# Whole Numbers

Whole numbers are the collection of positive integers and zero. They are included in the real numbers that do not include fractions, decimals, or negative integers (numbers). However, ‘0’ (zero), the smallest whole number, is an undefined identity representing a null set or no result. We can represent the set as {0, 1, 2, 3, . . . .}.

Why do we need whole numbers?

A number is an idea that we use to represent that quantity which we count. Whole numbers are the numbers that represent any quantity as a whole. Therefore, it is easy for us to count any items or quantities in terms of whole units. So we need whole numbers for counting chocolates, gifts, or even to analyze our bills.

## Symbol

The symbol used 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, …

## Properties

### Closure Property

The sum and product of any 2 whole numbers is always a whole number.

⇒ a × b = c, and, a + b = c, here a, b, and c are different whole numbers.

Solve 2 + 3, and  5 × 2.

Solution:

As we know,
2 + 3 = 5
And
5 × 2 = 10

### Associative Property

The sum or product of any 3 whole numbers remains the same even if we change the order of numbers.

⇒ a + (b + c) = (a + b) + c = (a + c) + b, and, a × (b × c) = (a × b) × c = (a × c) × b

Demonstrate the associative property of 5 + (7 + 3)

Solution:

As we know, 5 + (7 + 3) = (5 + 7) + 3 = (5 + 3) + 7 = 15
Also, when we multiply the following numbers we get the same product irrespective of their order: 3 × (4 × 5) = (3 × 4) × 5 = (3 × 5) × 4 = 60.

### Commutative Property

The commutative property states that a change in the order of addition does not change the value of the sum. It means the sum and the product of any 2 whole numbers remain the same even if we interchange their order. This property also applies to multiplication. However, it does not apply to subtraction or division.

⇒ a + b = b + a, here a and b are 2 whole numbers.

Show the commutative property of 8 + 9.

Solution:

As we know,
8 + 9 = 17 = 9 + 8.

Show commutative property of 6 × 7

Solution:

As we know,
6 × 7 = 42 = 7 × 6

When we add a whole number to 0, we get the same value

⇒ a + 0 = a, here a is a whole number.

Demonstrate the additive identity of 14 + 0.

Solution:

As we know,
14 + 0 = 14

### Multiplicative Identity

When we multiply a whole with 1, its value remains unchanged.

⇒ a × 1 = a, here a is a whole number

Demonstrate the multiplicative Identity of 12 × 1

Solution:

As we know,
12 × 1 = 12

### Distributive Property

According to this property, the multiplication of a whole number is distributed over the sum of the whole numbers.

⇒ a × (b + c) = (a × b) + (a × c)

For example: a = 2, b = 3 and c = 4

⇒ 2 × (3 + 4) = 14

Also, (2 × 3) + (2 × 4) = 6 + 8 = 14.

The same property holds true for subtraction as well.

⇒ a × (b − c) = (a × b) − (a × c)

Demonstrate the distributive property of 8 × (7 − 3).

Solution:

As we know,
8 × (7 − 3) = 32, here a = 8, b = 7 and c = 3
Also, (8 × 7) − (8 × 3) = 56 − 24 = 32

As we know, ‘0’ is also a whole number; let us learn about the multiplication and division with zero.

Multiplication by zero: When a whole number is multiplied to 0, the result is always 0

⇒ a × 0 = 0 × a = 0. For example: 5 × 0 = 0

Division by zero: A whole number, when divided by 0, gives an undefined result.

⇒ a ÷ 0 = undefined

Show division by zero: 8/0

Solution:

As we know,
8/0 = undefined

Points to Remember

• Zero (0) is a whole number, but it is NOT a natural number.
• Negative numbers, fractions, and decimals are neither natural nor whole numbers unless simplified as natural numbers or whole numbers.

Can Whole Numbers be Negative?

Whole numbers only include positive integers and zero. Negative numbers are not included in whole numbers.

## Whole Numbers vs Natural Numbers

Although whole numbers and natural numbers mostly have common numbers in their respective sets, there are some precise differences. Let us learn them from the table below.

### Similarities

1. The common number set for whole numbers and natural numbers are 1, 2, 3, 4, 5 . . .∞
2. Both include all the positive integers (numbers) till infinity.

## Whole Numbers on the Number Line

We can represent the set of whole and natural numbers on a number line as given below. All the positive integers (integers on the right-hand side of 0) represent the natural numbers. All the positive integers including zero, represent the whole numbers.

The whole and natural numbers can be represented on the number line as follows: