Natural Numbers : Definition, Concept, Types, Properties, and Number Line

Natural Numbers Overview

All positive numbers from 1 to infinity are considered to be natural numbers, which make up the entire number system. Because they don't contain zero or negative numbers, natural numbers are also known as counting numbers. They are a subset of real numbers, which only include positive integers and exclude negative, zero, fractional, and decimal numbers.

What are Natural Numbers?

Natural numbers are all whole numbers with the exception of zero. We frequently use these numbers in our speech and daily activities. Everywhere we look, numbers are used to count things, represent or exchange money, measure temperature, tell the time, etc. These numbers are referred to as "natural numbers" because they are used to count objects. For example, while counting objects, we say 5 cups, 6 books, 1 bottle, and so on.

Natural Numbers Definition

The numbers that are used for counting and are a subset of real numbers are known as natural numbers. Only positive integers, such as 1, 2, 3, 4, 5, 6,.........., are included in the set of natural numbers.

Natural Numbers Examples

The positive integers, also referred to as non-negative integers, are a subset of the natural numbers. A few examples are 1, 2, 3, 4, 5, 6,... To put it another way, the set of all whole numbers excluding 0 is known as the "natural numbers."

Natural numbers include 23, 56, 78, 999, 100202, etc.

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Is "0" a Natural Number?

The answer to this question is ‘No'. Natural numbers are positive integers that range from 1 to infinity, as we already know. However, when we add 0 to a positive integer like 10, 20, etc., it turns into a natural number. In fact, 0 is a whole number which has a null value.

Types of Natural numbers

The natural numbers are of two types-

1. Odd natural numbers

Natural numbers that are odd and fall under the category of set N are known as odd numbers. The collection of odd natural numbers is therefore 1, 3, 5, 7, etc.

2. Even natural numbers

The numbers that are even, precisely divisible by 2, and part of the set N are known as even natural numbers. So the set of even natural numbers is {2,4,6,8,...}.

Set of Natural Numbers

The set of natural numbers is represented in mathematics as 1, 2, 3,... The letter N stands for the collection of natural numbers. N = {1, 2, 3, 4, 5, … ∞}. The smallest natural number is one (1). A set, which in this case refers to numbers, is a grouping of elements. The smallest element in N is 1, and the next element for any element in N is defined in terms of 1 and N. 2 is 1 greater than 1, 3 is 1 greater than 2, and so on. The table below explains the various set forms of natural numbers.

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Natural Numbers and Whole Numbers

All whole numbers, with the exception of zero, are considered to be natural numbers. In other words, all whole numbers are not natural numbers, but all natural numbers are whole numbers.

1. Natural Numbers are 1, 2, 3, 5, 6, 7, 8, 9, etc.
2. Whole Numbers = 0 through 9, including 0 and 1.

 "A part of integers consisting of all natural numbers, except zero," is what is meant by a whole number.

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Difference Between Natural Numbers and Whole Numbers

Positive numbers like 1, 2, 3, 4, and so forth make up all natural numbers. They are the numbers you typically count, and they go on forever. All natural numbers, including 0 (i.e., 0, 1, 2, 3, 4, etc.), are considered whole numbers. All whole numbers and their negative equivalent are included in integers. For instance, -4, -3, -2, -1, 0, 1, 2, 3, and so forth. The distinction between a natural number and a whole number is displayed in the following table.

Every Natural Number is a Whole Number. True or False?

Every whole number is a natural number. The claim is accurate because whole numbers also include all positive integers, including 0, and natural numbers are the positive integers that range from 1 to infinity.

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Properties of Natural Numbers

Four main characteristics of natural numbers are divided into groups, and they are as follows:

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

The details of each of these attributes are provided below-

Closure Property

Under addition and multiplication, natural numbers are always closed. A natural number will always result from the addition and multiplication of two or more natural numbers. Natural numbers do not adhere to the closure property when it comes to subtraction and division, which means the result of subtracting or dividing two natural numbers might not be a natural number.

1. Addition: 1 + 2 = 3, 1 + 3 + 4 = 7, etc. The outcome is always a natural number in each of these examples.
2. Multiplication: 2 x 3 = 6, 5 x 4 = 20, etc. In each of these instances, the outcome is a natural number.
3. Subtraction yields results like: 9 - 5 = 4, 3 - 5 = -2, etc., which may or may not be natural numbers.
4. Division: 10 ÷ 5 = 2, 10 ÷ 3 = 3.33, etc. In this case also, the resultant number may or may not be a natural number.

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Note: If any of the numbers in the case of multiplication and division are not natural numbers, the closure property is not valid. Only the closure property applies to addition and subtraction, though, if the result is a positive number.

For instance: 

Not a natural number: -2 x 3 = -6

6/2 = 3, which is not a natural number.

Associative Property

In the case of addition and multiplication of natural numbers, a + (b + c) = (a + b) + c and a  (b  c) = (a b)  c, the associative property holds true. OOn the other hand, the associative property does not apply to the operations of subtracting and dividing natural numbers. n example of this is given below.

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

Commutative Property

1. Natural number addition and multiplication demonstrate the commutative property, as in the examples x + y = y + x and a b = b a
2. Natural number division and subtraction do not exhibit the commutative property, as seen in the examples x-y-y-x and x-y-y-x-x.

Distributive Property

1. Natural number multiplication is always distributive over addition; for instance, a  (b + c) = ab + ac
2. Natural number division is also distributive over multiplication. For instance, a  (b - c) = ab - ac

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Operations With Natural Numbers

The table below provides a summary of the addition, subtraction, multiplication, and division algebraic operations with natural numbers as well as each operation's associated properties. 

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Important Points

1. Since it is a whole number, zero is not a natural number.
2. The terms "natural numbers" and "whole numbers" do not apply to negative numbers, fractions, or decimals.