Properties of Whole Numbers

The fundamental arithmetic operations such as addition, subtraction, multiplication, and division on whole numbers lead to four main properties of whole numbers, such as closure property, commutative property, associative property, and distributive property.

Closure property

The sum and product of two whole numbers is also a whole number and can be closed under addition and multiplication.

For example, the sum of two whole numbers, 14 and 18, is 32, which is also a whole number [14+18 = 32], and their product is 252, which is also a whole number [14 × 18 = 252].

Commutative Property

The sum and product of whole numbers remain the same even if the order of the numbers is interchanged. Let us consider two whole numbers, ‘a’ and ‘b’. Then, according to the commutative property,

• a + b = b + a 
• a × b = b × a

Note that the commutative property does not hold true in the case of subtraction and division of whole numbers.

Additive Identity

When any whole number is added to 0, then its value remains unchanged. For example, let us consider a whole number “a”, then

• a + 0 = 0 + a = a

Example: 5 + 0 = 0 + 5 = 5

Multiplicative Identity

When any whole number is multiplied by 1, then its value remains unchanged. For example, let us consider a whole number “a”, then 

• a × 1 = 1 × a = a

Example: 8 × 1 = 1 × 8 = 8

Associative property

The sum or product of any three whole numbers remains unchanged even if the grouping of numbers is changed. Let us consider three whole numbers, ‘a’, ‘b’, and ‘c’. Then, according to the associative property,

• a + (b + c) = (a + b) + c
• a × (b × c) = (a × b) × c

Distributive Property

The distributive property states that the multiplication of a whole number is distributed over the sum or difference of the whole numbers. Let us consider three whole numbers, ‘a’, ‘b’, and ‘c’. Then the distributive property states that,

• a × (b + c) = (a × b) + (a × c)
• a × (b – c) = (a × b) – (a × c)

A number system refers to a system that represents numbers, where a number is defined as the mathematical value that helps to count, measure, or label and perform various mathematical calculations. We have various types of numbers based on their properties, such as natural numbers, whole numbers, integers, rational and irrational numbers, real numbers, etc. In a number system, these numbers are used as digits. A number system also helps us to perform various arithmetic operations such as addition, subtraction, and division. A binary number system, an octal number system, a decimal number system, and a decimal number system are different types of number systems.