# Natural Numbers

**Understanding Natural Numbers**

Before we begin explaining Natural Numbers, let us first get an idea of the concept with the help of this simple video:

**Introduction**

We see numbers everywhere around the world.

- For counting the objects, we use numbers.
- For representing money, we use numbers.
- For measuring the temperature, we use numbers, etc.

Especially, the numbers that are used for counting objects are called “**natural numbers**”.

While counting objects, we say like \(5\) cups, \(6\) leaves, \(1\) tiger, etc.

**Natural Numbers**

**Definition of Natural Numbers**

**Natural numbers are the numbers that are used for counting.**

**The set of natural numbers in Mathematics is the set** \(\textbf{ \{1,2,3,...\} } \).

Recall that a set** **is a collection of elements (numbers in this context).

The set of natural numbers is denoted by the symbol, \(N\).

N = \(\{1,2,3,4,5,...\}\) |

**Smallest Natural Number**

We note that the smallest element in \(N\) is 1 and that for every element in \(N\), we can talk about the next element in terms of 1 and \(N\) (which is 1 more than that element).

For example, two is one more than one, three is one more than two and so on.

**Natural Numbers from 1 to 100**

The **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 and 100

**Enter a number in the box below and you will see those many natural numbers.**

**Is 0 a Natural Number?**

No, \(0\) is NOT a natural number because natural numbers are counting numbers. For counting any number of objects, we start counting from \(1\) and not from \(0\).

**Odd Natural Numbers**

The odd natural numbers are the numbers which are odd and belong to the set \(N\). So the set of odd natural numbers is \(\{1,3,5,7,...\}\).

**Even Natural Numbers**

The even natural numbers are the numbers which are even and belong to the set \(N\). So the set of even natural numbers is \(\{2,4,6,8,...\}\).

**Whole Numbers**

The set of whole numbers is as same as the set of natural numbers, except it includes an additional number which is \(0\).

**The set of whole numbers in Mathematics is the set **\(\textbf{ \{0,1,2,3,...\} } \). It is denoted by the letter, \(W\)

W = \(\{0,1,2,3,4…\}\) |

**Natural Numbers and Whole Numbers**

From the above definitions, we can understand that every natural number is a whole number.

Also, every whole number other than \(0\) is a natural number.

The set of natural numbers and the set of whole numbers can be shown on the number line as follows:

We can say that the set of natural numbers is a subset of the set of whole numbers.

**Comparison between Natural numbers and Whole numbers**

Natural Number | Whole Number |
---|---|

The set of natural numbers is, \(N= \{1,2,3,...\}\) | The set of whole numbers is, \(W=\{0,1,2,3,...\}\) |

The smallest natural number is \(1\) | The smallest whole number is \(0\) |

We cannot determine the largest natural number | We cannot determine the largest whole number |

Each natural number is obtained by adding \(1\) to its previous number | Each whole number is obtained by adding \(1\) to its previous number |

\(0\) is NOT a natural number | \(0\) is a whole number |

**Properties of Natural Numbers**

**1. Closure Property:**

The sum and product of two natural numbers is always a natural number.

The closure property of \(N\) is stated as follows:

For all \(a,b \in N\)

\[a + b \in N\\ {\text{ and }}\\a \times b \in N\] |

So the set of natural numbers, \(N\) is closed under addition and multiplication.

**2. Associative Property:**

The sum or product of any three natural numbers remains the same though the grouping of numbers is changed.

The associative property of \(N\) is stated as follows:

For all \(a,b,c \in N\)

\[\begin{gathered} a + \left( {b + c} \right) = \left( {a + b} \right) + c \\ {\text{and}} \\ a \times \left( {b \times c} \right) = \left( {a \times b} \right) \times c \\ \end{gathered} \] |

So the set of natural numbers, \(N\) is associative under addition and multiplication.

**3. Commutative Property:**

The sum or product of two natural numbers remains the same even after interchanging the order of the numbers.

The commutative property of \(N\) is stated as follows:

For all \(a,b \in N\)

\[\begin{gathered} a + b = b + a \\ {\text{and}} \\ a \times b = b \times a \\ \end{gathered} \] |

So the set of natural numbers, \(N\) is commutative under addition and multiplication.

Let us summarise these three properties of natural numbers in a table.

Operation | Closure Property | Associative Property | Commutative Property |
---|---|---|---|

Addition | yes | yes | yes |

Subtraction | no | no | no |

Multiplication | yes | yes | yes |

Division | no | no | no |

**4. Distributive Property:**

The distributive property of multiplication over addition is

\[\begin{align}a \times (b+c) &= a \times b + a \times c\end{align} \] |

The distributive property of multiplication over subtraction is

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

To learn more about the properties of natural numbers, click here.

- Is \(N\) closed under subtraction and division?
- Is \(N\) associative under subtraction and division?
- Is \(N\) commutative under subtraction and division?

**Solved Examples**

Example 1 |

Identify the natural numbers among the following numbers.

\( -1, 0, 3, \dfrac{1}{2}, 5\).

**Solution: **

The set of natural numbers in Mathematics is the set \(\textbf{1, 2, 3, ...}\)

So among the given numbers, the natural numbers are, \(3 \text{ and }5\).

Example 2 |

Identify the natural numbers among the following numbers.

\( 7, 2, -3, -\dfrac{3}{5}, 0\).

**Solution: **

The set of natural numbers in Mathematics is the set \(\textbf{1, 2, 3, ...}\)

So among the given numbers, the whole numbers are, \(7, \text{ and }2\).

- \(0\) is Not a natural number but it is a whole number.
- Negative numbers, fractions and decimals are neither natural numbers nor whole numbers unless they can be simplified as a natural number or whole number.
- \(N\) is closed, associative and commutative under both addition and multiplication (but not under subtraction and division).

**Practice Questions**

**Here are few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.**

**Important Topics**

**Given below are the list of topics that are closely connected to natural numbers. These topics will also give you a glimpse of how such concepts are covered in Cuemath.**

**Frequently Asked Questions (FAQs)**

## 1. Is the number \(0\) a Natural number?

No, \(0\) is NOT a Natural number.

## 2. What is an example of a Natural number?

The set of Natural numbers in Mathematics is the set \(\textbf{1, 2, 3, ...}\). So one example can be \(5\).

## 3. Is \(23\) a Natural number?

The set of Natural numbers in Mathematics is the set \(\textbf{1, 2, 3, ...}\). So \(23\) is a Natural number.

## 4. Why are Natural numbers called Natural?

Natural numbers are called Natural because they are used for counting naturally. In a sense, the set of **Natural numbers** is the most basic system of numbers, because it is intuitive, or natural, and hence the name.

We use Natural numbers in our everyday life, in counting discrete objects, that is, objects which can be counted.

## 5. What are the first five Natural numbers?

The **first five Natural numbers** are \(1, 2, 3, 4 \text{ and } 5\).