2nd Lesson: Whole Numbers

• by Jay krishna Pathak
• May 18th, 2022
• Maths

Whole Numbers Definition

The whole numbers are the numbers without fractions and it is a collection of positive integers and zero. It is represented by the symbol “W” and the set of numbers are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,……………}. Zero as a whole represents nothing or a null value.

Whole Numbers: W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10……} Natural Numbers: N = {1, 2, 3, 4, 5, 6, 7, 8, 9,…} Integers: Z = {….-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,…} Counting Numbers: {1, 2, 3, 4, 5, 6, 7…}

These numbers are positive integers including zero and do not include fractional or decimal parts (3/4, 2.2 and 5.3 are not whole numbers). Also, arithmetic operations such as addition, subtraction, multiplication and division are possible on whole numbers.
 

Symbol

The symbol to represent whole numbers is the alphabet ‘W’ in capital letters.

W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10…}

Thus, the whole numbers list includes 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12…

Facts:

All the natural numbers are whole numbers All counting numbers are whole numbers All positive integers including zero are whole numbers All whole numbers are real numbers

If you still have doubt, what is a whole number in maths? A more comprehensive understanding of the whole numbers can be obtained from the following chart:

Whole Numbers Properties

The properties of whole numbers are based on arithmetic operations such as addition, subtraction, division and multiplication. Two whole numbers if added or multiplied will give a whole number itself. Subtraction of two whole numbers may not result in whole numbers, i.e. it can be an integer too. Also, the division of two whole numbers results in getting a fraction in some cases. Now, let us see some more properties of whole numbers and their proofs with the help of examples here.
 

Closure Property

They can be closed under addition and multiplication, i.e., if x and y are two whole numbers then x. y or x + y is also a whole number.

Example:

5 and 8 are whole numbers.

5 + 8 = 13; a whole number

5 × 8 = 40; a whole number

Therefore, the whole numbers are closed under addition and multiplication.
 

Commutative Property of Addition and Multiplication

The sum and product of two whole numbers will be the same whatever the order they are added or multiplied in, i.e., if x and y are two whole numbers, then x + y = y + x and x . y = y . x

Example:

Consider two whole numbers 3 and 7.

3 + 7 = 10

7 + 3 = 10

Thus, 3 + 7 = 7 + 3.

Also,

3 × 7 = 21

7 × 3 = 21

Thus, 3 × 7 = 7 × 3

Therefore, the whole numbers are commutative under addition and multiplication.
 

Additive identity

When a whole number is added to 0, its value remains unchanged, i.e., if x is a whole number then x + 0 = 0 + x = x

Example:

Consider two whole numbers 0 and 11.

0 + 11 = 11

11 + 0 = 11

Here, 0 + 11 = 11 + 0 = 11

Therefore, 0 is called the additive identity of whole numbers.
 

Multiplicative identity

When a whole number is multiplied by 1, its value remains unchanged, i.e., if x is a whole number then x.1 = x = 1.x

Example:

Consider two whole numbers 1 and 15.

1 × 15 = 15

15 × 1 = 15

Here, 1 × 15 = 15 = 15 × 1

Therefore, 1 is the multiplicative identity of whole numbers.
 

Associative Property

When whole numbers are being added or multiplied as a set, they can be grouped in any order, and the result will be the same, i.e. if x, y and z are whole numbers then x + (y + z) = (x + y) + z and x. (y.z)=(x.y).z

Example:

Consider three whole numbers 2, 3, and 4.

2 + (3 + 4) = 2 + 7 = 9

(2 + 3) + 4 = 5 + 4 = 9

Thus, 2 + (3 + 4) = (2 + 3) + 4 

2 × (3 × 4) = 2 × 12 = 24

(2 × 3) × 4 = 6 × 4 = 24

Here, 2 × (3 × 4) = (2 × 3) × 4

Therefore, the whole numbers are associative under addition and multiplication.
 

Distributive Property

If x, y and z are three whole numbers, the distributive property of multiplication over addition is x. (y + z) = (x.y) + (x.z), similarly, the distributive property of multiplication over subtraction is x. (y – z) = (x.y) – (x.z)

Example: 

Let us consider three whole numbers 9, 11 and 6.

9 × (11 + 6) = 9 × 17 = 153

(9 × 11) + (9 × 6) = 99 + 54 = 153

Here, 9 × (11 + 6) = (9 × 11) + (9 × 6)

Also,

9 × (11 – 6) = 9 × 5 = 45

(9 × 11) – (9 × 6) = 99 – 54 = 45

So, 9 × (11 – 6) = (9 × 11) – (9 × 6)

Hence, verified the distributive property of whole numbers.

Multiplication by zero

When a whole number is multiplied to 0, the result is always 0, i.e., x.0 = 0.x = 0

Example:

0 × 12 = 0

12 × 0 = 0

Here, 0 × 12 = 12 × 0 = 0

Thus, for any whole number multiplied by 0, the result is always 0.

Division by zero

The division of a whole number by o is not defined, i.e., if x is a whole number then x/0 is not defined.

Difference Between Whole Numbers and Natural Numbers

Difference Between Whole Numbers & Natural Numbers
Whole Numbers	Natural Numbers
Whole Numbers: {0, 1, 2, 3, 4, 5, 6…}	Natural Numbers: {1, 2, 3, 4, 5, 6 …}
Counting starts from 0	Counting starts from 1
All whole numbers are not natural numbers	All Natural numbers are whole numbers

 

The below figure will help us to understand the difference between the whole number and natural numbers:

Can Whole Numbers be negative?

The whole number can’t be negative!

As per definition: {0, 1, 2, 3, 4, 5, 6, 7 …till positive infinity} are whole numbers. There is no place for negative numbers.

Is 0 a whole number?

Whole numbers are the set of all the natural numbers including zero. So yes, 0 (zero) is not only a whole number but the first whole number.
 

Solved Examples

Example 1: Are 100, 227, 198, 4321 whole numbers?

Solution: Yes. 100, 227, 198, and 4321 are all whole numbers.

Example 2: Solve 10 × (5 + 10) using the distributive property.

Solution:  Distributive property of multiplication over the addition of whole numbers is:

x × (y + z) = (x × y) + (x × z)

10 × (5 + 10) = (10 × 5) + (10 × 10)

= 50 + 100

= 150

Therefore, 10 × (5 + 10) = 150

However, we can show several examples of whole numbers using the properties of the whole numbers.
 

Practice Problems

1. Write whole numbers between 12 and 25.
2. What is the additive inverse of the whole number 98?
3. How many whole numbers are there between -1 and 14?

 

Curriculum

• Whole Numbers

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