Natural Numbers- Definition, Example, Properties, Solved Questions

Natural Numbers

Natural Numbers: We use numbers for counting various things around us like the number of students in class, the number of coins in your piggy bank, or the number of days, months, etc. Do you know that the numbers used to count these days and months or things are called natural numbers? Yes, we use natural numbers to count, like 10 bags, 5 pencils, 20 books, 16 eggs, 9 bottles, 130 students, etc. Let’s go through this article to understand natural numbers in more detail. We will learn about definitions, properties, and examples of Natural Numbers.

Natural Numbers Definition

A natural number is a portion of the number system that includes all the positive integers from 1 to infinity (∞). Natural numbers are also called counting numbers because they do not include zero or negative numbers.

What are Natural Numbers?

Natural numbers are part of real numbers that are used for counting.

Natural numbers are denoted by N.

N = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,.............. ∞

Natural numbers are only positive numbers, not any negative integers, fractions, or decimals. For example,

N = 1, 2, 3, 55, 1000, 15888, 1568, 10456, 1235654651, 6513546, 158, 150, etc. 

N ≠1, -2, -5, -100, 1/2, 5/2, 150/29, 290/53, 0.333, 1.295, 99.52, 59.45, etc.

Properties of Natural Numbers

We know that basic mathematical operations like addition, subtraction, multiplication, and division can be performed with natural numbers.

So there are 4 properties of natural numbers mentioned below:

1. Closure Property
2. Associative Property
3. Commutative Property
4. Distributive Property
5. Closure Property

1. Closure Property

This property of Natural numbers states that the addition and multiplication of two natural numbers is always a natural number.

1. Closure property of addition: a+b = c 

e.g. -  2+3 = 5 (Here, 2 and 3 both are natural numbers, and their sum 5 is also a natural number.)

Likewise, 3+5 = 8, 5+10= 15, 100+50 = 150 

Here all additional results are a natural number.

2. Closure property of multiplication: ab= c

e.g. -  45=20 (Here, 4 and 5 both are natural numbers and their product 20 is also a natural number.)

Likewise, 62= 12, 93=27, 35=15 

Here all multiplication produced are natural numbers

3. But in the case of Subtraction and Division results may or may not be a natural number.

For example, 

7-5= 2 is a natural number but 2-7= -5 is not a natural number.

102= 5 is a natural number but, 210= 0.2 is not a natural number.

2. Associative Property

This property of Natural numbers states that the addition or multiplication of any three natural numbers remains the same even if the grouping of numbers is changed.

1. Associative property of addition: a+(b+c)=(a+b)+c

E.g. -  3+(6+4)= 3+10 = 13 and the same result is obtained from the (3+6)+4= 9+4 = 13

2. Associative property of multiplication: a(bc)=(ab)c

E.g. -  5(21)= 52 = 10 and the same result is obtained from the (52)1 = 101 = 10

Here all the numbers are natural numbers including results.

3. But, in the case of Subtraction and Division, the results may or may not be a natural number and the same.

For example, 

5-(4-1) = 5-3 = 2 is a natural number but, (5-4)-1 = 1-1 = 0 is not a natural number and results are also not the same.

12(42) = 122 = 6 is a natural number but, (124)2 = 32 = 1.5 is not a natural number and the results are also not the same.

3. Commutative Property

This property of natural numbers states that the addition or multiplication of two natural numbers remains the same even after interchanging the order of the numbers. 

It means for all a,b∈N

1. Commutative property of addition: a+b=b+a

E.g.:    6+4 = 10 and 4+6 = 10

           5+3 = 8 and 3+5 = 8

2. Commutative property of multiplication: ab=ba

 E.g.:   52 = 10 and 25 = 10

            73 = 21 and 37 = 21

3. But in the case of Subtraction and Division, results may or may not be a natural number and same.

For example, 

9-2 = 7 but 2-7 = -5, results are not the same and -5 is also not a natural number.

102 = 5 but 210 = 0.2, results are not the same and 0.2 is also not a natural number.

4. Distributive Property

This property of natural numbers states that the multiplication of natural numbers is always distributive over addition and the multiplication of natural numbers is also distributive over subtraction.

1. Multiplication over addition: a(b+c)=ab+ac

E.g.:  2(3+6) = 23+26 

                              29 = 6+12

                                    18 = 18

2. Multiplication over subtraction: a(b-c)=ab-ac 

E.g:  6(4-3) = 64-63

                   61 = 24-18

                           6 =6

Natural Numbers from 1 to 100

Odd Natural Numbers

Odd natural numbers are those numbers that are not divisible by 2 and are left with some remainder and belong to the set of N. Let's have a look at the list of Odd natural numbers from 1 to 50.  

Even Natural Numbers

Even Natural Numbers are those numbers that are exactly divisible by 2 and belong to the set of N. Let's have a look at the list of Even natural numbers from 1 to 50.  

Difference between Natural numbers and Whole numbers

All positive numbers from 1 to infinity are natural numbers like 1, 2, 3, 4, 5, 6, and so on. As we know that these numbers are usually used in counting. Whereas, the whole numbers are the numbers that include all-natural numbers with 0, for example, 0, 1, 2, 3, 4, 5, 6, and continue to infinity. 1 is considered the smallest natural number, while, (0) zero is the smallest whole number. We must keep in mind that every natural number is a whole number, but all whole numbers are not natural numbers. Or we can also say that each whole number is a natural number, excluding (0) zero.

But when we combine zero with positive integers then the resulting number will be a natural number like when 1 is combined with zero it becomes 10, which is a natural number.

Is (0) Zero a natural number?

Zero is NOT a natural number, as the natural number is the counting number that starts from 1 and goes to infinity. When we include zero in counting then it is considered as a whole number. So zero is not a natural number.

Solved Question on Natural Numbers

Q1: Find out all the natural numbers among the given numbers: -2, 6, 0,  4, -1/4, 11, 0.5.

Answer: All counting numbers excluding zero are the number that is considered a natural number only. Other than this neither a negative integer nor fraction or decimals fall under the category of natural numbers.

So, only 6, 4, and 11 are natural numbers. As  Zero is a whole number -2 is a negative integer, -¼ is a fraction and 0.5 is a decimal number.

Q2: What is the largest natural number?

Answer: The largest natural number is very hard to identify as the counting of a number goes to infinity and still counting.

Q3: Solve the following operation using the suitable arrangement and properties of natural numbers? 140 x 25 + 24 x 14 x 5 x 2 

Answer: Arranging the given number using distributive property of multiplication then

=140 x 25 + 24 x 14 x 10

=140 x 25 + 24 x 140

=140 (25 + 24)

=140 x 49

=6860 is the answer.

Q4: Solve the following operation using the suitable arrangement and properties of natural numbers? 120 x 2 x 65 - 55 x 12 x 10 x 2. 

Answer: The given number can be rearranged using the distributive property of multiplication over subtraction then we get,

=120 x 2 x 65 - 55 x 12 x 20

=120 x 2 x 65 - 55 x 240

=240 x 65 - 55 x 240

=240 (65 - 55)

=240 (10)

=2400 is the answer

Q5: State whether the given statement is True or False: There is always a natural number between any two consecutive natural numbers.

Answer: False, The above statement is not true in between two consecutive natural numbers if other numbers will appear then it is not a natural number, for example, In between 3 and 4, no natural number exists as 3.1, 3.2, 3.3, 3.5, 3.9, etc are then decimal numbers.

Q6: What is the Successor of 5199?

Answer: The successor is the just next number of any natural number here, the successor of 5199 is

=5199+1=5200

Q7: Find out four consecutive predecessors of 8001?

Answer: The predecessor of any number is just one number before that particular number, so here in 8001, the following are the four predecessors:

8001-1= 8000

8000-1=7999

7999-1=7998

7998-1=7997

So, the four consecutive predecessors of 8001 are: 8000, 7999, 7998, 7997

Q8: Satyam brought 50 new phones for his shop and 22 Headphones also. How many total gadgets does Satyam bring to his shop?

Answer: The number of the new phones is 50 and the headphones are 22, then by simply adding these two natural numbers we will get to know the total new gadgets Satyam brings so 50+22=72

Q9: What is the smallest 5-digit number that can be formed by the digits 8, 0, 9, 2, and 4?

Answer: The smallest 5-digit number that can be formed by using the given digits 8, 0, 9, 2, and 4 is 20489.

Q10: Check out how many natural numbers are there in the set of given numbers: 55, 106, 400, -4, 0.01, 0, 225, 87.250.

Answer: There are 4-natural numbers present in the given set of numbers; those are 55, 106, 400, and 225 all other numbers are fulfilling the criteria of a natural number.