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Positive numbers from 1 onwards to infinity are known as natural numbers. Natural numbers are also called counting numbers. Here we will learn about natural numbers in detail....Read MoreRead Less

We use numbers in our daily life to indicate the count of a quantity. For example, the count of number of students in a class, measure of distance, expenses, measure of weight and height, and so on are represented using numbers. A number is represented using numerical figures like 1, 2, …, 10, 11, .. and so on. The figures from 0 to 9 are called digits. A number can be 1 digit, 2 digits, 3 digits and so on.

**Definition**: The set of positive integers from 1 to infinity are called natural numbers. In other words, whole numbers excluding zero are called natural numbers.

*[Note: Natural numbers are also called counting numbers and we start counting from the number 1. Zero is not a part of natural numbers.]*

A set of natural numbers is denoted by “N”.

Therefore, N = 1, 2, 3, 4, 5, 6, 7………………∞

As we have discussed above, all positive integers are natural numbers, so on the number line the values on the right of the zero (0) mark are natural numbers. The positive integers including zero are whole numbers.

The properties of natural numbers are based on the 4 basic arithmetic operations – addition, subtraction, multiplication and division.

- Commutative property
- Associative property
- Distributive property
- Closure property

**1. Commutative property: **The commutative property states that the order of terms does not matter in basic arithmetic operations like addition and multiplication. This implies that addition or multiplication of any two natural numbers is not altered by a change in the order of numbers. For example,

5 + 7 = 12 and also 7 + 5 = 12

\(5~\times~7=35\) and also \(7~\times~5=35\)

In general If there are two natural numbers x and y then,

- x + y = y + x (commutative property of addition)
- \(x~\times~y=y~\times~x\) (commutative property of multiplication)

*[Note: Commutative property does not apply to subtraction and division operation.]*

**2. Associative property: **The associative property of natural numbers states that addition or multiplication of more than two natural numbers is not altered by a change in the order in which the numbers are added or multiplied. This property is only valid for addition and multiplication. For example,

3 + 6 + 7 = (3 + 6) + 7 = 9 + 7 = 16

Also, 3 + 6 + 7 = 3 + (6 + 7) = 3 + 13 = 16

\(3~\times~6~\times~7=(3~\times~6)~\times~7=18~\times~7=126\)

\(3~\times~6~\times~7=3~\times~(6~\times~7)=3~\times~42=126\)

In general, if there are three natural numbers x , y and z then;

- (x + y) + z = x + (y + z) (Associative property of addition)
- \((x~\times~y)~\times~z=x~\times~(y~\times~z)\) (Associative property of multiplication)

*[Note: Associative property does not apply to subtraction and division operations.]*

**3. Distributive property of multiplication:** The distributive property states that the product of numbers can be distributed over addition and subtraction. In general,

\(x~\times~(y+z)=xy+xz\) (Distributive property over addition/sum)

\(x~\times~(y-z)=xy-xz\) (Distributive property over subtraction)

**4. Closure property of addition and multiplication:** This property states that the addition and multiplication of two or more natural numbers results in a natural number.

*[Note: Closure property does not apply to subtraction and division operation.]*

**Example 1:** List the first 10 even natural numbers.

**Solution:** FIrst 10 even natural numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.

**Example 2:** Is the closure property of natural numbers valid for subtraction?

**Solution:** Let us take two natural numbers 3 and 9

3 – 9 = -6

Here -6 is not a natural number, so the closure property of natural numbers is not valid for subtraction.

**Example 3:** Write down the first 5 natural numbers which are multiples of 4.

**Solution: **

4 x 1 = 4

4 x 2 = 8

4 x 3 = 12

4 x 4 = 16

4 x 5 = 20

So, the first 5 natural numbers which are multiple of 4 are 4, 8, 12, 16 and 20.

Frequently Asked Questions on Natural Numbers

Natural numbers start from 1, so the smallest natural number is 1.

The numbers 0, 1, 2, 3 and so on are whole numbers. Natural numbers are whole numbers excluding zero, that is, 1, 2, 3, and so on.

2 is the smallest even natural number.

No, integers include all positive and negative numbers along with zero but natural numbers are only positive integers, so every integer is not a natural number.

Yes, we can classify natural numbers as odd natural numbers and even natural numbers.