Natural Numbers and Real Numbers

Before we look at the difference between natural numbers and real numbers, we will first see how natural numbers and real numbers are defined. A real number is defined as a quantity that can represent a distance in a line. This means that any point on the number line can be a real number. This set of numbers can include both rational and irrational numbers. The real number system can be split into natural numbers, whole numbers, and rational and irrational numbers. We can conclude that natural numbers are considered a subcategory of real numbers. Natural numbers include numbers starting with one and can be extended up to infinity. This set of numbers does not include zero. Hence the number 1 is the smallest natural number. Natural numbers exclude decimals, fractions, and negative numbers. In the further sections, we will explore the properties of natural and real numbers and see how they differ from each other.

The Difference between Natural Numbers and Real Numbers is based on the types

Natural Numbers 

In mathematics, counting and ranking items are done using natural numbers. This unique set of numbers has been used in daily activities. The alphabet N is used as a symbol to address natural numbers. This set of numbers is represented as N = {1,2,3,4,……….}. They are also known for their synonym counting numbers. There can be no largest natural number as the natural number set extends to infinity.

The unique types of natural numbers are: 

• Even natural numbers 

It is expressed as A = B x 2 

Here, A is an even natural number 

Examples – 2,4,6,… and so on. 

• Odd natural numbers 

It is expressed as A = (B x 2) + 1 

Here, A is an odd natural number 

Examples – 1,3,5,…and so on. 

• Composite natural numbers 

It is expressed as A = B x C, where B and C should not be equal to one or zero 

Examples – 4,6,8,…. and so on. 

• Prime natural numbers 

A natural number is called a prime number if it is not a composite number and not 1 or 0. 

Examples – 2,3,5,7,…. so on. 

• Square natural numbers 

Expressed as A = B x B

Here, A is the natural square number of B 

Examples – 1,9,25… so on. 

Real Numbers 

Real numbers can be defined as a set with an infinite number of numbers. This is why there can be no smallest or largest number. Hence, we can also say that real numbers cannot be counted as all the numbers cannot be put into a logical sequence. The alphabet R is used to address real numbers. 

The various types of real numbers are 

• Natural numbers – values above zero and excluding negative, fraction, and decimal numbers 

• Integers – values that have a complete number. This set of numbers includes both negative and positive numbers

Examples: -1, -2, 0, 1, 2, …

• Rational numbers – values that can be represented as p/q. Where P is the numerator of any value and q is a non-zero value

Examples: 25/3, 2/9, …

• Irrational numbers – values that cannot be represented as p/q

Examples -3, 5,…

• Whole numbers – values starting from zero and extending up to infinity. They cannot be negative, fractional, or decimal numbers

The Difference between Natural Numbers and Real Numbers is based on their properties

Properties of Natural Numbers

Four predominant properties of natural numbers are: 

• Closure property 

When any two natural numbers are multiplied or added, they always result in a natural number. It is true for addition and multiplication operations only.

The expression

1. For addition – C + D = E

2. For multiplication – C x D = E

• Commutative property 

When the places of any two natural numbers are interchanged in addition and multiplication operations, the result will remain the same. The expressions are as follows:

1. For Addition – C + D = D + C  

2. For Multiplication – C x D = D x C  

• Distributive property 

When an expression is of the form C x ( D + E) or C x ( D – E), the C operand can be distributed over the other two operands D and E. The expressions are as follows:

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 

• Associative property

When the grouping of three natural numbers is changed, the answer will remain the same. The expressions are as follows:

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

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

Properties of Real Numbers 

The properties of real numbers are elucidated below: 

• Closure property 

When any two real numbers are multiplied or added, they always result in a real number. 

1. For addition – L + M = N 

2. For multiplication – L x M = N

• Commutative property 

If the places of any two real numbers are interchanged in addition and multiplication operations, the result will remain the same. 

1. For Addition – L + M = M + L

2. For Multiplication – L x M = M x L  

• Distributive property 

If there is an expression of the form L x ( M + N) or L x ( M – N), the L operand can be distributed over the other two operands M and N. 

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

• Associative property

If the grouping of three real numbers is interchanged, the answer will remain the same. 

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

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

• Inverse property 

1. Additive inverse – M + ( -M) = 0 

When we add a real number with the negative counter value, we get a zero 

  2. Multiplicative inverse – M x ( 1/M) = 1

When we multiply a real number with its reciprocal, we always get one. 

Conclusion 

A real number is defined as a quantity that can represent a distance in a line. The real number system can be split into natural numbers, whole numbers, and rational and irrational numbers. Natural numbers include numbers starting with one and can be extended up to infinity. The types of natural numbers are odd, even, prime, composite, and square natural numbers. The natural number properties are – closure, associative, commutative, and distributive. The properties of real numbers are – closure, associative, commutative, distributive, and inverse properties.