Natural Numbers

Normal numbers are a component of the number structure that contains all the positive integers from 1 till infinity. It is an integer that is often larger than zero(0). This should be remembered that the natural numbers only contain the positive entities i.e. group of all counting numbers such as 1, 2, 3, …… The percentages, decimals, and negative numbers are omitted. For real numbers, natural figures are composed.

Note: The real numbers are rarely negative numbers or empty.

In this article, you’ll learn more about natural numbers in terms of their definition, comparison with whole numbers, number line representation, properties etc.

Natural Number Definition

Natural numbers are, as explained in the introduction part, the numbers that are positive in nature and include numbers from 1 to infinity (almost). These numbers are countable and are usually used for the purpose of measurement. The set of natural numbers is seen in the letter “N”

Natural Numbers and Whole Numbers

Natural numbers include all the whole numbers excluding the number 0. In other words, all natural numbers are whole numbers, but all whole numbers are not natural numbers. Check out the difference between natural and whole numbers to know more about the differentiating properties of these two sets of numbers.

The above representation of sets shows two regions,

A ∩ B ie. intersection of natural numbers and whole numbers (1, 2, 3, 4, 5, 6, ……..) and the green region showing A-B, i.e. part of the whole number (0).

Thus, a whole number is “a part of Integers consisting of all the Natural number including 0.

Is ‘0’ a Natural Number?

‘No’ is the answer to this query. Natural numbers, as we already know, range from 1 to infinity, and are optimistic in nature. But it becomes a natural number when we mention 0 with a positive integer such as 10, 20 etc. In addition, 0 is an integer that has a null value.

Representing Natural Numbers on a Number Line

Natural numbers representation on a number line is as follows:

The line of numbers above reflects real numbers and all numbers on a line of numbers. All of the integers on the right side of 0 represent the natural numbers, forming an infinite number set. Such numbers are entire numbers when 0 is added and are therefore an endless series of numbers.

Set of Natural Numbers

In set notation, the symbol of natural number is “N” and it is represented as given below.

Statement:

N = Set of all numbers starting from 1.

In Roster Form:

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

In Set Builder Form:

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

Natural Numbers Examples

The natural numbers include the positive integers (also known as non-negative integers) and a few examples include 1, 2, 3, 4, 5, 6, ..∞.  In other words, natural numbers are a set of all the whole numbers excluding 0.

Properties of Natural Numbers

Natural numbers properties are segregated into four main properties which include closure property, commutative property, associative property, and distributive. Each of these properties is explained below in detail.

Closure Property

The natural numbers are always closed under addition and multiplication i.e. the addition and multiplication of two or more natural numbers will always yield a natural number. In the case of subtraction and division, natural numbers are not closed which means subtracting or dividing two natural numbers might not give a natural number as a result.

  • Addition: 1 + 2 = 3, 3 + 4 = 7, etc. In each of these cases, the resulting number is always a natural number.
  • Multiplication: 2 × 3 = 6, 5 × 4 = 20, etc. In this case also, the resultant is always a natural number.
  • Subtraction: 9 – 5 = 4, 3 – 5 = -2, etc. In this case, the result may or may not be a natural number.
  • Division: 10 ÷ 5 = 2, 10 ÷ 3 = 3.33, etc. In this case also, the resultant number may or may not be a natural number.

Note: Closure property does not hold, if any of the number in case of multiplication and division, is not a natural number. But for addition and subtraction, if the result is a positive number, then only closure property exist.

For example: 

  • -2 x 3 = -6; Not a natural number
  • 6/-2 = -3; Not a natural number

Associative Property

The associative property holds true in 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, for subtraction and division of natural numbers, the associative property does not hold true. An example of this is given below.

  • Addition: a + ( b + c ) = ( a + b ) + c => 3 + (15 + 1 ) = 19 and (3 + 15 ) + 1 = 19.
  • Multiplication: a × ( b × c ) = ( a × b ) × c => 3 × (15 × 1 ) = 45 and ( 3 × 15 ) × 1 = 45.
  • Subtraction: a – ( b – c ) ≠ ( a – b ) – c => 2 – (15 – 1 ) = – 12 and ( 2 – 15 ) – 1 = – 14.
  • Disivion: a ÷ ( b ÷ c ) ≠ ( a ÷ b ) ÷ c => 2 ÷( 3 ÷ 6 ) = 4 and ( 2 ÷ 3 ) ÷ 6 = 0.11.

Commutative Property

  • Addition and multiplication of natural numbers show the commutative property. For example, x + y = y + x and a × b = b × a.
  • Subtraction and division of natural numbers does not show the commutative property. For example, x – y ≠ y – x and x ÷ y ≠ y ÷ x.

Distributive Property

  • Multiplication of natural numbers is always distributive over addition. For example, a × (b + c) = ab + ac.
  • Multiplication of natural numbers is also distributive over subtraction. For example, a × (b – c) = ab – ac.

Operations With Natural Numbers

An overview of algebraic operation with natural numbers i.e. addition, subtraction, multiplication, and division along with their respective properties are summarized in the table given below.

Properties and Operations on Natural Numbers
OperationClosure PropertyCommutative PropertyAssociative Property
AdditionYesYesYes
SubtractionNoNoNo
MultiplicationYesYesYes
DivisionNoNoNo

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