Number System : Natural, Whole, Integer, Rational, Irrational and Real Number

by Wonder Boy · Published April 9, 2020 · Updated May 9, 2020

Number : A number is a mathematical object to measure, count and label.

Natural Number : The natural numbers are also called the counting numbers. The numbers 1,2,3,4 and so on are natural numbers. Basically the positive numbers which we use to count things are natural numbers.
Important : Zero ‘0’ is not considered as a natural number.

Whole Numbers : The natural numbers and zero makes a set of whole numbers. The whole numbers are the numbers 0, 1, 2, 3, 4, and so on. Whole numbers does not include any Negative numbers.

So by definition all natural numbers will come under whole numbers, but not all whole numbers are natural numbers, since zero is a whole number but not a natural number.
Important : Zero ‘0’ is whole number but not a natural number.

Integers : All the natural numbers, their negatives and zero makes a set of integers. So an integer is any number in the set (…,-3,-2,-1,0,1,2,3,….). Integers does not include any fraction or decimals.

Important : All Natural numbers are integers. But not all integers are natural numbers.

All Whole numbers are integers, but not all integers are whole numbers.

Rational Numbers : The rational numbers include all the integers, all fractions, terminating decimals and repeating decimals (We will discuss about fractions, and decimals later).

Every rational number can be written in the form of fraction $\frac{p}{q}$ , where p and q are integers.

For e.g. 7 can be written as $\frac{7}{1}$ .
0.875 can be written as $\frac{7}{8}$ .
$3\frac{5}{16}$ can be written as $\frac{53}{16}$ .

Important : All natural numbers are rationals, but not all rational numbers are natural numbers.
All whole numbers are rationals, but not all rational numbers are whole numbers.
All integers are rationals, but not all integers are whole numbers.

Irrational Numbers: The numbers which cannot be written in the form of $\frac{p}{q}$ , where p and q are integers. These numbers are non-terminating and non-recurring.

For e.g. $\tau$ , $\sqrt{2}$ , $\sqrt{11}$
$\tau$ = 3.1415926…..
$\sqrt{2}$ = 1.4142135….
$\sqrt{11}$ = 3.3166247….