## What Are the Properties of Natural Numbers?

**Properties of natural numbers in mathematics are certain rules that can be applied or the characteristics of natural numbers that can be used when we perform arithmetic operations on them.**

The four basic properties of natural numbers are:

- Closure Property
- Associative Property
- Commutative Property
- Distributive Property

Natural numbers are 1, 2, 3, 4, 5, 6, and go on till infinity. They are also called **counting numbers** as they are used to count objects. Natural numbers do not include 0 or negative numbers. The alphabet **ℕ** is used as a symbol to refer to natural numbers.

The basic operation of addition, subtraction, multiplication, and division give rise to four main properties of natural numbers.

The image shown above simply gives a quick overview of each property. Let’s understand each property in detail with examples.

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## Closure Property of Natural Numbers

**Closure property of addition**

When you add two natural numbers, the result will always be a natural number.

Examples of closure property of addition: $2 + 2 = 4, 3 + 4 = 7, 5 + 5 = 10$

In each case, the result of the addition of natural numbers is a natural number.

**Closure property of multiplication**

When you multiply two natural numbers, the result will always be a natural number.

Examples of closure property of multiplication: $2 \times 2 = 4, 3 \times 2 = 6, 5 \times5 = 25$

In each case, the result of the multiplication of natural numbers is a natural number.

**Thus, we say that the natural numbers are closed under addition and multiplication.**

**However, in the case of division and subtraction, this property does not hold true. Subtracting or dividing two natural numbers will not always give us a natural number. **

Examples: $4 \;-\; 6 = \;-\;2,\; 5 \;-\; 3 = 2,\; 6 \;-\; 9 = \;-\;3$

The second case resulted in a natural number but the first and third ones did not.

Examples of division: $10 \div 3 = 3.33,\; 9 \div 3 = 3,\; 15 \div 4 = 3.75$

The first and third cases did not result in natural numbers.

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## Associative Property of Natural Numbers

**Associative property of addition**

The sum of natural numbers remains unchanged even if the grouping of numbers is changed.

It is expressed in the form of an equation as $a + (b + c) = (a + b) + c$

Examples of associative property of addition: $2 + (5 + 6) = 13$ and $(2 + 5) + 6 = 13$

**Associative property of multiplication**

The product of natural numbers remains unchanged even if the grouping of numbers is changed. It is expressed in the form of an equation as $a \times (b \times c) = (a \times b) \times c$

Examples of associative property of multiplication: $2 \times (3 \times 4) = 24$ and $(2 \times 3) \times 4 = 24$

Let us now look at the nature of subtraction and division with respect to this property.

**Associative property does not hold true for subtraction and division.**

$a \;-\; (b \;-\; c) \neq (a \;-\; b) \;-\; c$

$a \div (b \div c) \neq (a \div b) \div c$

Examples of subtraction: $4 \;-\;(10 \;-\; 2) = \;-\; 4$ and $(4 \;-\; 10) \;-\; 2 = \;-\; 8$

Examples of division: $5 \div (6 \div 3) = 2.5$ and $(5 \div 6) \div 3 = 0.27$

## Commutative Property of Natural Numbers

**Commutative property of addition**

If we change the order of natural numbers during addition, the result does not change.

It is expressed in the form of an equation as $(a + b) = (b + a)$

Examples of commutative property of addition: $6 + 5 = 11$ and $5 + 6 = 11$

**Commutative property of multiplication**

If we change the order of natural numbers during multiplication, the result does not change.

It can be expressed in the form of an equation as $(a \times b) = (b \times a)$

Examples of commutative property of multiplication: $2 \times 4 = 8$ and $4 \times 2 = 8$

**The commutative property does not apply to subtraction and division of natural numbers. **

$a \;-\; b \neq b \;-\; a$

$a \div b \neq b \div a$

Examples for subtraction: $5 \;-\; 3 = 2$ and $3 \;-\; 5 = \;-\; 2$

Examples for division: $6 \div 3 = 2$ and $3 \div 6 = 0.5$

## Distributive Property of Natural Numbers

According to the distributive property of multiplication over addition, if we multiply the total of two addends by a number or multiply each addend individually and then add them, the result will be the same.

**Distributive property of multiplication over addition:** $a(b + c) = (a \times b) + (a \times c)$

Example: $2 \times (5 + 3) = 16$

$(2 \times5) + (2 \times 3) = 16$

This property also holds true in the case of multiplication over subtraction.

**Distributive property of multiplication over subtraction: **$a(b \;-\; c) = (a \times b) \;-\; (a \times c)$

Example: $2 \times (5 \;–\; 3) = 4$

$(2 \times 5) \;–\; (2 \times 3) = 4$

## Facts about Properties of Natural Numbers

- There are an infinite number of natural numbers. There is no largest natural number. Natural numbers go on forever. The smallest natural number is 1.

- By simply adding 1 to a given natural number, you will get the next natural number.

- For the natural number 1, there is no “predecessor” or a previous natural number.

($1 = 0 + 1$, but we know that 0 is not a natural number).

- Natural numbers are also called counting numbers as they are used to count objects.

- The properties of natural numbers do not hold true in the case of division and

subtraction.

