Whole Numbers
Whole Numbers are obtained when 0 is included with the
natural numbers. Understanding the role of zero and the
basic properties of whole numbers is important for Number System
questions in JSSC, SSC, Railway and other competitive examinations.
1. What are Whole Numbers?
The set containing 0 and all natural numbers is called
the set of Whole Numbers.
W = {0, 1, 2, 3, 4, 5, 6, ...}
Thus, 0, 1, 2, 3, 4, 5, ... are whole numbers.
Important:
Natural Numbers: N = {1, 2, 3, 4, ...}
Whole Numbers: W = {0, 1, 2, 3, 4, ...}
2. Relationship Between Natural and Whole Numbers
Every natural number is a whole number, but 0 is a whole number
that is not a natural number under the convention used in this
study material.
N ⊂ W
This means that the set of natural numbers is a subset of the set
of whole numbers.
| Natural Numbers |
Whole Numbers |
| {1, 2, 3, 4, ...} |
{0, 1, 2, 3, 4, ...} |
| Start from 1 |
Start from 0 |
| 0 is not included under our convention |
0 is included |
3. Smallest and Largest Whole Number
| Smallest Whole Number |
0 |
| Largest Whole Number |
Does not exist |
There is no largest whole number because for every whole number
n, the number n + 1 is a larger
whole number.
4. Whole Numbers on the Number Line
Whole numbers can be represented on a number line beginning from 0:
0 → 1 → 2 → 3
→ 4 → 5 → ...
As we move towards the right, the value of the numbers increases.
- 0 lies to the left of 1.
- 1 lies to the left of 2.
- If a < b, then a lies to the left of b.
The distance between two consecutive whole numbers on the number
line is one unit.
5. Successor of a Whole Number
The whole number immediately after a given whole number is called
its successor.
Successor of n = n + 1
Examples:
- Successor of 0 = 1
- Successor of 49 = 50
- Successor of 999 = 1000
Every whole number has a successor.
6. Predecessor of a Whole Number
The number immediately before a given number is called its
predecessor.
Predecessor of n = n − 1
For example:
- Predecessor of 10 = 9
- Predecessor of 1 = 0
- Predecessor of 1000 = 999
0 has no predecessor in the set of whole numbers.
Its predecessor would be −1, which is not a whole number.
7. Closure Property of Whole Numbers
Whole numbers are closed under an operation when applying that
operation to any two whole numbers always gives another whole number.
| Operation |
Closed? |
Example / Counterexample |
| Addition |
Yes |
7 + 5 = 12 |
| Multiplication |
Yes |
7 × 5 = 35 |
| Subtraction |
No |
5 − 8 = −3 |
| Division |
No |
5 ÷ 2 = 2.5 |
Exam Rule: Whole numbers are closed under
addition and multiplication, but not under
subtraction and division.
8. Commutative Property
Whole numbers satisfy the commutative property for addition and
multiplication.
Addition: a + b = b + a
Multiplication: a × b = b × a
Example:
6 + 9 = 9 + 6 = 15
6 × 9 = 9 × 6 = 54
Subtraction and division are not commutative.
9 − 4 ≠ 4 − 9
8 ÷ 2 ≠ 2 ÷ 8
9. Associative Property
Whole numbers are associative under addition and multiplication.
Addition:
(a + b) + c = a + (b + c)
Multiplication:
(a × b) × c = a × (b × c)
Example:
(2 + 5) + 7 = 2 + (5 + 7) = 14
(2 × 5) × 7 = 2 × (5 × 7) = 70
Subtraction and division are not associative.
10. Distributive Property
Multiplication is distributive over addition.
a × (b + c) = ab + ac
Example:
8 × (10 + 5)
= (8 × 10) + (8 × 5)
= 80 + 40
= 120
The algebraic distributive identity over subtraction is:
a × (b − c) = ab − ac
When discussing whole numbers specifically, remember that
b − c may not itself be a whole number if b < c.
The identity remains an algebraic rule, but closure under subtraction
does not hold for whole numbers.
