Natural Numbers and Whole Numbers

Before we look at the difference between natural and whole numbers, we will see the fundamental definition of natural and whole numbers. Both natural and whole numbers are categorised under the real number system. 

Natural numbers are numbers that do not include zero, negative numbers, fractions, or decimals. Whole numbers are described as numbers that do not include negative numbers, fractions, or decimals. They can include zero. 

Hence, we can say that whole numbers are a group of positive (non-negative) integers and zero. We can say that natural numbers are a subset of whole numbers from the above statements. In the further sections, we will see how natural and whole numbers differ based on their definitions, properties, and applications.

Definition of natural and whole numbers 

Whole numbers 

Whole numbers are addressed by the symbol W. They are a set of positive (non-negative integers), including zero. They can be mathematically represented as W = { 0,1,2,3,4,……}

The set does not have any last term as it continues to infinity. The smallest whole number value is one, and the largest cannot be determined as the set continues to infinity.

Natural numbers 

Natural numbers can be described as a set of positive integers. It is addressed by the symbol N. The set of natural numbers is illustrated as N = { 1,2,3,4,….}

The natural numbers set is also an infinite set. The smallest value is 1, and the largest value is not determined as it is an infinite set.

Properties of whole numbers and natural numbers

Whole number 

The properties of whole numbers are elucidated below 

• The closure property 

The summation or multiplication of any two whole numbers will result in a whole number. The generalised form 

1. For addition operation – L + M = N – where L, M, and N are all whole numbers

2. For multiplication – L x M = N – where L, M, and N are whole numbers 

• Associative property 

If the grouping of any three whole numbers is changed, the addition and multiplication results will not change. The general form 

1. For Addition – (L + M) + N = L + (M + N)

2. For multiplication – (L x M) x N = L x (M x N) 

• Commutative property 

If the places of any two whole numbers are interchanged in the addition and multiplication operations, the result will be the same. The general form 

1. For addition – L + M = M + L = N

2. For multiplication – L x M = M x L = N

• Distributive property  

The result will remain the same when the multiplication operation is distributed over the addition and subtraction of any three whole numbers. 

1. For addition – L x ( M + N ) = (L x M) + (L x N)

2. For subtraction – L x ( M – N ) = (L x M) – (L x N)

• Additive identity 

This property says that when zero is added to any whole number, the result will always be a whole number (unchanged). General form: L + 0 = L 

• Multiplicative identity 

This property says that when any whole number is multiplied by 1, the value of the whole number remains the same. General form: L x 1 = L 

Natural numbers

• The closure property 

The summation or multiplication of any two natural numbers will result in a whole number. The generalised form 

1. For addition – C + D = E 

2. For multiplication – C x D = E 

where C, D, and E are natural numbers 

• Associative property 

If the grouping of any three natural numbers is changed, the addition and multiplication results will not change. The general form 

1. For Addition – (C + D) + E = C + (D + E)

2. For multiplication – (C x D) x E = C x (D x E) 

• Commutative property 

If the places of any two natural numbers are interchanged in the addition and multiplication operations, the result will be the same. The general form 

1. For addition – C + D = D + C = E

2. For multiplication – C x D = D x C = E

• Distributive property  

The result will remain the same when the multiplication operation is distributed over the addition and subtraction of any three whole numbers. 

1. For addition – C x ( D + E ) = (C x D) + (C x E)

2. For subtraction – C x ( D – E ) = (C x D) – (C x E)

Uses of natural and whole numbers 

Uses of natural numbers 

The predominant applications of natural numbers are: counting and ranking (ordering)

• Counting: is assigning the first object of the set with number one and then assigning the next object from the set as number two 

• Ranking: is assigning a natural number to an object based on the extreme values (biggest, smallest, tallest, etc.) 

Uses of whole numbers 

The predominant uses of whole numbers are the fundamentals of understanding complex concepts like real numbers and rational and irrational numbers. Approximating numbers to whole numbers makes calculations easier.

Conclusion 

In this article, we have looked at the definition of natural and whole numbers. Both natural and whole numbers are categorised under the real number system. Natural numbers are numbers that do not include zero, negative numbers, fractions, or decimals. In contrast, whole numbers are numbers that do not include negative numbers, fractions, or decimals. The difference between natural and whole numbers has been elucidated. The properties of whole numbers are closure, associative, distributive, multiplicative identity, additive identity, and commutative. The properties of natural numbers are closure, associative, distributive, and commutative. The uses of whole numbers are – helpful in rounding off and understanding concepts of rational, irrational and complex numbers. The uses of natural numbers are – counting and ranking.