Whole Numbers

In our daily lives, we keep count of various things, such as age, quantities, and items. The counting numbers in mathematics are referred to as natural numbers. Zero, along with natural numbers, is included in the collection of whole numbers. For instance, 0, 2, 77, and 9999 are some examples of whole numbers, and the set of whole numbers goes up to infinity. The capital letter ‘W’ is used to represent whole numbers. 

Understanding the differences between whole and natural numbers helps us easily distinguish between numbers and solve calculations efficiently. 
 

Whole numbers can be represented visually using a number line. It is a horizontal line that includes all positive integers and zero, arranged in order. The starting point of the number line is zero, and it consists of whole numbers and the natural numbers, as seen below:  
 

Whole numbers are a fundamental aspect of mathematics, consisting of natural numbers along with zero. Understanding the key properties of whole numbers helps in solving complex mathematical problems more easily and strengthens the foundation of arithmetic knowledge. The properties of whole numbers include: 

• Closure property: When we add or multiply two whole numbers, the result is always a whole number. This means whole numbers are closed under addition and multiplication because these operations always give whole numbers. This property can be represented as follows:
  
  If x and y are whole numbers, then x + y ∈ W and x × y ∈ W.
  
  Here, W represents whole number. 
  
  For example, 2 + 3 = 5, which is a whole number.

2 × 4 = 8, which is a whole number.

• Commutative property: When we switch the order of two whole numbers in addition or multiplication, the result remains the same regardless of order. I
  
  The commutative property of addition is: x + y = y + x.
  
  The commutative property of multiplication is: x × y = y × x.
  
  For instance, 3 + 1 = 1 + 3 = 4

4 × 3 = 3 × 4 = 12

• Additive identity: If we add a whole number with zero the result is always the same whole number, i.e., x + 0 = x
  
  For example, 8 + 0 = 8 

• Multiplicative identity: If we multiply a whole number by 1 the result is always the same whole number. It is represented as: x × 1 = x
  
  For instance, 4 × 1 = 4

• Associative property: If the grouping of three whole numbers is changed in addition or multiplication, the result stays the same. This is represented as:
  
  x + (y + z) = (x + y) + z (associative property of addition).
  
  x × (y × z) = (x × y) × z (associative property of multiplication).
  
  For instance, 1 + (5 + 2) = 1 + 7 = 8 

(1 + 5) + 2 = 6 + 2 = 8

Likewise, 1 × (5 × 2) = 1 × 10 = 10
(1 × 5) × 2 = 5 × 2 = 10

• Distributive property: The multiplication of a whole number is distributed over the sum or difference of the whole numbers.
  
  It is represented as: x × (y + z) = (x × y) + (x × z).
  
  For example, take a look at this: 

2 × (3 + 6) = 2 × 9 = 18
(2 × 3) + (2 × 6) = 6 + 12 = 18

Thus, 2 × (3 + 6) = (2 × 3) + (2 × 6)

• Multiplication by zero: If we multiply a whole number by zero, the result is always zero, i.e., a × 0 = 0

For instance, 14 × 0 = 0

• Division by zero: Division by zero is undefined. This is expressed as: 

a/0 is undefined. 
 

In our daily lives, we count objects and items like fruits, vehicles, people, and ages using
whole numbers. Whole numbers are vital in various fields to indicate and represent counts. Whole numbers are widely applied in finance, construction, manufacturing, and population studies.

• Whole numbers are used in the fields of banking and finance to indicate deposit interest rates and withdrawal amounts of customers. It makes the transactions easy to understand and it reduces confusion.

• In engineering and construction, whole numbers are used to represent the measured distance, area, volume, and time. For example, the distance between two cities might be 30 km, which is represented using whole numbers.

• Manufacturers and producers use whole numbers to keep track of the number of products available in stock or currently being produced.

• Whole numbers are used in demographic studies to represent the total population of a city or country. For example, the population of a country is recorded in a census.