11. Identity Elements in Whole Numbers
Additive Identity
Adding 0 to any whole number leaves the number unchanged.
a + 0 = 0 + a = a
Therefore, 0 is the additive identity for whole numbers.
Multiplicative Identity
Multiplying any whole number by 1 leaves the number unchanged.
a × 1 = 1 × a = a
Therefore, 1 is the multiplicative identity.
12. Important Properties of Zero
Zero plays an especially important role in whole numbers.
| a + 0 |
a |
| a − 0 |
a |
| 0 × a |
0 |
| 0 ÷ a |
0, provided a ≠ 0 |
| a ÷ 0 |
Not defined |
| 0 ÷ 0 |
Indeterminate / not defined as ordinary division |
Very Important: Division by zero is not defined.
Never write a ÷ 0 = 0.
13. Multiplication by Zero
The product of any whole number and zero is always zero.
a × 0 = 0 × a = 0
Examples:
784 × 0 = 0
0 × 99999 = 0
14. Comparing Whole Numbers
Whole numbers can be compared using the symbols
>, < and =.
- 27 > 19
- 8 < 15
- 25 = 25
- 0 < every positive whole number
On the number line, the number lying farther to the right is greater.
15. Consecutive Whole Numbers
Whole numbers occurring one after another are called
consecutive whole numbers.
If the first number is n, then three consecutive whole numbers are:
n, n + 1, n + 2
Examples:
0, 1, 2 are consecutive whole numbers.
48, 49, 50 are consecutive whole numbers.
16. Counting Whole Numbers in an Interval
For whole numbers a and b, where a ≤ b, the number of whole numbers
from a to b when both endpoints are included is:
b − a + 1
Example: How many whole numbers are there from
0 to 50, including both endpoints?
= 50 − 0 + 1
= 51
This is a common source of error: from 0 to n, inclusive, there are
n + 1 whole numbers, not n.
Solved Examples
Example 1. What is the smallest whole number?
The smallest whole number is 0.
Example 2. What is the predecessor of 1
in the set of whole numbers?
Predecessor of 1 = 1 − 1 = 0.
Example 3. Are whole numbers closed under
subtraction?
No. For example, 3 − 7 = −4 and −4 is
not a whole number.
Example 4. How many whole numbers are there
from 0 to 100, including both endpoints?
Number = 100 − 0 + 1
= 101.
Example 5. Which whole number has no
predecessor in the set of whole numbers?
0, because its predecessor would be −1,
which is not a whole number.
Important Exam Points
- Whole numbers are 0, 1, 2, 3, ...
- The smallest whole number is 0.
- There is no largest whole number.
- Every natural number is a whole number.
- 0 is a whole number but not a natural number under our convention.
- Every whole number has a successor.
- 0 has no predecessor within the set of whole numbers.
- Whole numbers are closed under addition and multiplication.
- They are not closed under subtraction and division.
- 0 is the additive identity.
- 1 is the multiplicative identity.
- Any whole number multiplied by 0 gives 0.
- Division by 0 is not defined.
- From 0 to n inclusive, there are n + 1 whole numbers.
Quick Revision
| Whole Numbers |
{0, 1, 2, 3, 4, ...} |
| Symbol |
W |
| Smallest Whole Number |
0 |
| Largest Whole Number |
Does not exist |
| Relationship |
N ⊂ W |
| Successor of n |
n + 1 |
| Predecessor of n |
n − 1, for n ≥ 1 |
| Additive Identity |
0 |
| Multiplicative Identity |
1 |
| Closed Under |
Addition and Multiplication |
| Not Closed Under |
Subtraction and Division |
| a ÷ 0 |
Not defined |
| Count from a to b, inclusive |
b − a + 1 |
Practice important previous-year and exam-oriented questions based on
whole numbers, zero, successor and predecessor, identity elements and
fundamental properties of whole numbers.
Solve these additional exam-oriented questions to strengthen the
fundamental concepts of whole numbers. Try each question before
revealing its answer.