# What are Natural Numbers?

Number system is the system established to systematically define and place numbers, the most common system that is used in mathematics and in daily lives is the decimal system and natural numbers are also defined for the same. Natural numbers, which include all positive integers from 1 to infinity, are a component of the number system. Natural numbers are only positive integers, not zero, fractions, decimals, or negative numbers, and they are part of real numbers.

### Natural Numbers

Natural numbers are those integers that generate from 1 and go upto infinity. Numbers may be found everywhere, used for counting items, representing or trading money, calculating temperature, telling time, and so on. These numbers are referred to as “natural numbers” since they are used to count items. When counting items, it can be 5 glasses, 6 books, 1 bottle, and so on. A collection of all whole integers except 0 is referred to as natural numbers. These figures play an important role in the daily actions and communication.

**Definition **

Natural numbers are those that can be counted and are a component of real numbers. Only positive integers, such as 1, 2, 3, 4, 5, 6, etc., are included in the set of natural numbers.

**Examples**

Non-negative integers are also known as natural numbers (all positive integers). 24, 57, 88, 979, 120502, and so forth are only a few instances. Will -4 be a Natural number? No. Since, it is a negative integer. Will 3.6 be a natural number? No. Since, it is not an Integer.

**Natural Numbers Array**

A collection of elements is referred to as a set (numbers in this context). In mathematics, the set of natural numbers is expressed as 1,2,3,… The set of natural numbers is represented by the symbol N. N = {1,2,3,4,5,…∞}. One (1) is the lowest natural number. The smallest element in N is 1 and the next element in terms of 1 and N for any element in N. 2 is 1 greater than 1, 3 is 1 greater than 2, and so on.

**Natural Even Numbers**

Even natural numbers are those that are even, divisible by 2 precisely, and belong to the set N. So 2,4,6,8,… is the set of even natural numbers.

**Natural Odd Numbers**

Natural numbers that are odd and belong to the set N are known as odd natural numbers, not divisible by 2 precisely. So 1,3,5,7,… is the set of odd natural numbers.

### Does the number zero belong to the Natural numbers?

Natural numbers are counting numbers, 0 is not a natural number. Since, the counting begins 1 instead of 0 when counting any number of items. The number 0 precisely belongs to the Whole number, 0 is also a part of Integers and is represented on the number line, however, even on the number line, everything from +1 and its right side belongs to Natural numbers.

**Whole Numbers**

The set of whole numbers is identical to the set of natural numbers, with the exception that it includes a 0 as an extra number. In mathematics, the set of whole integers is expressed as 0,1,2,3,… The letter W stands for it. It is clear from the definitions that any natural number is a whole number. Furthermore, all whole numbers other than 0 are natural numbers.

**Difference Between Natural Numbers and Whole Numbers**

Natural numbers, such as 1, 2, 3, 4, and so on, are all positive numbers. They’re the numbers used for counting, and they go on indefinitely. The entire numbers, on the other hand, are all natural numbers, excluding zero, such as 1, 2, 3, 4, and so on. All whole numbers and their negative counterparts are considered integers. -4, -3, -2, -1, 0,1, 2, 3, 4, and so on are some examples.

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

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

**Representing Natural Numbers on a Number Line**

On the number line, the set of natural numbers and whole numbers is shown below. 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.

**Properties of Natural numbers**

The four operations on natural numbers, addition, subtraction, multiplication, and division, resulting in four main characteristics of natural numbers, which are illustrated below:

- Closure property
- Commutative property
- Associative property
- Distributive property

**1. Closure Property**

When two or more natural numbers are added and multiplied, the result is always a natural number. Addition Closure Property is **a+ b= c** i.e. 3+ 2= 5, 9+ 8= 17. The sum of natural numbers is always a natural number, as this demonstrates. Multiplication Closure Property is** ab= c** i.e. 2x 4= 8, 7x 8= 56, etc. This demonstrates that a natural number is always the product of two natural numbers.

Note:Natural numbers may not obey the closure property when it comes to subtraction and division, which implies that subtracting or dividing two natural numbers may not result in a natural number.

**2. Associative Property:**

When adding and multiplying natural integers, the associative condition is true, i.e. a+( b+ c)= ( a+ b) + c and a( b c) = ( a b) c. Associative Property of Addition is** a+(b+ c)= (a+ b)+c** i.e; 1+(3+5)=1+8=9 and the same result is obtained in (1+3)+5=4+5=9. Associative Property of Multiplication is **a×(b× c)= (a× b)×c** i.e; 2× (2× 1)= 2× 2= 4 and the same result is obtained in (a× b)× c= (2× 2)× 1= 4× 1= 4.

Note:The associative property, on the other hand, does not hold true for natural number subtraction and division.

**3. Commutative Property:**

Even if the sequence of the numbers is changed, the sum or product of two natural numbers stays the same. The commutative property of N says that a+ b= b+ a and ab= ba for any a, b ∈ N.

Commutative Property of Addition:** a+ b=b+ a** ⇒ 4+5=9 and b+ a= 5+ 4= 9.

Commutative Property of Multiplication: **a× b= b× a **⇒ 3× 2= 6 and 2× 3= 6.

**4. Distributive Property: **

Multiplication over addition has the distributive property: **a× (b c) = ab + ac.**

Multiplication over subtraction has the distributive property: **a× (b– c) = ab – ac**

### Conceptual Questions

**Question 1: Every Whole Number is a Natural Number. Is this statement true or false?**

**Answer:**

True. Every whole number is a natural number. The assertion is correct because natural numbers are positive integers that begin at 1 and run all the way to infinity, whereas whole numbers contain all positive integers plus 0.

**Question 2: List first 10 natural numbers.**

**Answer:**

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