Natural Numbers (original) (raw)

Last Updated : 8 Oct, 2025

Natural numbers are the set of numbers that we use for counting and ordering things. They start from 1 and end at infinity.

Natural Numbers = {1, 2, 3, 4, 5, 6, … ∞}

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**Characteristics of Natural Numbers:

**Note: Some definitions (especially in computer science or set theory) include **0 as a natural number:Is Zero a Natural Number?

Types of Natural Numbers

Natural Numbers vs Whole Numbers

Natural Numbers Whole Numbers
The smallest natural number is 1. The smallest whole number is 0.
All natural numbers are whole numbers. All whole numbers are not natural numbers.
Representation of the set of natural numbers is N = {1, 2, 3, 4, ...} Representation of the set of whole numbers is W = {0, 1, 2, 3, ...}

Set of Natural Numbers

Set Form Explanation
Statement Form N = Set of numbers generated from 1.
Roaster Form N = {1, 2, 3, 4, 5, 6, ...}
Set-builder Form N = {x: x is a positive integer starting from 1}

The set of whole numbers is identical to the set of natural numbers, with the exception that it includes 0 as an extra number.

**W = {0, 1, 2, 3, 4, 5, ...} and **N = {1, 2, 3, 4, 5, ...}

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**Natural Numbers on Number Line

Natural numbers are represented by all positive integers or integers on the right-hand side of 0, whereas whole numbers are represented by all positive integers plus zero.

Natural Numbers on Number Line

Representation of Natural Numbers on Number Line

**Properties of Natural Numbers

All the natural numbers have these properties in common :

  1. Closure property
  2. Commutative property
  3. Associative property
  4. Distributive property

Let's learn about these properties in the table below.

**Property **Description **Example
**Closure Property
Addition Closure The sum of any two natural numbers is a natural number. 3 + 2 = 5, 9 + 8 = 17
Multiplication Closure The product of any two natural numbers is a natural number. 2 × 4 = 8, 7 × 8 = 56
**Associative Property
Associative Property of Addition Grouping of numbers does not change the sum. 1 + (3 + 5) = 9, (1 + 3) + 5 = 9
Associative Property of Multiplication Grouping of numbers does not change the product. 2 × (2 × 1) = 4, (2 × 2) × 1 = 4
**Commutative Property
Commutative Property of Addition The order of numbers does not change the sum. 4 + 5 = 9, 5 + 4 = 9
Commutative Property of Multiplication The order of numbers does not change the product. 3 × 2 = 6, 2 × 3 = 6
**Distributive Property
Multiplication over Addition Distributing multiplication over addition. a(b + c) = ab + ac
Multiplication over Subtraction Distributing multiplication over subtraction. a(b - c) = ab - ac

**Note:

Operations With Natural Numbers

We can add, subtract, multiply, and divide the natural numbers together, but the result of the subtraction and division is not always a natural number.

Let's understand the operations on natural numbers:

Operation Description Symbol Examples
**Addition Combines two or more numbers to find their total. + 3 + 4 = 7, 11 + 17 = 28
**Subtraction Finds the difference between two natural numbers; can result in natural or non-natural numbers. - 5 - 3 = 2, 17 - 21 = -4
**Multiplication Finds the value of repeated addition. × or * 3 × 4 = 12, 7 × 11 = 77
**Division Dividing the number into equal parts may result in a quotient and a remainder. ÷ or / 12 ÷ 3 = 4, 22 ÷ 11 = 2
**Exponentiation Raises a number to a certain power. ^ 23 = 8
**Square Root The value that, when multiplied by itself, gives the original number. √ √25 = 5
**Factorial The product of all positive integers up to and including that number. ! 5! = 120

Mean of First n Natural Numbers

As mean is defined as the ratio of the sum of observations to the number of total observations.

Mean Formula for the first **n terms of natural numbers:

**Mean = S/n = (n+1)/2

where,

Sum of Squares of First n Natural Numbers

The sum of the squares of the first n natural numbers is given as follows:

**S = n(n + 1)(2n + 1)/6

Where n is the number taken into consideration.

Solved Examples of Natural Numbers

Let's solve some example problems on Natural Numbers.

**Question 1: Identify the natural numbers among the given numbers: 23, 98, 0, -98, 12.7, 11/7, 3.

**Solution:

Since negative numbers, 0, decimals, and fractions are not a part of natural numbers.
Therefore, 0, -98, 12.7, and 11/7 are not natural numbers.
Thus, natural numbers are 23, 98, and 3.

**Question 2: Prove the distributive law of multiplication over addition with an example.

**Solution:

Distributive law of multiplication over addition states: a(b + c) = ab + ac

For example, 4(10 + 20), here 4, 10, and 20 are all natural numbers and hence must follow distributive law
4(10 + 20) = 4 × 10 + 4 × 20
4 × 30 = 40 + 80
120 = 120
Hence, proved.

**Question 3: Prove the distributive law of multiplication over subtraction with an example.

**Solution:

Distributive law of multiplication over addition states: a(b - c) = ab - ac.

For example, 7(3 - 6), here 7, 3, and 6 are all natural numbers and hence must follow the distributive law. Therefore,
7(3 - 6) = 7 × 3 - 7 × 6
7 × -3 = z1 - 42
-21 = -21
Hence, proved.

**Question 4: List the first 10 natural numbers.

**Solution:

1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 are the first ten natural numbers.

Natural Numbers Question

**Question 1: What is the Smallest Natural Number?

**Question 2: What is the Biggest Natural Number?

**Question 3: Simplify, 17(13 - 16)

**Question 4: Simplify, 11(9 - 2)

**Question 5: Find the sum of the first 20 natural numbers.

**Question 6: Is 97 a prime natural number?

**Question 7: What is the smallest natural number that is divisible by both 12 and 18?

**Question 8: Find the product of the first 5 natural numbers.

**Question 9: How many natural numbers are there between 50 and 100 (inclusive)?

**Question 10: Discuss whether 0 is included in the set of natural numbers based on its definition.

**Answer Key:

  1. **1
  2. **Not defined
  3. **-51
  4. **77
  5. **210
  6. **Yes
  7. **36
  8. **120
  9. **51
  10. **No