Natural numbers: Concepts, Properties and Examples

This article will help you to learn more about natural numbers concerning their definition, their comparison with whole numbers, their representation in the number line, and their properties.
 
Numbers that are used to count are called Natural numbers. The natural numbers are the numbers used for counting for example - there are ten coins in the purse or order for example - Delhi is the third-largest city in the country in terms of population. 

Also Read:

• Rational Numbers
• Integers
• Whole Numbers
• Complex Numbers

Natural numbers: Concepts, Properties, and Examples -PDF

 In mathematics, numbers colloquially used for counting are "cardinal numbers", and words used for ordering are "ordinal numbers".  Here is a downloadable PDF to explore more.

Set of Natural Numbers and Introduction for Whole Numbers and Integers 

Natural numbers are the part of the number system that include all positive integers from \(1\) uptill infinity. They are also used for counting purpose. 
Whole numbers include all the natural numbers along with the number \(0\). 

In other words, all natural numbers can be whole numbers, but all whole numbers are not natural numbers. For example zero \((0)\) is a whole number but not a natural number.

Set of Natural Numbers \(\begin{align} = 1,2,3,4,5,6,7,8,9,.....\end{align}\)

Set of Whole Numbers \(\begin{align}= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,….}\end{align}\)

         
The above representation of sets shows two regions natural numbers and whole numbers .Thus a whole number is “a number system consisting of all the natural number including 0.”
 
 

Properties of Natural Numbers with Examples

Closure Property  

Natural numbers are closed under addition and multiplication. 

This means when two or more natural numbers are added or multiplied it always results in a natural number. In the case of subtraction and division, natural numbers do not follow closure property, which means subtracting or dividing two natural numbers may not necessarily result in a natural number.

• Addition:\(\begin{align}4 + 2 = 6, 23 + 44 = 67,\end{align}\) etc. In each of these cases, the resulting number is a natural number.
• Multiplication:\( \begin{align}2 × 3 = 6, 25 × 4 = 100\end{align}\), etc. In this case also, the resultant is a natural number.
• Subtraction: \(\begin{align}9 – 1 = 8, 4 \,– 7 = –3,\end{align}\) etc. In this case, we see that the result may or may not be a natural number.
• Division:\(\begin{align}20 ÷ 4 = 5, 10 ÷ 3 = 3.33\end{align}\), etc. This case also shows that the resultant number may or may not be a natural number.

Commutative Property  

Addition and multiplication of natural numbers follow the commutative property. 

  \(\begin{align}a + b +  = b + a\,\,\text{and}\,a\, \times b = b \times a\end{align}\)

Whereas, subtraction and division of natural numbers do not follow the commutative property. 

    \(\begin{align}a - b \ne b - a\,\,\text{and}\,\,a \div b \ne b \div a\end{align}\)

Associative Property 

The associative property holds true in case of addition and multiplication of natural numbers 

For addition               \(\begin{align}a + \left( {b + c} \right) = \left( {a + b} \right) + c\end{align}\)         

For multiplication        \(\begin{align}a \times (b \times c) = (a \times b) \times c\end{align}\).
 

For addition               \(\begin{align}30 + (25 + 11) = 66\,\text{and}\,(30 + 25) + 11 = 66\end{align}\)
 
For multiplication       \(\begin{align}3 \times (15 \times 1) = 45\,\text{and}\,(3 \times 15) \times 1 = 45\end{align}\)
 
On the other hand, the associative property does not hold true for subtraction and division of natural numbers.

Let us understand this with an example: 

Subtraction: 

\(\begin{align}a – ( b – c ) ≠ ( a – b ) – c \end{align}\)

\(\begin{align}48 - (15 - 11) = 44\,\,\text{and}\,\,(48 - 15) - 11 = 22\end{align}\)

Division:  

\(\begin{align}a \div (b \div c) \ne (a \div b) \div c\end{align}\)

\(\begin{align}2 \div (3 \div 6) = 4\,\,\text{and}\,\,(2 \div 3) \div 6 = 0.11\end{align}\)

Distributive Property 

Multiplication of natural numbers is distributive over addition    

 \(\begin{align}a \times (b + c) = a \times b + a \times c\end{align}\)

Multiplication of natural numbers is distributive over subtraction also.

\(\begin{align}a \times (b - c) = a \times b - a \times c\end{align}\)

 \(\begin{align}
&5 \times (7 + 3) = 5 \times 10 = 50\,\text{and}\,5 \times 7 + 5 \times 3 = 35 + 15 = 50\\
&9 \times (54 - 45) = 81\,\,\text{and}\,\,9 \times 54 - 9 \times 45 = 486 - 405 = 81
\end{align}\)

Read More Here:

Under the study of natural numbers you will study about various other related terms such as odd/even numbers, prime/composite numbers, factors/multiples, HCF/LCM, rounding off/estimation, square/square roots etc.

Solved Examples

From the following list of numbers find the natural numbers:

\(\begin{align} -20, 105, 1535, 63.99, 5/2, 69, −78, 0, −2, −3/2\end{align}\)

Solution:

Write the first \(10\) natural numbers?

Solution:

Is the number \(0\) a natural number?

Solution:

What is the sum of first \(10\) natural numbers?

Solution:

Add the numbers as follows:-

FAQs (FREQUENTLY ASKED QUESTIONS)