Table of Contents

Last modified on August 3rd, 2023

Natural numbers are the real numbers that consist of all the positive numbers from 1 to infinity, excluding zero (0), decimals, fractions, and negative numbers. We use natural numbers to count.

Another name for natural numbers is integers. However, all integers, including negative numbers, are not natural numbers.

In other words, natural numbers include all whole numbers excluding zero (0).

we can understand the representation of natural numbers on a number line from the link given below:

So **are natural numbers whole numbers?**

NO. All Whole numbers are natural numbers but all natural numbers are not whole numbers since they do not include ‘0’.

*The symbol used to represent *natural numbers is the alphabet ‘N’ in capital letters.

The **set of natural numbers** is represented as below:

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

As we know, that natural numbers include only the positive integers; here are some examples of natural numbers:

2, 23, 54, 742, 8651, 99726, 6564132, etc.

As discussed above, Natural numbers are the positive numbers till infinity, we can write the list of natural numbers from 1 to 100.

Closure property means that when added or multiplied, 2 natural numbers give a natural number. However, it does not hold for subtraction or division. When subtracted or divided, 2 natural numbers may or may not give a natural number as a result.

- Addition: 5 + 2 = 7, 11 + 5 = 16; for both the examples, the result is a natural number.
- Multiplication: 8 × 2 = 16, 4 × 5 = 20; both the examples, the result is a natural number.
- Subtraction: 15 – 5 = 10, 9 – 12 = -3; here, the result may or may not be a natural number.
- Division: 18 ÷ 6 = 3, 20 ÷ 3 = 6.667; also here, the result may or may not be a natural number.

Commutative property means that the sum and the product of any 2 natural numbers remain the same even if we interchange their order. The property does not apply to subtraction or division.

⇒ a + b = b + a, a × b = b × a here a and b are 2 whole numbers

⇒ a – b ≠ b – a, a ÷ b ≠ b ÷ a

The sum or product of any 3 natural numbers remains the same even if we change the order of numbers. The property does not apply to subtraction or division.

- Addition: a + (b + c) = (a + b) + c; ⇒ 3 + (4 + 5) = (3 + 4) + 5 = (5 + 3) + 4 = 12
- Multiplication: a × ( b × c ) = ( a × b ) × c; ⇒2 × (4 × 6) = (2 × 4) × 6 = (6 × 4) × 2 = 48
- Subtraction: a – ( b – c ) ≠ ( a – b ) – c; ⇒ 3 -(11 -1 ) = – 17 and ( 3 – 11 ) – 1 = – 9
- Division: a ÷ ( b ÷ c ) ≠ ( a ÷ b ) ÷ c ⇒ 9 ÷( 6 ÷ 2 ) = 3 and ( 9 ÷ 6 ) ÷ 2 = 0.75

According to this property, the multiplication of a natural number is distributed over the sum of the whole numbers.

- a × (b + c) = (a × b) + (a × c); ⇒ 2 × (3 + 4) = (2 × 3) + (2 × 4) = 14

The same property holds true for subtraction as well.

⇒ a × (b − c) = (a × b) − (a × c); ⇒ 4 × (7 − 2) = (4 × 7) − (4 × 2) = 20

The properties of natural numbers with the operations in tabular form are given below.

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

Addition | Yes | Yes | Yes |

Subtraction | No | No | No |

Multiplication | Yes | Yes | Yes |

Division | No | No | No |

Let us solve some examples involving natural numbers.

**Find two consecutive natural numbers whose sum is 87.**

Solution:

As we know,

When added or multiplied, 2 natural numbers give a natural number as a result

Let us take the number as 86

Dividing it by 2

⇒ 86/2 = 43

Now, the number consecutive to 43 is 44.

Adding 43 + 44 = 87

∴Two consecutive natural numbers whosesum is 87 are 43 and 44.

**Find out the natural numbers from the following list: 15, 200, 72.85, 7/5, 90, −24, 0, −9, −3/2.**

Solution:

As we know,

Natural numbers do not include zero, decimals, fractions, and negative numbers,

The natural numbers here are 15, 200, 90.

**Ans. **No. ‘0’ is a whole number.

**Ans**. No. all natural numbers are positive integers.

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Last modified on August 3rd, 2023