- Negative numbers, fractions, and decimals are neither natural numbers nor whole numbers.

- Even natural numbers $= E = \left\{2, 4, 6, 8, 10, …\right\}$

Odd natural numbers $= O = \left\{1, 3, 5, 7, 9, 11, …\right\}$

## Conclusion

In this article, we explored natural numbers and its properties. These properties make the natural number set unique. In this process, we also understand that division and subtraction of natural numbers is not guaranteed to be a natural number, but that there are incidences where the result is a natural number. Now let’s apply this knowledge to solve some examples.

## Solved Examples on Properties of Natural Numbers

**1. Which of the following numbers are natural numbers? **

$2, 18, \;-\;5, 25.5, 1, \;-\;20$**. **

**Solution: **

Natural numbers are given by 1, 2, 3, 4, 5, … and so on.

The numbers 2, 18, and 1 are natural numbers.

The numbers $\;-\;5$ and $\;-\;20$ are not natural numbers because they are negative.

25.5 is not a natural number because it is a decimal.

**2. State whether each of the given statements is True or False.**

**There is always a natural number between any two consecutive natural numbers.****Subtraction of two natural numbers is always a natural number.****Natural numbers do not include 0.**

**Solution:**

- The statement is false. Between two consecutive natural numbers, there is no natural number.

For example, between 1 and 2, no natural number exists as 1.1, 1.2, 1.3, 1.7, 1.9, etc. are decimal numbers.

- The statement is false. The natural numbers are not closed under subtraction. It means that the subtraction of two natural numbers may or may not be a natural number.

- True. 0 is not a natural number. It is a whole number.

**3. Solve the expression 2 (20 + 15) using the distributive property of multiplication over addition.**

**Solution: **

According to distributive property,

$a(b + c) = (a \times b) + (a \times c)$

$2 (20 + 15) = (2 \times 20) + (2 \times 15)$

$= 40 + 30$

$= 70$

Thus, $2 (20 + 15) = 70$

**4. Identify the properties of the natural numbers based on the expressions given below**

**a) **$2 + (5 + 6) = (2 + 5) + 6$

**b) **$(10 + 15) = (15 + 10)$

**c) **$4 \times (6 \times 8) = (4 \times 6) 8$

**Solution: **

a) $2 + (5 + 6) = (2 + 5) + 6$

This is an example of the associative property of addition.

b) $(10 + 15) = (15 + 10)$

This is an example of the commutative property of addition.

c) $4 \times (6 \times 8) = (4 \times 6) \times 8$

This is an example of the associative property of multiplication.

## Practice Problems on Properties of Natural Numbers

## Properties of Natural Numbers: Definitions, Examples, Facts

### The smallest natural number is _______.

The smallest natural number is 1.

### The natural numbers do not follow the closure property under _______.

The subtraction of two natural numbers may not be a natural number. The natural numbers are not closed under subtraction.

### The sum and product of two natural numbers is always a natural number. This property is called ______.

According to the closure property, when you add or multiply two natural numbers, the result will always be a natural number.

### If a, b and c are natural numbers then $(a + b) + c = a + (b + c)$. This property is called ______.

According to associative property, the sum or product of natural numbers remains unchanged even if the grouping of numbers is changed. It is given in the form of equation as $(a + b) + c = a + (b + c)$

### Which of the following is an example of the commutative property of multiplication?

According to commutative property, if we change the order of natural numbers during multiplication, the result does not change. Thus, $2 \times 4 = 4 \times 2 = 8$

## Frequently Asked Questions on Properties of Natural Numbers

**Is **$-1$** a natural number?**

The set of natural numbers in mathematics is the set {1, 2, 3, …}. Since **–**1 is a negative number, it is not a natural number.

**Are natural numbers whole numbers?**

All natural numbers are whole numbers. Whole numbers include natural numbers and 0.

**Is 0 a natural number ?**

Natural numbers are a subset of real numbers that only include positive integers like 1, 2, 3, 4, 5, 6, and so on, while excluding negative numbers, zero, decimals, and fractions. They do not comprise negative numbers or zero.

**What are cardinal numbers ?**

Cardinal numbers are the numbers used for the purpose of counting. Cardinal numbers are natural numbers or positive integers. The smallest cardinal number is 1. Examples of cardinal numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, and so on. The smallest cardinal number is 1.

**Is associative property true for subtraction and division of natural numbers ?**

The associative property holds true in case of addition and multiplication of natural numbers.

But for subtraction and division of natural numbers, the associative property does not hold true.

For example, $5 \;-\; (3 \;-\; 2 ) = 4$ and $(5 \;-\; 3) \;-\; 2 = 0$

$30 \div (10 \div 5 ) = 15$ and $(30 \div 10) \div 5 = \frac{3}{5}$

**Is a square of a natural number also a natural number?**

A square of a natural number is a natural number multiplied by itself. Since natural numbers follow the closure property, the square of a natural number is also a natural number. For example, $5^2 = 5 \times 5 = 25$, and $25$ is a natural number.

**Is the square root of a natural number always a natural number?**

The square root of a natural number may or may not be a natural number.

For example, the square root of 9 is 3, square root of 16 is 4, which are natural numbers, but the square root of 6 is approximately 2.449, which is not a natural number.