Properties of Natural Numbers

Properties of natural numbers refer to the result of four main arithmetic operations on natural numbers. Natural numbers are a set of whole numbers, excluding zero. These numbers are used in our day-to-day activities and speech. Natural numbers are one of the classifications under real numbers, that include only the positive integers i.e. 1, 2, 3, 4,5,6, ………. excluding zero, fractions, decimals, and negative numbers. Remember that the set of natural numbers do not include negative numbers or zero.

In this article, you will learn about the properties of natural numbers in detail.

Natural numbers are the numbers that are positive integers and include numbers from 1 till infinity(∞). These numbers are countable and are generally used for calculation purposes. The set of natural numbers in Mathematics is the set starting from 1, that is {1,2,3,...}. The set of natural numbers is denoted by the symbol, N. The four properties of natural numbers are as follows:

1. Closure Property
2. Associative Property
3. Commutative Property
4. Distributive Property

Let's explore them in detail.

Closure property of natural numbers states that the addition and multiplication of two or more natural numbers always result in a natural number. Let's check for all four arithmetic operations and for all a, b ∈ N.

• Addition: 1 + 5 = 6, 7 + 4 = 11, etc. Clearly, the resulting number or the sum is a natural number. Thus, a + b ∈ N, for all a, b ∈ N.
• Multiplication: 2 × 5 = 10, 6 × 4 = 24, etc. Clearly, the resulting number or the product is a natural number. Thus, a × b ∈ N, for all a, b ∈ N.
• Subtraction: 8 – 5 = 3, 7 - 2 = -5, etc. Clearly, the result may or may not be a natural number. Thus, a - b or b - a ∉ N, for all a, b ∈ N.
• Division: 15 ÷ 5 = 3, 10 ÷ 3 = 3.33, etc. Clearly, the resultant number may or may not be a natural number. Thus, a ÷ b or b ÷ a ∉ N, for all a, b ∈ N.

Therefore, we can conclude that the set of natural numbers is always closed under addition and multiplication but the case is not the same for subtraction and division.

Associative property of natural numbers states that the sum or product of any three natural numbers remains the same though the grouping of numbers is changed. Let's check for all four arithmetic operations and for all a, b, c ∈ N.

• Addition: a + ( b + c ) = ( a + b ) + c. 3 + (15 + 1 ) = 19 and (3 + 15 ) + 1 = 19.
• 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.
• Division: a ÷ ( b ÷ c ) ≠ ( a ÷ b ) ÷ c. 2 ÷( 3 ÷ 6 ) = 4 and ( 2 ÷ 3 ) ÷ 6 = 0.11.

Therefore, we can conclude that the set of natural numbers is associative under addition and multiplication but the case is not the same for subtraction and division. So, the associative property of N is stated as follows: For all a, b, c ∈ N, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c

Commutative property of natural numbers states that the sum or product of two natural numbers remains the same even after interchanging the order of the numbers. Let's check for all four arithmetic operations and for all a, b ∈ N.

• Addition: a + b = b + a.
• Multiplication: a × b = b × a
• Subtraction: a – b ≠ b – a
• Division: a ÷ b ≠ b ÷ a

Therefore, we can conclude that the set of natural numbers is commutative under addition and multiplication but the case is not the same for subtraction and division. So, the commutative property of N is stated as follows: For all a, b ∈ N, a + b = b + a and a × b = b × a

Distributive property of natural numbers states any expression with three numbers a, b, and c, given in the form a (b + c) then it is resolved as a × (b + c) = ab + ac or a (b - c) = ab - ca, meaning that the operand a is distributed among the other two operands, b, and c.

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

Example: 3 × (2 + 5) = 3 × 2 + 3 × 5

3 × (2 + 5) = 3 × 7 = 21

3 × 2 + 3 × 5 = 6 + 15 = 21

3 × (2 + 5) = 3 × 7 = 21

3 × 2 + 3 × 5 = 6 + 15 = 21

a × (b − c) = a × b − a × c

Example: 3 × (2 − 5) = 3 × 2 − 3 × 5

3 × (2 −5) = 3×(−3) = −9

3 × 2 − 3 × 5 = 6 − 15 = −9

Related Articles

Check out these interesting articles related to the properties of natural numbers for an in-depth understanding.

• Distributive Property of Multiplication
• Commutative Property
• Associative Property
• Distributive Property Calculator

What Are the Properties of Natural Numbers in Math?

The properties of natural numbers are:

1. Closure property
2. Associative property
3. Commutative property
4. Distributive property

Is the Set of Natural Numbers Associative under Division?

The set of natural numbers is NOT associative under division. For example, let us consider three natural numbers 6,4, and 2. Then: (6÷4)÷2 = 3÷2=1. 6÷(4÷2) = 6÷2 = 3. Thus, (6÷4)÷2 ≠ 6÷(4÷2).

What Do You Mean by Commutative Property of Addition?

As per the commutative property of addition, natural numbers can be added in any order, and their answer will remain the same answer. The formula for this property is a + b = b + a, which holds true for any a, b ∈ N. For example, 1 + 2 or 2 + 1, both will give the same answer.

What Does Associative Property of Addition Mean?

The associative property of addition is the property of natural numbers which states that the sum of three or more numbers will not change even though the grouping of numbers is changed. The corresponding equation is a + ( b + c ) = ( a + b ) + c . Here, grouping refers to how the given numbers are arranged in parentheses.

Which Equation Shows the Commutative Property of Addition?

The equation that shows the commutative property of addition is 'a + b = b + a'. Let's take an example, 4 + 3 = 3 + 4. Here, the sum at both sides of the equation is the same, that is 7.

Which Equation Shows the Distributive Property of Multiplication?

The equation which shows the distributive property of multiplication is 'a ( b + c ) = a b + a c'. Here, the terms within the parentheses can't be simplified because of one or more variables.