Natural Numbers and Whole Numbers

Definition of Whole Number and Natural Number:

Numbers are the symbols used to represent the amount of things in Mathematical calculations. There are several types of numbers used in the Number system of Mathematics. Numbers play a very vital role in almost all the Mathematical calculations. There are several types of numbers which include natural numbers, whole numbers, prime numbers, odd numbers, even numbers, composite numbers, rational numbers, irrational numbers, decimal numbers, fractions, real numbers, imaginary numbers, etc.

What are Natural Numbers?

The numbers which are used to count are called the natural numbers. Generally counting starts with 1 and increments. Counting can never be a negative number. So natural numbers are positive numbers starting from 1. Natural numbers are denoted as ‘N”. The natural numbers are 1, 2, 3…….

Numbers which can be regarded as natural numbers include:

• Whole numbers other than zero

• Positive integers

• Odd and even positive numbers

• Square numbers

• Prime numbers 

• Composite numbers

The following types of numbers do not have properties of whole number and natural number.

• Negative integers

• Fractions and decimals

• Irrationational numbers

• Imaginary numbers

What are Whole Numbers?

All natural numbers along with zero are categorized as whole numbers. Whole numbers are denoted by the symbol ‘W’. Zero is not categorized as a natural number because zero is not used in counting. Both natural numbers and whole numbers are rational numbers because they can be expressed in the form of a fraction in which denominators are not equal to zero. 

Properties of Whole Number and Natural Number:

Property 1: Closure Property:

Natural numbers are closed under addition and multiplication. However, they are not closed under subtraction and division. 

• When two natural numbers are added, the result obtained as a sum is also a natural number.

• Product of 2 natural numbers is a natural number.

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• Difference between two natural numbers need not be a natural number. 

Ex: 2 - 5 = -3 which is not a natural number.

• Division of 2 natural numbers does not give a natural number as a result always.

Ex: If 5 is divided by 2, the answer is 2.5 which is not a natural number.

Property 2: Commutative Property

Natural numbers satisfy commutative law for addition and multiplication. However, they are not commutative for subtraction and division.

i.e. If m and n are two natural numbers, then

• m + n = n + m

• m x n = n x m

• m - n ≠ n - m

• m ÷ n ≠ n ÷ m

Property 3: Associative Property

Addition and multiplication of natural numbers are associative. But, subtraction and division of natural numbers are not associative. If E, F and G are 3 natural numbers, then

• (E + F) + G = E + (F + G)

• (E x F) x G = E x (F x G)

• (E - F) - G ≠ E - (F - G)

• (E ÷ F) ÷ G ≠ E ÷ (F ÷ G)

Property 4: Distributive property

Natural numbers obey distribution of multiplication over addition. If k, l and m are two natural numbers, then

k (l + m) = (k x l) + (k x m)

Property 5: Identity Property

The additive identity of a number is that number which when added to a number gives the same number as the sum. Additive identity of any natural number is 0. i.e. If zero is added to any natural number, the sum is the natural number itself.

If ‘a’ is any natural number, then a + 0 = a

Multiplicative identity of a number is that number which when multiplied by the given number gives the given number as the product. Multiplicative identity of all natural numbers is 1. i.e. when any natural number is multiplied by 1, the product is the number itself.

If ‘a’ is a natural number, then a x 1 = a

Property 6: Inverse Property

Additive inverse of any natural number is that number which when added to a number yields the sum of additive identity. Additive inverse of any natural number is the negative of that number. 

If ‘a’ is a natural number, then its additive inverse is -a.

Multiplicative inverse of a natural number is that number which when multiplied by the given number yields the product equal to multiplicative identity. Multiplicative inverse of any natural number is the reciprocal of the number.

If ‘a’ is a natural number, then its multiplicative inverse is 1/a.

All the above-mentioned statements are true for both the properties of the whole number and natural number. 

Fun Facts:

• All the natural numbers are whole numbers. But all whole numbers are natural numbers except ‘0’. i.e. all the numbers except zero are natural numbers and whole numbers examples.

• All the natural numbers and whole numbers examples are real numbers. However, all real numbers are not natural numbers and whole numbers examples. 

• The product of any number multiplied by 0 is 0. The quotient of 0 divided by any number is 0 and that of any number divided by 0 is undefined.