + 25 = 105, etc.
Thus, the whole numbers are closed
under addition.
Closure property
of multiplication: 5 × 7 = 35; 6 × 8
= 48; 8 × 9 = 72; 5 × 9 = 45, etc.
Thus, the whole numbers are closed
under addition and multiplication.
Closure property
of subtraction: Let us take a few
numbers from the group of numbers: 6, 9, 25, 7, 12, 18, 14 and subtract.
25 – 9 = 16
18 – 12 = 6
7 – 14 = ?
6 – 9 = ?
In the last two cases, the difference
is not a whole number. Thus, the whole numbers
are not closed under subtraction.
Closure property
of division: Take a few
numbers from the group of numbers: 5, 8, 20, 9, 16, 18, 12 and divide.
20 ÷ 5 = 4
16 ÷ 8 = 2
12 ÷ 9 = ?
8 ÷ 5 = ?
In the last two cases, the quotients
are not whole numbers. Therefore, the whole numbers are not closed under division.
Commutative Property
The sum and the product of two whole
numbers remains the same even if we change the order of the whole numbers. In
general, a + b = b + a and a × b = b × a.
Commutative
property of addition: 7 + 9 = 16
and 9 + 7 = 16. Thus, 7 + 9 = 9 + 7.
Commutative
property of multiplication: 6 × 4 = 24
and 4 × 6 = 24. Thus, 6 × 4 = 4 × 6.
Thus, the whole numbers are
commutative under addition and multiplication.
Commutative Property of Subtraction: Consider
the numbers 64 and 35.
We know that 64 − 35 = 29.
But 35 − 64 ≠ 29
So, 64 − 35 ≠ 35 − 64
Hence, we can say that commutative
property does not hold true for subtraction of whole numbers.
Commutative Property of Division: We
know 16 ÷ 4 = 4, but 4 ÷16 is a fraction.
So, 16 ÷ 4 ≠ 4 ÷16
Hence, we can say that commutative
property does not hold true for division.
Associative Property
The sum and the product of three whole
numbers remains the same even if we change the grouping of the numbers. In
general, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.
Associative
property of addition: 4 + (3 + 5)
= 4 + 8 = 12 and (4 + 3) + 5 = 7 + 5 = 12. Thus, 4 + (3 + 5) = (4 + 3) + 5.
Associative
property of multiplication: 5 × (2 × 6)
= 5 × 12 = 60 and (5 × 2) × 6 = 10 × 6 = 60. Thus, 5 × (2 × 6) = (5 × 2) × 6.
Thus, the whole number are associative
under addition and multiplication.
Associative Property of Subtraction: Let
us consider an example.
(15 − 6) − 4 = 9 − 4 = 5
But 15 − (6 − 4) = 15 − 2 = 13
So, (15 − 6) − 4 ≠ 15 − (6 − 4)
Verify this result using some other
numbers also. We can say that associative property is not satisfied for the
subtraction of whole numbers.
Associative Property of Division: We
know that (30 ÷ 6) ÷ 5 ≠ 30 ÷ (6 ÷ 5)
Verify this result with some other
numbers also.
Thus, we can say that the associative
property is not satisfied for the division of whole numbers.
Additive Identity
12 + 0 = 12
45 + 0 = 45
134 + 0 = 134
This is true for all the whole
numbers. If you add zero to any whole number, the value of the number will not
change. So, zero is called the additive identity for whole numbers.
Multiplicative Identity
8 × 1 = 8
67 × 1 = 67
156 × 1 = 156
This result is true for all whole numbers.
When a number is multiplied by 1, we
get the same number as the product. 1 is called the multiplicative identity for
whole numbers.
Distributive Property
According to the distributive property
of multiplication over addition and subtraction, a × (b + c) = a × b + a × c
and a × (b – c) = a × b – a × c.
Distributive
property of multiplication over addition:
6 × (3 + 5) = 6 × 3 + 6 × 5
6 × 8 = 18 + 30
48 = 48
Distributive
property of multiplication over subtraction:
4 × (7 – 3) = 4 × 7 – 4 × 3
4 × 4 = 28 – 12
16 = 16
Multiplication by Zero
23 × 0 = 0 and 0 × 23 = 0
We can see that 23 × 0 = 0 = 0 × 23.
It is true for all whole numbers.
When we multiply a whole number by 0,
the product is always 0.
Dividing 0 by a Whole Number
If 0 is divided by a whole
number, we get 0 as the quotient.
For example, 0 ÷ 5 = 0
Similarly, 0 ÷ 12 = 0 and 0 ÷ 100 = 0
and so on.
Dividing a Whole Number by 1
If a whole number is divided by 1, the
quotient is the whole number itself.
For example, 47 ÷ 1 = 47
Similarly, 146 ÷ 1 = 146 and 485 ÷ 1 =
485 and so on.
Division by Zero
Division by 0 is not possible. So,
division of a whole number by 0 is not defined.
Solved Examples
on Whole Numbers
Example 1: Fill in the blanks.
a. The smallest whole number is _____.
b. ______ is a whole number but not a natural
number.
c. The first 5 whole numbers are
_____________.
Solution:
a. The smallest whole number is 0.
b. 0 is a whole number but not
a natural number.
c. The first 5 whole numbers are 0,
1, 2, 3, 4.
Example 2: Identify the whole numbers in the following
numbers.
15, -4, 0, 7.2, 9.8, 1/3, 106, -5, 123,
545, ½, -3/8, 65, 4/7, 2156, 25007
Solution: Whole numbers are:
15, 0, 106, 123, 545, 65, 2156, 25007
Example 3: Which property holds true in the following
statement:
5 + (8 + 6) = (5 + 8) + 6
Solution: In the given statement, the associative
property of addition holds true.
Example 4: Which property holds true in the following
statement:
25 × 14 = 14 × 25
Solution: In the given statement, the commutative
property of multiplication holds true.