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, . . . .}.
Whole Numbers
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
Additive Identity
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.
Difference between Whole Numbers and Natural Numbers
Whole Numbers
Natural Numbers
Whole Numbers has the set: {0, 1, 2, 3, 4, 5, …}
Natural numbers has the set: N = {1, 2, 3, 4, 5, …}
0 is the smallest whole number.
1 is the smallest natural number.
All whole numbers are not natural numbers as zero is excluded from the set
All natural numbers are whole numbers
Similarities
The common number set for whole numbers and natural numbers are 1, 2, 3, 4, 5 . . .∞
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:
