Natural Numbers: Definition, Properties, Examples, and Practice Problems

Natural Numbers

Natural Numbers: We rely on numbers every day to count objects around us—like students in a classroom, coins in a piggy bank, or the days of the month. These counting numbers are known as natural numbers. Whether we are counting 10 bags, 5 pencils, or 130 students, we are utilizing the natural number system. In this guide, we will explore the definition, essential properties, and practical examples of natural numbers.

Natural Numbers Definition

A natural number is a positive integer that forms a core part of the number system, ranging from 1 to infinity (∞). Often referred to as "counting numbers," natural numbers explicitly exclude zero and all negative integers.

What are Natural Numbers?

Natural numbers are a subset of the real numbers specifically used for counting purposes.

In mathematics, natural numbers are denoted by the symbol N.

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

Natural numbers include only positive integers. They do not include negative numbers, fractions, or decimal values. 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

Basic mathematical operations such as addition, subtraction, multiplication, and division can be performed using natural numbers, though the results may vary.

There are four fundamental properties of natural numbers:

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

1. Closure Property

The closure property states that the addition and multiplication of any two natural numbers always result in another natural number.

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

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

Similarly, 3+5 = 8, 5+10 = 15, 100+50 = 150. 

In these cases, all sums are natural numbers.

2. Closure property of multiplication: a × b = c

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

Similarly, 6 × 2 = 12, 9 × 3 = 27, 3 × 5 = 15. 

All products are natural numbers.

3. In the case of subtraction and division, results may not always be natural numbers.

For example: 

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

10 ÷ 2 = 5 is a natural number, but 2 ÷ 10 = 0.2 is not.

2. Associative Property

The associative property states that the addition or multiplication of any three natural numbers remains the same, regardless of how they are grouped.

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

e.g., 3+(6+4) = 3+10 = 13, and (3+6)+4 = 9+4 = 13.

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

e.g., 5(2 × 1) = 5 × 2 = 10, and (5 × 2) × 1 = 10 × 1 = 10.

Here, all intermediate and final results are natural numbers.

3. Subtraction and division do not satisfy the associative property.

For example: 

5-(4-1) = 5-3 = 2, while (5-4)-1 = 1-1 = 0. The results are not equal, and 0 is not a natural number.

12÷(4÷2) = 12÷2 = 6, while (12÷4)÷2 = 3÷2 = 1.5. The results are not equal, and 1.5 is not a natural number.

3. Commutative Property

The commutative property states that the addition or multiplication of two natural numbers remains consistent even if their order is swapped. 

This applies to 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: a × b = b × a

 e.g., 5 × 2 = 10 and 2 × 5 = 10.

            7 × 3 = 21 and 3 × 7 = 21.

3. Subtraction and division do not satisfy the commutative property.

For example: 

9-2 = 7, but 2-9 = -7. The results are different, and -7 is not a natural number.

10÷2 = 5, but 2÷10 = 0.2. The results are different, and 0.2 is not a natural number.

4. Distributive Property

The distributive property states that multiplication of natural numbers is distributive over both addition and subtraction.

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

e.g., 2(3+6) = 2(9) = 18 and 2 × 3 + 2 × 6 = 6 + 12 = 18.

                           18 = 18

 

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

e.g., 6(4-3) = 6(1) = 6 and 6 × 4 - 6 × 3 = 24 - 18 = 6.

                           6 = 6

 

Natural Numbers from 1 to 100

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

Odd Natural Numbers

Odd natural numbers are numbers not divisible by 2, meaning they leave a remainder of 1. Examples include 1, 3, 5, 7, and 9.

Odd Natural numbers from 1 to 50
1	3	5	7	9
11	13	15	17	19
21	23	25	27	29
31	33	35	37	39
41	43	45	47	49

Even Natural Numbers

Even natural numbers are numbers exactly divisible by 2. Examples include 2, 4, 6, 8, and 10.

Even Natural numbers from 1 to 50
2	4	6	8	10
12	14	16	18	20
22	24	26	28	30
32	34	36	38	40
42	44	46	48	50

Difference between Natural numbers and Whole numbers

All positive integers (1, 2, 3...) are natural numbers. Whole numbers, however, include all natural numbers plus zero (0, 1, 2, 3...). While 1 is the smallest natural number, 0 is the smallest whole number. Every natural number is a whole number, but not every whole number is a natural number.

When you append zero to a positive integer (e.g., 1 becomes 10), the resulting number is a natural number.

Is (0) Zero a natural number?

Zero is NOT a natural number because natural numbers are defined as counting numbers starting from 1. Once zero is included, the set is categorized as whole numbers.

Solved Question on Natural Numbers

Q1: Identify the natural numbers in the following list: -2, 6, 0, 4, -1/4, 11, 0.5.

Answer: Natural numbers are positive counting integers greater than zero. Negative numbers, fractions, and decimals do not qualify.

Therefore, 6, 4, and 11 are the natural numbers in this set.

Q2: What is the largest natural number?

Answer: There is no largest natural number, as the sequence continues infinitely.

Q3: Solve using properties: 140 × 25 + 24 × 14 × 5 × 2

Answer: Using the distributive property:

= 140 × 25 + 24 × 14 × 10

= 140 × 25 + 24 × 140

= 140(25 + 24)

= 140 × 49

= 6860.

Q4: Solve: 120 × 2 × 65 - 55 × 12 × 10 × 2.

Answer: Rearranging using the distributive property:

= 120 × 2 × 65 - 55 × 12 × 20

= 240 × 65 - 55 × 240

= 240(65 - 55)

= 240(10)

= 2400.

The answer is 2400.

Q5: True or False: There is always a natural number between any two consecutive natural numbers.

Answer: False. Between 3 and 4, there is no natural number, as only decimals exist in that range.

Q6: What is the successor of 5199?

Answer: The successor is found by adding 1.

5199 + 1 = 5200.

Q7: Find four consecutive predecessors of 8001.

Answer: Predecessors are found by subtracting 1 repeatedly.

8001 - 1 = 8000

8000 - 1 = 7999

7999 - 1 = 7998

7998 - 1 = 7997

The predecessors are: 8000, 7999, 7998, 7997.

Q8: Satyam bought 50 phones and 22 headphones. What is the total?

Answer: 50 + 22 = 72.

Q9: What is the smallest 5-digit number using 8, 0, 9, 2, and 4?

Answer: The smallest 5-digit number is 20489.

Q10: Identify natural numbers in: 55, 106, 400, -4, 0.01, 0, 225, 87.250.

Answer: The natural numbers are 55, 106, 400, and 225.