130 LearnersLast updated on July 22nd, 2025
Natural Numbers
The number system includes all positive numbers from 1 to infinity, which are known as natural numbers. Since they exclude zero and negative integers, natural numbers are often referred to as counting numbers. They are a subset of real numbers, consisting of only positive integers, and do not include all real numbers.
What Are Natural Numbers?
All numbers that begin at 1 and go on infinitely without ending are known as natural numbers. In our daily lives, these numbers play a significant role, as we use them for counting, handling money, measuring distances, and keeping track of time. Natural numbers are used in various activities, such as counting fruits, books, or toys. For example, we say 3 apples, 7 chairs, 2 pens, and so on.
Now, let’s quickly understand the characteristics of natural numbers:
Natural numbers are whole numbers, excluding fractions or decimals.
The natural numbers start from 1 and go on infinitely.
Each natural number increases by 1 from the previous number (e.g., 1, 2, 3, 4, 5,...).
Natural numbers are mainly used to count objects in real life.
Difference Between Natural Numbers and Whole Numbers
The main difference between the natural numbers and whole numbers is that natural numbers start from 1, while whole numbers include all natural numbers along with zero.
| Feature | Natural Numbers | Whole Numbers |
| Set Notation | N = {1, 2, 3,..} | W = {0,1, 2, ..} |
| Smallest Number | 1 | 0 |
| Includes Zero | No | Yes |
| Includes Fractions or Decimals | No | No |
| Includes Negative Numbers | No | No |
| Successor Property | Each number has a next number | Each number has a next number |
| Relation Between Sets | All natural numbers are whole numbers, but all whole numbers are not natural numbers. |
Whole numbers include all the natural numbers and zero. |
| Usage | Counting objects | Counting and representing nothing (zero) |
Types of Natural Numbers
Natural numbers can be classified into two types: odd natural numbers and even natural numbers.
- Odd natural numbers are greater than zero and are not divisible by 2, leaving a remainder of 1 when divided by 2. Examples include 1, 3, 5, 7, 9, 11, and so on.
- Even natural numbers are greater than zero and are divisible by 2. They can be written as 2n, where n is an integer. Examples include 2, 4, 6, 8, 10, and so on.
How to Represent Natural Numbers on a Number Line
Whole numbers and natural numbers can be represented on a number line. All positive integers to the right of 0 represent natural numbers, and whole numbers include 0 along with the natural numbers.
Properties of Natural Numbers
| Property | Definition | Example |
|
Closure Property |
The sum and product of two natural numbers are always natural numbers. This applies to addition and multiplication, but not to subtraction and division. |
Addition: 1 + 2 = 3, 7 + 8 = 15 Multiplication: 2 × 3 = 6, 7 × 8 = 56 15 and 56 are both natural numbers. |
| Associative Property |
The result of any three natural numbers remains, regardless of the order of grouping. This applies to addition and multiplication, but not to subtraction and division. |
Addition: 2 + (3 + 1) = (2 + 3) + 1 = 6 Multiplication: 2 × (3 × 1) = (2 × 3) × 1 = 6 |
|
Commutative Property |
The sum or product of any natural numbers remains the same after changing the order of the numbers. This applies to addition and multiplication, but not to subtraction and division. |
Addition: 8 + 9 = 9 + 8 = 17 Multiplication: 8 × 9 = 9 × 8 = 72 |
| Distributive Property |
The distributive law states that multiplication distributes over addition and subtraction: Addition: a × (b + c) = (a × b) + (a × c) Subtraction: a × (b - c) = (a × b) - (a × c). |
Addition: 2 × (3 + 4) = (2 × 3) + (2 × 4) = 6 + 8 = 14 Subtraction: 2 × (5 – 3) = (2 × 5) – (2 × 3) = 10 – 6 = 4 |
Operations With Natural Numbers
Basic mathematical operations can be operated on natural numbers. These operations help in solving real-life problems and understanding numerical relationships.
