Natural Numbers

What are natural numbers? – An introduction

Natural numbers are the rudimentary numbers starting from 1exending to infinity and thus also called an infinite set. This set of numbers does not include negative numbers, fractions, or decimal numbers. 

Natural numbers are also counting numbers. The reason behind calling it so is that this set of numbers can and is used to count objects by humans. It is called counting numbers as it includes numbers that humans can count. This set of numbers does not consist of the number zero. The letter N is used to address the set of natural numbers. The Roster form of the natural number is N = {1, 2, 3, 4, 5, …). 

History 

It is assumed that the history of natural numbers started when human beings began to count items around them. The Babylonians established a place value system beginning from the numeral 1 to numeral 10. Another system was added by the Egyptians that included numbers up to powers of 10 to 1 million. Pythagoras and Archimedes were the first Greek philosophers and mathematicians that studied natural numbers intensely.  

Properties with examples 

There are four predominant operations for the set of natural numbers. They are – addition, multiplication, division, and subtraction. These four operations have led to the four properties. 

• Closure property 

• Commutative property 

• Distributive property 

• Associative property 

Closure property 

This property of natural numbers states that – a sum or a product of two natural numbers will always result in a natural number. 

• Additive closure property

( P + Q = R), where P, Q, and R are all natural numbers. 

This property says that adding two natural numbers will always give a natural number.

Example – 1 + 4 = 5 . Here, 1, 4, and 5 are natural numbers.

• Multiplicative closure property 

( P x Q = R) 

This property of natural numbers says that when you multiply a natural number with another natural number, it will always give a natural number. 

Example – 3 x 7 = 21 

Here, 3, 7, and 21 are all natural numbers. Hence, this property is true. 

Commutative property 

The commutative property of natural numbers states – that the sum or product of any two natural numbers continues to result in the same answer even when the places of the two natural numbers are interchanged. 

• Additive commutative property 

( P + Q = Q + P) 

This property states that for all natural numbers when the places of A and B are interchanged, the sum will remain the same.

Example – 8 + 1 = 1 + 8 = 9

• Multiplicative commutative property

 ( P x Q = Q x P) 

This property states that when the places of A and B are interchanged, the product of these two natural numbers will remain the same for all natural numbers. 

Associative property 

This property of natural numbers states that when the grouping of any three natural numbers is altered, the summation or product of these numbers does not change. 

• Additive associative property of natural numbers

( P + Q ) + R = P + ( Q + R) 

The result of altered grouping in addition to any three natural numbers will always result in the same answer. 

Example – ( 3 + 4 ) + 5 = 3 + ( 4 + 5 ) = 12 

• Multiplicative associative property of natural numbers

The result of altered grouping in multiplication to any three natural numbers will always result in the same product. 

( P x Q ) x S = P x ( Q x S) 

Example – ( 2 x 5 ) x 3 = 2 x ( 5 x 3) = 30 

Distributive property 

The distributive property of natural numbers is applied with addition and subtraction. 

• The distributive property of multiplication with the addition of natural numbers

P x ( Q + S ) = P x Q + P x S

Example – 3 x (1 + 5 ) = 3 x 1 + 3 x 5 = 18 

• The distributive property of multiplication with subtraction of natural numbers

P x ( Q – S ) = P x Q – P x S 

Example – 3 x (1 – 5 ) = 3 x 1 – 3 x 5 = 12

The relation between natural numbers and whole numbers  

The only difference in the whole numbers set is the number zero, making natural numbers a subset of whole numbers. The whole number set is addressed by the letter W. The roster form of this set is W = { 0,1,2,3,4,………}. Zero is considered the smallest whole number. 

From the above statements, we can say that all natural numbers can be whole numbers, but vice versa is not true. 

The topics one can also go through as they are related to natural numbers and whole numbers:

• LCM 

To find the least common factor among two or more numbers

• HCF 

To find the highest common factor among two or more numbers 

• Problems on ages 

To find about the age of a person in the given problem 

• Profit and loss 

To calculate the profit or loss of a business 

• Time and work 

To calculate the time needed to complete a given work

• Calculating speed, time, and distances 

To calculate the speed of a vehicle, time of the entire travel and total distance covered

• Boat and stream problems

To calculate the upstream and downstream speeds of the boat 

Conclusion 

This article has addressed the question of what are natural numbers. The history of natural numbers began with the ancient Babylonians, who used numerals to count objects around them. Natural numbers are also known by another name – counting numbers. The letter N is used to address the set of natural numbers. These four operations work around four fundamental properties – closure, commutative, distributive, and associative properties. 

In whole numbers, the number zero is included. The relation between natural numbers and whole numbers is that natural numbers are a subset of whole numbers. The whole number set is addressed by the letter W. We can say that all natural numbers can be whole numbers, but vice versa is not true.