Definition properties and examples of natural numbers
The concept of natural numbers plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding what natural numbers are helps students master counting, number properties, and solve day-to-day problems quickly and accurately. This page breaks down the definition, properties, solved examples, and common questions about what are natural numbers for easy learning and fast revision.
What Is a Natural Number?
Natural numbers are positive integers that start from 1 and go on without end. In simple words, any number you use for counting objects—like 1, 2, 3, 4, 5, etc.—is a natural number. These are sometimes called counting numbers. The set of natural numbers is usually shown by the symbol N, so N = {1, 2, 3, 4, ...}. In everyday situations such as counting books, measuring time, or numbering days, you are using natural numbers.
Natural Numbers on the Number Line
You can easily spot natural numbers on a number line. They start from 1 and continue rightwards: 1 → 2 → 3 → 4 → 5 → ... Each step on the number line is a natural number. Numbers like 0, negative numbers, fractions, or decimals do not count as natural numbers because you cannot use them for counting whole objects.
Properties of Natural Numbers
| Property | Meaning & Example |
|---|---|
| Closure | Addition and multiplication of any two natural numbers will always give another natural number. Example: 2 + 3 = 5, 2 × 4 = 8 |
| Commutative | Changing the order in addition or multiplication does not change the answer. Example: 3 + 5 = 5 + 3, 2 × 7 = 7 × 2 |
| Associative | Grouping of numbers does not affect sum or product. Example: (1 + 2) + 3 = 1 + (2 + 3) |
| Distributive | Multiplication distributes over addition. Example: 2 × (3 + 4) = 2 × 3 + 2 × 4 |
| Identity | There is no natural number (except 0, which is not natural here) that can be added to a natural number to leave it unchanged; likewise for multiplication, 1 is the identity. Example: 5 × 1 = 5 |
Difference Between Natural and Whole Numbers
| Natural Numbers | Whole Numbers |
|---|---|
| Start from 1 | Start from 0 |
| N = {1, 2, 3, ...} | W = {0, 1, 2, 3, ...} |
| Do NOT include 0 | Include 0 |
Students often confuse zero as a natural number, but it is included only in whole numbers, not in natural numbers.
Examples and Non-Examples of Natural Numbers
| Natural Numbers (✓) | Not Natural Numbers (✗) |
|---|---|
| 1, 2, 3, 4, 7, 15, 28, 100, 205, 1820 | 0, -3, 2.5, 9/2, -1, 1/2, -7 |
Use Cases of Natural Numbers
Natural numbers appear everywhere in daily life and maths:
- Counting objects (e.g., 12 apples)
- Ranking positions (e.g., 1st, 2nd, 3rd place)
- Scoring in games
- Numbering chapters or exercises
- Marking days on a calendar
Solved Examples: Natural Numbers
- Write the first ten natural numbers.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10 - Is 0 a natural number?
No, 0 is not included in natural numbers. - Identify the natural numbers among: -2, 7, 0, 3.5, 1
7 and 1 are natural numbers. - List all natural numbers between 10 and 15.
11, 12, 13, 14, 15 - Is 23 a natural number?
Yes, 23 is a natural number because it’s a positive integer used for counting.
Try These Yourself
- Write the first five natural numbers.
- Check if 48 is a natural number.
- Find all natural numbers between 30 and 60.
- Identify non-natural numbers from the list: 12, 15, 18, 0.5, -7.
Frequent Errors and Misunderstandings
- Mistaking 0 or negative numbers as natural numbers
- Thinking decimals or fractions can be natural numbers (like 2.5 or 9/2)
Relation to Other Number Sets
Mastering natural numbers helps in understanding whole numbers, integers, and even more advanced topics like real numbers and prime numbers. Many properties, such as closure or commutativity, are best understood by first learning about natural numbers. Practice with solved examples and quizzes boosts your clarity for any class or exam.
Classroom Tip
An easy trick to remember: natural numbers are the numbers we naturally use to count things, and they always start from 1. Vedantu teachers often use number lines and real-life examples in live classes to help students remember this quickly and avoid confusion in exams.
We explored natural numbers—definition, key properties, solved examples, and tips to avoid mistakes. To get even more confident, try Vedantu’s free live sessions or check out their full set of solved NCERT solutions for Class 6 Maths Chapter 1 and factors and multiples explained.
FAQs on What Are Natural Numbers in Mathematics
1. What are natural numbers?
Natural numbers are the counting numbers starting from 1 and increasing by 1 each time. They are used for counting objects and are written as:
- N = {1, 2, 3, 4, 5, ...}
2. Do natural numbers include 0?
In most school-level definitions, natural numbers start from 1, but in some mathematical contexts they include 0. There are two common conventions:
- N = {1, 2, 3, ...} (without 0)
- N = {0, 1, 2, 3, ...} (with 0)
3. What is the difference between natural numbers and whole numbers?
The main difference is that whole numbers include 0, while natural numbers usually start from 1.
- Natural numbers: {1, 2, 3, 4, ...}
- Whole numbers: {0, 1, 2, 3, 4, ...}
4. What is the symbol for natural numbers?
The symbol used for natural numbers is N. It is commonly written as:
- N = {1, 2, 3, ...}
5. What are the properties of natural numbers?
Natural numbers follow important arithmetic properties such as closure, commutative, and associative properties. Key properties include:
- Closure: The sum and product of natural numbers are natural numbers.
- Commutative: a + b = b + a and a × b = b × a
- Associative: (a + b) + c = a + (b + c)
- Distributive: a × (b + c) = ab + ac
6. Are negative numbers natural numbers?
No, negative numbers are not natural numbers. Natural numbers are positive counting numbers such as 1, 2, 3, .... Numbers like -1, -5, -10 belong to the set of integers, not natural numbers.
7. What is the smallest natural number?
The smallest natural number is usually 1 in standard school mathematics. If 0 is included in the definition being used, then 0 becomes the smallest natural number. Always confirm which definition applies.
8. How do you represent natural numbers on a number line?
Natural numbers are represented on a number line as equally spaced points starting from 1 (or 0 in some definitions) and moving to the right. Steps:
- Draw a horizontal line.
- Mark equal intervals.
- Label them as 1, 2, 3, 4, ...
9. What is an example of natural numbers in real life?
Natural numbers are used in real life for counting objects. For example:
- 3 apples
- 10 students in a class
- 25 books on a shelf
10. What is the sum of the first n natural numbers?
The sum of the first n natural numbers is given by the formula n(n + 1) / 2. For example:
- Sum of first 5 natural numbers = 5(5 + 1)/2
- = 5 × 6 / 2
- = 15
