**Natural numbers** are a part of the number system which includes all the positive integers from 1 till infinity. It is an integer which is always greater than zero(0). It should be noted that the natural numbers include only the positive integers i.e. set of all the counting numbers like 1, 2, 3, ………. excluding the fractions, decimals, and negative numbers. It is a part of real numbers.

* Note:* Natural numbers will never consist of negative numbers or zero.

In this article, we are going to learn more about natural numbers with respect to its definition, comparison with whole numbers, representation in the number line, properties, etc.

Natural numbers are the numbers which are positive in nature and includes numbers from 1 till infinity(∞). These numbers are countable and are usually used for calculations purpose. The set of natural numbers are represented by the letter “N”.

Natural numbers include all the whole numbers excluding the number 0. In other words, all natural numbers are whole numbers, but all whole numbers are not natural numbers. Check out the difference between natural and whole numbers to know more about the differentiating properties of these two sets of numbers.

The above representation of sets shows two regions,

A ∩ B ie. intersection of natural numbers and whole numbers (1, 2, 3, 4, 5, 6, ……..) and the green region showing A-B, i.e. part of the whole number (0).

Thus, a whole number is **“a part of Integers consisting of all the Natural number including 0.”**

The answer to this question is ‘No’. As we know already, natural numbers start with 1 to infinity and are positive in nature. But when we mention 0 with a positive integer such as 10, 20, etc. it becomes a natural number. In fact, 0 is a whole number which has a null value.

Natural numbers representation on a number line is as follows:

The above number line represents natural numbers and the whole numbers on a number line. All the integers on the right-hand side of 0 represent the natural numbers, thus forming an infinite set of numbers. When 0 is included, these numbers become whole numbers which are also an infinite set of numbers.

In set notation, the symbol of natural number is “N” and it is represented as given below.

**Statement: **

N = Set of all numbers starting from 1.

**In Roster Form:**

N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ………………………………}

**In Set Builder Form:**

N = {x : x is an integer starting from 1}

The natural numbers include the positive integers (also known as non-negative integers) and a few examples include 1, 2, 3, 4, 5, 6, ..∞. In other words, natural numbers are a set of all the whole numbers excluding 0.

Natural numbers properties are segregated into four main properties which include **closure property, commutative property, associative property, **and **distributive. **Each of these properties is explained below in detail.

The **natural numbers are always closed under addition and multiplication** i.e. the addition and multiplication of natural numbers will always yield a natural number. In the case of **subtraction and division, natural numbers are not closed** which means subtracting or dividing two natural numbers might not give a natural number as a result.

**Addition:**1 + 2 = 3, 3 + 4 = 7, etc. In each of these cases, the resulting number is alwasy a natural number.

**Multiplication:**2 × 3 = 6, 5 × 4 = 20, etc. In this case also, the resultant is always a natural number.

**Subtraction:**9 – 5 = 4, 3 – 5 = -2, etc. In this case, the result may or may not be a natural number.

**Division:**10 ÷ 5 = 2, 10 ÷ 3 = 3.33, etc. In this case also, the resultant number may or may not be a natural number.

The **associative property holds true in case of addition and multiplication of natural numbers **i.e. a + ( b + c ) = ( a + b ) + c and a × ( b × c ) = ( a × b ) × c. On the other hand, for **subtraction and division of natural numbers, the associative property does not hold true**. An example of this is given below.

a + ( b + c ) = ( a + b ) + c => 3 + (15 + 1 ) = 19 and (3 + 15 ) + 1 = 19.**Addition:**

**Multiplication:**a × ( b × c ) = ( a × b ) × c => 3 × (15 × 1 ) = 45 and ( 3 × 15 ) × 1 = 45.

**Subtraction:**a – ( b – c ) ≠ ( a – b ) – c => 2 – (15 – 1 ) = – 12 and ( 2 – 15 ) – 1 = – 14.

**Disivion:**a ÷ ( b ÷ c ) ≠ ( a ÷ b ) ÷ c => 2 ÷( 3 ÷ 6 ) = 4 and ( 2 ÷ 3 ) ÷ 6 = 0.11.

For commutative property,

- Addition and multiplication of natural numbers show the commutative property. For example, x + y = y + x and a × b = b × a.
- Subtraction and division of natural numbers does not show the commutative property. For example, x – y ≠ y – x and x ÷ y ≠ y ÷ x.

- Multiplication of natural numbers is always distributive over addition. For example, a × (b + c) = ab + ac.
- Multiplication of natural numbers is also distributive over subtraction. For example, a × (b – c) = ab – ac.

An overview of algebraic operation with natural numbers i.e. addition, subtraction, multiplication, and division along with their respective properties are summarized in the table given below.

Properties and Operations on Natural Numbers | |||
---|---|---|---|

Operation |
Closure Property |
Commutative Property |
Associative Property |

Addition | Yes | Yes | Yes |

Subtraction | No | No | No |

Multiplication | Yes | Yes | Yes |

Division | No | No | No |

**Questions: **Sort out the natural numbers from the following list

20,1555, 63.99, 5/2, 60, −78, 0, −2, −3/2

**Solution: **Natural numbers from the above list are 20,1555 and 60.

**Question 2:** What are the first 10 natural numbers?

**Solution: **The first 10 natural numbers on the number line are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

**Question 3: **Is the number 0 a natural number?

**Solution: **0 is not a natural number. It is a whole number. Natural numbers only include positive integers and since zero does not have a positive or negative sign, it is not considered as a natural number.