Natural numbers are a part of the number system, including all the positive integers from 1 to infinity. Natural numbers are also called counting numbers because they do not include zero or negative numbers. They are a part of real numbers including only the positive integers, but not zero, fractions, decimals, and negative numbers.

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 1 Introduction to Natural Numbers 2 What Are Natural Numbers? 3 Natural Numbers and Whole Numbers 4 Difference Between Natural Numbers and Whole Numbers 5 Natural Numbers on Number Line 6 Properties of Natural Numbers 7 FAQs on Natural Numbers

## Introduction to Natural Numbers

We see numbers everywhere around us, for counting objects, for representing or exchanging money, for measuring the temperature, telling the time, etc. These numbers that are used for counting objects are called “natural numbers”. For example, while counting objects, we say 5 cups, 6 books, 1 bottle, etc.

## What Are Natural Numbers?

Natural numbers refer to a set of all the whole numbers excluding 0. These numbers are significantly used in our day-to-day activities and speech.

### Natural Numbers Definition

Natural numbers are the numbers that are used for counting and are a part of real numbers. The set of natural numbers include only the positive integers, i.e., 1, 2, 3, 4, 5, 6, ……….∞.

### Examples of Natural Numbers

Natural numbers, also known as non-negative integers(all positive integers). Few examples include 23, 56, 78, 999, 100202, and so on.

### Set of Natural Numbers

A set is a collection of elements (numbers in this context). The set of natural numbers in Mathematics is written as {1,2,3,...}. The set of natural numbers is denoted by the symbol, N. N = {1,2,3,4,5,...∞}

 Statement Form N = Set of all numbers starting from 1. Roaster Form N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ………………………………} Set Builder Form N = {x : x is an integer starting from 1}

### Smallest Natural Number

The smallest natural number is 1. We know that the smallest element in N is 1 and that for every element in N, we can talk about the next element in terms of 1 and N (which is 1 more than that element). For example, two is one more than one, three is one more than two, and so on.

### Natural Numbers from 1 to 100

The natural numbers from 1 to 100 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99 and 100.

### Is 0 a Natural Number?

No, 0 is NOT a natural number because natural numbers are counting numbers. For counting any number of objects, we start counting from 1 and not from 0. ### Odd Natural Numbers

The odd natural numbers are the numbers that are odd and belong to the set N. So the set of odd natural numbers is {1,3,5,7,...}.

### Even Natural Numbers

The even natural numbers are the numbers that are even, exactly divisible by 2, and belong to the set N. So the set of even natural numbers is {2,4,6,8,...}.

The set of whole numbers is the same as the set of natural numbers, except that it includes an additional number which is 0. The set of whole numbers in Mathematics is written as {0,1,2,3,...}. It is denoted by the letter, W.

W = {0,1,2,3,4…}

From the above definitions, we can understand that every natural number is a whole number. Also, every whole number other than 0 is a natural number. We can say that the set of natural numbers is a subset of the set of whole numbers. Natural numbers are all positive numbers like 1, 2, 3, 4, and so on. They are the numbers you usually count and they continue till infinity. Whereas, the whole numbers are all natural numbers including 0, for example, 0, 1, 2, 3, 4, and so on. Integers include all whole numbers and their negative counterpart. e.g, -4, -3, -2, -1, 0,1, 2, 3, 4 and so on. The following table shows the difference between a natural number and a whole number.

Natural NumberWhole Number
The set of natural numbers is N= {1,2,3,...∞}The set of whole numbers is W={0,1,2,3,...}
The smallest natural number is 1.The smallest whole number is 0.
All natural numbers are whole numbers, but all whole numbers are not natural numbers.Each whole number is a natural number, except zero.

The set of natural numbers and whole numbers can be shown on the number line as given below. All the positive integers or the integers on the right-hand side of 0, represent the natural numbers, whereas, all the positive integers along with zero, represent the whole numbers. The four operations, addition, subtraction, multiplication, and division, on natural numbers, lead to four main properties of natural numbers as shown below:

Closure PropertyAssociative PropertyCommutative PropertyDistributive Property

### 1. Closure Property:

The sum and product of two natural numbers is always a natural number.

Closure Property of Addition: a+b=c ⇒ 1+2=3, 7+8=15. This shows that the sum of natural numbers is always a natural number.Closure Property of Multiplication: a×b=c ⇒ 2×3=6, 7×8=56, etc. This shows that the product of natural numbers is always a natural number.

So, the set of natural numbers, N is closed under addition and multiplication but this is not the case in subtraction and division.

### 2. Associative Property:

The sum or product of any three natural numbers remains the same even if the grouping of numbers is changed.

Associative Property of Addition: a+(b+c)=(a+b)+c ⇒ 2+(3+1)=2+4=6 and the same result is obtained in (2+3)+1=5+1=6.Associative Property of Multiplication: a×(b×c)=(a×b)×c ⇒ 2×(3×1)=2×3=6= and the same result is obtained in (a×b)×c=(2×3)×1=6×1=6.

So, the set of natural numbers, N is associative under addition and multiplication but this does not happen in the case of subtraction and division.

### 3. Commutative Property:

The sum or product of two natural numbers remains the same even after interchanging the order of the numbers. The commutative property of N states that: For all a,b∈N: a+b=b+a and a×b=b×a.

Commutative Property of Addition: a+b=b+a ⇒ 8+9=17 and b+a=9+8=17.Commutative Property of Multiplication: a×b=b×a ⇒ 8×9=72 and 9×8=72.

So, the set of natural numbers, N is commutative under addition and multiplication but not in the case of subtraction and division.Let us summarise these three properties of natural numbers in a table. So, the set of natural numbers, N is commutative under addition and multiplication.

OperationClosure PropertyAssociative PropertyCommutative Property
Subtractionnonono
Multiplicationyesyesyes
Divisionnonono

### 4. Distributive Property:

The distributive property of multiplication over addition is a×(b+c)=a×b+a×cThe distributive property of multiplication over subtraction is a×(b−c)=a×b−a×c

### Important Points

0 is not a natural number, it is a whole number.N is closed, associative, and commutative under both addition and multiplication (but not under subtraction and division).

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Example 2: Is N, as a set of natural numbers, closed under addition and multiplication?

Solution:

Natural numbers include only the positive integers and we know that on adding two or more positive integers, we get their sum as a positive integer, similarly, when we multiply two negative integers, we get their product as a positive integer. Thus, for any two natural numbers, their sum and the product will be natural numbers only. Therefore, N is closed under addition and multiplication.

Note: This is not the case with subtraction and division so, N is not closed under subtraction and division.

Example 3: Silvia and Susan collected seashells on the beach. Silvia collected 10 shells and Susan collected 4 shells. How many shells did they collect in all? Club all the natural numbers, given in the situation and perform the arithmetic operation accordingly.

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Solution:

Shells collected by Silvia = 10 and shells collected by Susan = 4. Thus, the total number of shells collected by them=10+4=14. Therefore, Silvia and Susan collected 14 shells in all.