Properties of Whole Numbers

There are various properties of whole numbers that help us to perform operations on whole numbers. These properties define the characteristics of operations. In this article, we are going to learn the properties of whole numbers under addition, subtraction, multiplication, and division.

Whole numbers are the natural numbers along with the number 0. The set of whole numbers in mathematics is the set {0,1,2,3,...}. It is denoted by the symbol, W. The four properties of whole numbers are as follows:

• Closure Property
• Associative Property
• Commutative Property
• Distributive Property

Let's explore all four properties of whole numbers in detail.

The closure property of the whole number states that "Addition and multiplication of two whole numbers is always a whole number." For example: 0+2=2. Here, 2 is a whole number. In the same way, multiply any two whole numbers and you will see that the product is again a whole number. For example, 3×5=15. Here, 15 is a whole number. Thus the set of whole numbers, W is closed under addition and multiplication.

The closure property of W is stated as follows:

For all a,b∈W, a+b∈W, and a×b∈W.

This property does not hold true in the case of subtraction and division operations on whole numbers. As, 0 and 2 are whole numbers, but 0 - 2 = -2, which is not a whole number. Similarly, 2/0 is not defined. Therefore, whole numbers are not closed under subtraction and division.

The associative property of whole numbers states that "The sum and the product of any three whole numbers remain the same regardless of how the numbers are grouped together or arranged".

Example 1: (1+2)+3 = 1+(2+3) because,

(1+2)+3 = 3+3 = 6

1+(2+3) = 1+5 = 6

Example 2: (1×2)×3 = 1×(2×3) because,

(1×2)×3 = 2×3 = 6

1×(2×3) = 1×6 = 6

Thus the set of whole numbers, W is associative under addition and multiplication. The associative property of W is stated as follows:

For all a,b,c∈W, a+(b+c)=(a+b)+c and a×(b×c)=(a×b)×c.

The associative property of whole numbers does not hold true for subtraction and division operations. It is because the arrangement of numbers is important in these operations. For example, 2, 3, and 4 are whole numbers, but 2 - (3 - 4) = 2 - (-1) = 3 and (2 - 3) - 4 = - 1 - 4 = -5. So, 3 ≠ -5. The same is with the division where 8 ÷ (4 ÷ 2) ≠ (8 ÷ 4) ÷ 2.

The commutative property of whole numbers states that "The sum and the product of two whole numbers remain the same even after interchanging the order of the numbers". It is the same as associative property, the only difference is that here we are only talking about two whole numbers.

Example 1: 2+3 = 3+2 because,

2+3 = 5

3+2 = 5

Example 2: 2×3 = 3×2 because,

2×3 = 6

3×2 = 6

Thus the set of whole numbers, W is commutative under addition and multiplication. The commutative property of W is stated as follows:

For all a,b∈W, a+b=b+a and a×b=b×a.

The commutative property of whole numbers does not hold true under subtraction and division.

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

The distributive property of multiplication over addition is a×(b+c)=a×b+a×c.

Example 1: 3×(2+5) = 3×2+3×5 because,

3×(2+5) = 3×7 = 21

3×2+3×5 = 6+15 = 21

The distributive property of multiplication over subtraction is a×(b−c)=a×b−a×c.

Example 2: 3×(5−2) = 3×5−3×2 as,

3×(5−2) = 3×3 = 9

3×5-3×2 = 15-6 = 9

To conclude, let us look at the chart of properties of whole numbers given below to understand which property is applicable to which operation.

Think Tank:

• Is W closed under subtraction and division?
• Is W associative under subtraction and division?
• Is W commutative under subtraction and division?

Challenging Questions on Properties of Whole Numbers:

• Find the product using the distributive property: 28×75.
• The set of whole numbers is commutative under which of the operations?
  (a) Addition (b) Subtraction
  (c) Multiplication (d) Division

► Also Check:

• Whole Numbers
• Properties of Integers
• Properties of Rational Numbers

What are the Properties of Whole Numbers?

Properties of whole numbers are a set of rules or laws that can be applied while doing basic arithmetic operations on whole numbers. For example, as per the commutative property of whole numbers, we can add 2 to 99 rather than adding 99 to 2.

What are the Four Properties of Whole Numbers?

The four properties of whole numbers are given below:

• Closure property
• Associative property
• Commutative property
• Distributive property

What are the Properties of Whole Numbers Under Addition and Multiplication?

The properties of whole numbers under addition are given below:

• Closure property ⇒ a + b ∈ W, ∀ a,b ∈ W.
• Associative property ⇒ a + (b + c) = (a + b) + c, ∀ a,b,c ∈ W.
• Commutative property ⇒ a + b = b + a, ∀ a,b ∈ W.
• Distributive property ⇒ a × (b + c) = (a × b) + (a × c), ∀ a,b,c ∈ W.
• Additive identity ⇒ 0 is the identity element for addtion of whole numbers as 0 + a = a + 0 = a, ∀ a ∈ W.

The properties of whole numbers under multiplication are mentioned below:

• Closure property ⇒ a × b ∈ W, ∀ a,b ∈ W.
• Associative property ⇒ a × (b × c) = (a × b) × c, ∀ a,b,c ∈ W.
• Commutative property ⇒ a × b = b × a, ∀ a,b ∈ W.
• Zero property ⇒ a × 0 = 0 × a = 0, ∀ a ∈ W.
• Multiplicative identity ⇒ 1 is the identity element for multiplication of whole numbers as 1 × a = a × 1 = a, ∀ a ∈ W.

What are the Properties for Division of Whole Numbers?

The properties of whole numbers under division are given below:

• 0 divided by any non-zero whole number always results in 0.
• 0 divided by 0 is not defined.
• Any non-zero whole number divided by 1 always results in the same whole number.
• Any non-zero whole number divided by itself always results in 1.
• The division of whole numbers satisfies the division algorithm which states "Dividend = Divisor × Quotient + Remainder".

What are the Properties for Subtraction of Whole Numbers?

The properties of whole numbers under subtraction are listed below:

• 0 subtracted from any whole number results in the same number.
• Any whole number subtracted from 0 results in its additive inverse.
• Closure, associative, and commutative properties do not hold true for subtraction.
• The distributive property of multiplication over subtraction satisfies. It implies, a × (b - c) = (a × b) - (a × c), ∀ a,b,c ∈ W.

What is the Associative Property of Addition in Whole Numbers?

The associative property of addition of whole numbers states that the order in which three numbers are arranged does not affect their sum. Mathematically, it can be expressed as a + (b + c) = (a + b) + c = (a + c) + b, ∀ a,b,c ∈ W.