Natural Numbers: Types, Properties, Solved Examples and More

Natural numbers are part of the number system and include all positive values from 1 to infinity. Natural numbers, often known as counting numbers, They are a subset of real numbers that only include positive integers and exclude zero, fractions, decimals, and negative numbers. On this page, we will look at the definition of natural numbers as well as examples of them.

What are Natural Numbers?

Natural numbers include all whole numbers, except the value zero. These numerical entities play a significant role in our daily lives. Numerals saturate our surroundings, serving a variety of functions such as tallying items, denoting money values, measuring temperature, indicating time, and so on. The precise subset of integers used for item counting is known as ‘natural numbers’. For example, when quantifying objects, we express quantities such as 5 cups, 6 books, 1 bottle, and so on.

The set of natural numbers is showcased by the letter “N”.

N= {1,2,3,4,5,6,….}

Natural Number Symbol

In mathematics, a set is an assembly of elements, with numbers being particularly essential in this context. The natural numbers are symbolically expressed as {1, 2, 3, …}, indicating an infinite sequence.

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}

Also Read: What is the Number System? Types and Solved Examples

Is 0 A Natural Number?

No, 0 is not a natural number because natural numbers are defined as counting numbers. When counting items, the enumeration starts at 1 and does not include 0. As a result, 0 is not considered a natural number.

Types of Natural Numbers

The types of natural numbers are as follows:

Odd Natural Numbers

Odd natural numbers are integers greater than zero that cannot be divided evenly by 2, leaving a remainder of one when divided by two. The odd natural numbers include 1, 3, 5, 7, 9, 11, and so on.

Even Natural Numbers

Even natural numbers are whole numbers that divide by two without leaving a remainder. In other words, they are integers bigger than zero that may be represented as 2n, where n is an integer. Even natural numbers include 2, 4, 6, 8, 10 and so on.

Natural Numbers from 1 to 100

Since natural numbers are counting numbers, the natural numbers from 1 to 100 include:

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, 100.

Natural Numbers on Number Line

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

Here’s how natural numbers and whole numbers are represented on the number line:

Also Read: Rational And Irrational Numbers: Differences, Examples

Properties of Natural Numbers

All the natural numbers have these properties:

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

Let’s see the properties mentioned below:

Property	Description	Example
Closure Property
Addition Closure	Sum of any two natural numbers is a natural number.	3 + 2 = 5, 9 + 8 = 17
Multiplication Closure	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	Order of numbers does not change the sum.	4 + 5 = 9, 5 + 4 = 9
Commutative Property of Multiplication	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:

1. Subtraction and division may not yield a natural number.
2. Subtraction and division do not follow the Associative Property.

Solved Examples

1. Find the natural numbers among the following: -1, 0, 3, 1/2, and 5.

Solution,

In mathematics, the set {1, 2, 3, …} represents the natural numbers. Now, since -1 is a negative integer, it is not a natural number. Zero is also not a natural number. 1/2, as a fractional number, is not a natural number. Therefore, among the above integers, the natural numbers are 3 and 5.

2. State true or false with respect to natural numbers.

a.) Zero is a natural number.

b.) All natural numbers are whole numbers.

c.) Every natural number is an integer.

Solution,

a.) False; 0 is not a natural number. It’s a whole number.

b.) True, every natural number is a whole number because it includes all positive counting numbers beginning with 0.

c.) True, all natural numbers are integers since they include all positive and negative numbers, as well as zero.

3. Give first 10 natural numbers.

Solution,

The first 10 natural numbers are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

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