| Operation | Definition | Symbol | Example |
| Addition | Adds two or more numbers to get a total. | + | 3 + 4 = 7, 11 + 17 = 28 |
| Subtraction | Finds the difference between two numbers; the result may or may not be a natural number. | - | 5 − 3 = 2, 21 − 17 = 4 |
| Multiplication | Repeated addition of a number. | × or * |
3 × 4 = 12, 7 × 11 = 77 |
| Division |
Splits a number into equal parts; may result in a quotient with or without a remainder, and the result may not always be a natural number. |
÷ or / | 12 ÷ 3 = 4, 22 ÷ 11 = 2 |
| Exponentiation |
Raises a number to a specified power. | ^ | 2³ = 8 |
| Square Root | Finds the number which, when multiplied by itself, gives the original number. | √ | √25 = 5 |
| Factorial | The product of all positive integers up to and including a given number. | ! | 5! = 120 |
The sum of the squares of the first n natural numbers is given by:
S = n(n + 1) (2n + 1) / 6
Common Mistakes of Natural Numbers and How to Avoid Them
While learning about natural numbers, students sometimes make mistakes that can lead to confusion. Here are some of the most common errors and simple ways to avoid them
Mistake 1
Thinking 0 is a natural number.
Remember that natural numbers start from 1, unlike whole numbers, which include 0.
Mistake 2
Including fractions or decimals as natural numbers
Students sometimes include fractions and decimals as natural numbers, which is wrong. Start with 1 and include only whole numbers.
Mistake 3
Forgetting that natural numbers go on forever.
Students tend to think that the natural numbers end, but it is not true. Natural numbers form an infinite sequence starting from 1, excluding zero.
Mistake 4
Assuming negative numbers can be natural numbers
Know that natural numbers are always positive and do not include negative values.
Mistake 5
Mixing up natural numbers with whole numbers.
Students tend to confuse natural and whole numbers. Remember that natural numbers start from 1, while whole numbers start from 0.
Real Life Applications of Natural Numbers
Natural numbers play an important role in our daily lives, helping us count, measure, and organize information. Here are some common ways we use natural numbers in real-life situations.
- Counting Objects: We use natural numbers to count things like books, chairs, apples, or people.
- Money and Financial Transactions: When buying or selling items, prices and currency notes are represented using natural numbers.
- Timekeeping: Hours, minutes, and days are counted using natural numbers. For example, we count hours on a clock from 1 to 12.
- Measurement: Natural numbers help measure distances, heights, weights, and temperatures in whole units.
Solved Examples for Natural Numbers
Problem 1
What are the first five natural numbers?
The first five natural numbers are 1, 2, 3, 4, and 5
Explanation
The natural numbers are whole numbers excluding 0.
Problem 2
Is 7.5 a natural number?
No, 7.5 is not a natural number.
Explanation
As natural numbers do not include decimals or fractions, 7.5 is not a natural number.
Problem 3
Can a negative number be a natural number?
No, negative numbers are not natural numbers.
Explanation
Natural numbers include only the positive numbers from 1 to infinity.
Problem 4
What is the smallest natural number?
The smallest natural number is 1.
Explanation
As the natural numbers start from 1, 1 is the smallest natural number.
Problem 5
What comes after the natural number 99?
The next natural number after 99 is 100.
Explanation
Natural numbers follow a sequence where each number increases by 1, so after 99 comes 100.
FAQs Related to Natural Numbers
1.Can natural numbers be negative?
2.What is the largest natural number?
3.Do natural numbers include fractions and decimals?
4.Are all whole numbers also natural numbers?
5.Do natural numbers include 0?
6.How can children in Philippines use numbers in everyday life to understand Natural Numbers ?
7.What are some fun ways kids in Philippines can practice Natural Numbers with numbers?
8.What role do numbers and Natural Numbers play in helping children in Philippines develop problem-solving skills?
9.How can families in Philippines create number-rich environments to improve Natural Numbers skills?
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Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
