Natural Numbers

Natural numbers are part of a numerical system, which includes all positive numbers from 1 to infinity. But, Zero (0) is not included. The natural numbers are 1,2,3,4,5,6,7,8,9…., and are also called as counting numbers.

Natural numbers are real numbers and include only positive numbers, i.e., 1, 2, 3, 4, 5, 6… except for zeros, fields, decimals and negative numbers.

Here, you will learn about some definitions and comparisons to other numbers, such as whole numbers, number lines, features, and more.

Natural Numbers Definition

These numbers are countable and are used for calculation purposes in general. The natural number set is represented by the “N” letter.

N = {1,2,3,4,5,6,7,8,9,10….…}

Natural Numbers and Whole Numbers

Real numbers include all numbers except 0. Thus, natural numbers are complete numbers, but not all numbers are natural numbers.

Whole numbers are natural numbers, but natural numbers are not all numbers.

  • Natural Numbers ={1,2,3,4,5,6,7,8,9…..}
  • Whole Numbers ={0,1,2,3,4,5,6,7,8,9……}

Discover the difference between whole numbers and integers to learn more about these two sets of numerical characteristics.

The above representation of the seats depicts two letters A ∩ B, i.e., Natural numbers and Whole numbers (0, 1, 2, 3, 4, 5, 6).

Therefore, an integer “contains all natural numbers 0. It is part of all the components. “

Is ‘0’ a Natural Numbers?

‘No.’ We know that natural numbers start at 1 to infinity and are positive numbers. But when we associate 0 with positive integers like 10, 20, etc., it becomes a natural number. Hence, 0 is a null-valued integer.

Representing a natural number on a number line

The representation of natural numbers on the number line is as follows.

The above number is a natural number and a whole number. All the numbers represent the actual number to the right of 0, thus creating an infinite number set. These numbers become real numbers and are often infinite numbers, where 0 is added.

Set of Natural Numbers

The natural number symbol in a given notation is “N” and is represented as shown below.


N = Set of all numbers with an initial value of 1.

In Roster Form:

N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

In Set Builder Form:

N={x: x, it is an integer starting from 1}

Natural Numbers Example

Natural numbers (also known as non-negative numbers) sometimes include numbers, such as 1, 2, 3, 4, 5 and 6. In other words, the natural number is a sum of all the numbers except 0. Examples of all natural numbers are 23, 56, 78, 999, 100202, and so on.

Properties of Natural Numbers

The properties of natural numbers are split into four key characteristics, such as:

  • Clouse property
  • Commutative property
  • Associative property
  • Distributive property

Closure Property

Natural numbers are always closed by addition and multiplication. Natural numbers is the result of the addition and multiplication of two or more real numbers. Real numbers do not correspond to the closing property in the case of subtraction and division.  It means that subtraction or division of two natural numbers is not actually a natural number.

Let’s consider some examples. 

  • Addition: 1+2 =3, 3+4=7, etc. In any of these, a natural number is always a result.
  • Multiplication: 2*3=6, 5*4=20, etc. In this case, a natural number could be the result.
  • Subtraction: 9-5=4, 3-5= -2, etc. In this case, a natural number could be the result.
  • Division: 10%5 =2, 10%3 =3.33, etc. In this case, too, the resulting number could not have been a natural number.

For Example:

  • -3*4 = -12; This is not a natural number
  • 8/-2 = -4; This is not a natural number

Associative Property

The associative property is valid in the case of addition and multiplication of natural numbers, i.e., a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c. On the other hand, the property attribute is not correct in distinguishing and distributing natural numbers. Below is an example of this.

  • Addition: a+ (b+c) = (a+b) +c => 3+ (13+1) =17 and (3+13) +1=17
  • Multiplication: a*(b*c) = (a+b) +c=> 3*(12*1) = 36 and (3*12)*1=36
  • Subtraction: a-(b-c) ≠ (a-b) – c=> 2-(15-1) = -12 and (2-15)-1= -14
  • Division: a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c => 2 ÷(3 ÷ 6) = 4 and (2 ÷ 3) ÷ 6 = 0.11

Commutative Property

For commutative property

  • The commutative property is indicated by the addition and multiplication of natural numbers.

For example, x + y = y + x and a × b = b × a

  • Subtracting the number of natural numbers and dividing does not commutative the property. 

For example, x – y ≠ y – x and x ÷ y ≠ y ÷ x

Distributive Property

  • In addition, the multiplication of natural numbers is always divisible. 

For example, a * (b + c) = ab + ac

  • The multiplication of natural numbers is also divided by subtraction. 

For example, a *(b – c) = ab – ac

Operation with Natural Numbers

The following table shows an overview of the algebraic operations on natural numbers, i.e., addition, subtraction, multiplication and division, and other related properties.

Properties and operation on Natural Numbers
OperationClouse PropertyCommutative PropertyAssociative Property