Natural Numbers | Definition, Examples & Properties

Natural numbers are the numbers that start from 1 and end at infinity. In other words, natural numbers are counting numbers and they do not include 0 or any negative or fractional numbers.

Here, we will discuss the definition of natural numbers, the types and properties of natural numbers, as well as other operations involving natural numbers.

Natural numbers are also called the counting numbers: 1, 2, 3, 4, 5, and so on. They begin at 1, unlike whole numbers, which start at 0.

Essential features of Natural Numbers:

• Natural numbers or counting numbers are those integers that begin with 1 and go up to infinity.
• Only positive integers, such as 1, 2, 3, 4, 5, 6, etc., are included in the set of natural numbers. Natural numbers start from 1 and go up to ∞.
• Natural numbers are the set of positive integers starting from 1 and increasing incrementally by 1.
• They are used for counting and ordering.
• The set of natural numbers is typically denoted by N and can be written as {1, 2, 3, 4, 5, …}

Check: Is Zero a Natural Number?

Characteristics of Natural Numbers

No Decimals or Fractions: One defining characteristic of natural numbers is that they are whole numbers. Unlike rational or real numbers, natural numbers do not include decimals or fractions. For example, numbers like 3.14 or 1/2 are not natural numbers​.

Starting from 1: Natural numbers always start from 1 and go up to infinity. This distinguishes them from whole numbers, which include zero​.

Non-Negative: By definition, natural numbers are always positive integers, which means they don’t include negative numbers like -1, -2, etc.​

Set of Natural Numbers

In mathematics, the set of natural numbers is expressed as 1, 2, 3, ... The set of natural numbers is represented by the symbol N. N = {1, 2, 3, 4, 5, ... ∞}. A collection of elements is referred to as a set (numbers in this context). The smallest element in N is 1, and the next element in terms of 1 and N for any element in N. 2 is 1 greater than 1, 3 is 1 greater than 2, and so on. The below table explains the different set forms of natural numbers.

Natural numbers are the subset of whole numbers, and whole numbers are the subset of integers. Similarly, integers are the subset of real numbers.

Types of Natural Numbers

Odd natural numbers

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

Even natural numbers

Even natural numbers are whole numbers that are divisible by 2 without leaving a remainder. In other words, they are integers greater than zero that can be expressed in the form 2n, where n is an integer. Examples of even natural numbers include 2, 4, 6, 8, 10, and so on.

Natural Numbers from 1 to 100

As Natural Numbers are also called counting numbers, thus natural numbers from 1 to 100 are:

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 and Whole Numbers

The set of whole numbers is identical to the set of natural numbers, with the exception that it includes a 0 as an extra number.

W = {0, 1, 2, 3, 4, 5, ...} and N = {1, 2, 3, 4, 5, ...}

Read more about: Whole Number

Difference Between Natural Numbers and Whole Numbers

Let's discuss the differences between natural numbers and whole numbers.

Natural Numbers on Number Line

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

Here is how we represent natural numbers and whole numbers on the number line:

Properties of Natural Numbers

All the natural numbers have these properties in common :

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

Let's learn about these properties in the table below.

Note:

• Subtraction and Division may not result in a natural number.
• Associative Property does not hold true for subtraction and division.

Operations With Natural Numbers

We can add, subtract, multiply, and divide the natural numbers together but the result in the subtraction and division is not always a natural number.

Let's understand the operations on natural numbers:

Mean of First n Natural Numbers

As mean is defined as the ratio of the sum of observations to the number of total observations.

Mean Formula for the first n terms of natural number:

Mean = S/n = (n+1)/2

where,

• S is the Sum of all Observations
• n is the Number of Terms Taken into Consideration

Sum of Square of First n Natural Numbers

The sum of the square of the first n natural numbers is given as follows:

S = n(n + 1)(2n + 1)/6

Where n is the number taken into consideration.

People Also Read:

• Number System
• Real Numbers
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Solved Examples of Natural Numbers

Let's solve some example problems on Natural Numbers.

Question 1: Identify the natural numbers among the given numbers: 23, 98, 0, -98, 12.7, 11/7, 3.

Solution:

Since negative numbers, 0, decimals, and fractions are not a part of natural numbers.
Therefore, 0, -98, 12.7, and 11/7 are not natural numbers.
Thus, natural numbers are 23, 98, and 3.

Question 2: Prove the distributive law of multiplication over addition with an example.

Solution:

Distributive law of multiplication over addition states: a(b + c) = ab + ac

For example, 4(10 + 20), here 4, 10, and 20 are all natural numbers and hence must follow distributive law
4(10 + 20) = 4 × 10 + 4 × 20
4 × 30 = 40 + 80
120 = 120
Hence, proved.

Question 3: Prove the distributive law of multiplication over subtraction with an example.

Solution:

Distributive law of multiplication over addition states: a(b - c) = ab - ac.

For example, 7(3 - 6), here 7, 3, and 6 are all natural numbers and hence must follow the distributive law. Therefore,
7(3 - 6) = 7 × 3 - 7 × 6
7 × -3 = 21 + 42
-21 = -21
Hence, proved.

Question 4: List the first 10 natural numbers.

Solution:

1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 are the first ten natural numbers.

Natural Numbers Question

Question 1: What is the Smallest Natural Number?

Question 2: What is the Biggest Natural Number?

Question 3: Simplify, 17(13 - 16)

Question 4: Simplify, 11(9 - 2)

Question 5: Find the sum of the first 20 natural numbers.

Question 6: Is 97 a prime natural number?

Question 7: What is the smallest natural number that is divisible by both 12 and 18?

Question 8: Find the product of the first 5 natural numbers.

Question 9: How many natural numbers are there between 50 and 100 (inclusive)?

Question 10: Discuss whether 0 is included in the set of natural numbers based on its definition.

Answer Key:

1. 1
2. Not defined
3. -51
4. 77
5. 210
6. Yes
7. 36
8. 120
9. 51
10. No

Conclusion

• Natural numbers form the foundation of the number system, containing all positive integers from 1 to infinity.
• They are important for counting and ordering, playing a crucial role in everyday mathematics and various advanced fields.
• Natural numbers exhibit properties like closure (the sum or product of two natural numbers is also a natural number), commutative, associative, and distributive properties.
• Basic operations with natural numbers include addition, subtraction, multiplication, division, exponentiation, square roots, and factorials.