Number System : Rational, Irrational & Real Numbers

by Wonder Boy · Published April 15, 2020 · Updated May 9, 2020

Rational Numbers: The rational numbers include all the integers, all fractions, terminating decimals and repeating decimals (We will discuss 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 $\frac{53}{16}$ .

Rational numbers are either Terminating or Recurring decimal numbers.

Terminating Decimal Numbers: These numbers have a finite number of digits after the decimal point.

For e.g. The fraction $\frac{1}{4}$ when expressed as a decimal equals 0.25
Similarly, $\frac{1}{5} = 0.2$
And, $\frac{1}{8} = 0.125$
So, these fractions can be expressed as terminating decimal numbers as they have a finite number of digits after decimal point.

Recurring Decimal Numbers: These numbers have one or more repeating numbers or sequence of numbers after the decimal point, which continue infinitely.

For e.g. $\frac{1}{3} = 0.3333 = 0.\overline{3}$
$\frac{1}{7} = 0.142857142857….. = 0.\overline{142857}$
$\frac{4}{11} = 0.36363636 = 0.\overline{36}$

So, these fractions can be expressed as non-terminating but repetition of digits after a certain point. Every recurring decimal number can be converted to a regular fraction.

Now let’s discuss how to convert a recurring decimal number to a regular fraction.

Full process is described below for understanding. After the discussion of the full process we will discuss a short process to directly convert the recurring decimal to a regular fraction.

Full Process
For e.g.: Convert $0.\overline{3}$ to regular fraction.
Let $x = 0.\overline{3} = 0.333333$ ….. …..(i)
$\therefore 10x = 3.333333$ …. ……..(ii)
From (ii)-(i), we get
$\rightarrow 9x = 3$
$\therefore x = \frac{3}{9} = \frac{1}{3}$

Convert $0.\overline{36}$ to regular fraction.
Let $x = 0.\overline{36} = 0.363636….$ …..(i)
$\therefore 100x = 36.363636….$ ……..(ii)
From (ii)-(i), we get,
$\rightarrow 99x = 36$
$\therefore x = \frac{36}{99} = \frac{4}{11}$

Short Process
~ First check the repeating digits. Note the number of repeating digits which is denoted by over-line.
~ Place numerator as the repeating digits.
~ In denominator, write $10^{(number\ of\ repeating\ digit)}-1$
~ Reduce to the lowest simple fraction by eliminating common factors.

For e.g. : Convert $0.\overline{3}$ to regular fraction.
Here there is only one repeating digit.
So, $0.\overline{3} = \frac{3}{10^1 -1} = \frac{3}{9} = \frac{1}{3}$

Convert $0.\overline{36}$ to regular fraction.
Here there are two repeating digits.
So, $0.\overline{36} = \frac{36}{10^2 -1} = \frac{36}{99} = \frac{4}{11}$

Now, What if the decimal number is repeating and non-terminating but it starts after a certain digit.
For e.g. 0.07555… , 0.0123123…..
Again we will discuss the full process for better understanding and then we will discuss the short process for fast calculation.

Full Process
For e.g. Convert $0.07\overline{5}$ to regular fraction.
Let $x = 0.07\overline{5} = 0.0755555….$ …..(i)
$\therefore 100x = 7.55555….$ ……..(ii)
Also, $1000x = 75.55555….$ ……..(iii)
From (iii)-(ii), we get
$\rightarrow 900x = 68$
$\therefore x = \frac{68}{900} = \frac{17}{225}$

Convert $0.0\overline{123}$ to regular fraction.
Let $x = 0.0\overline{123} = 0.123123….$ …..(i)
$\therefore 10x = 0.123123….$ ……..(ii)
Also, $10000x = 123.123123….$ ……..(iii)
From (iii)-(ii), we get
$\rightarrow 9990x = 123$
$\therefore x = \frac{123}{9990}$

Short Process
~ First multiple the number with power of 10 (say 10^p) to bring non repeating digits on the left of the decimal and repeating digits on the right of the decimal. Let say the left side of decimal part is A.
~ Now multiply the resulting number with power of 10 (say 10^q) to bring first set of repeating digit on the left hand side of decimal. Now lets say the left side of decimal part is B.
~ Now, Numerator = B-A
~ Now convert the repeating digits to decimal as usual and take the denominator part of it. Multiply it with 10^p. This will give the required denominator.

For e.g. Convert $0.07\overline{5}$ to regular fraction.
Here A = 7, B = 75, $10^p = 10^2$
So, $0.07\overline{5} = \frac{75-7}{(10^1-1)(10^2)} = \frac{68}{900} =\frac{17}{225}$

Convert $0.0\overline{123}$ to regular fraction.
Here, A = 0, B = 123, $10^p = 10^1$
So, $0.0\overline{123} = \frac{123-0}{(10^3-1)(10^1)} = \frac{123}{9990}$

Convert $0.0768\overline{123}$ to regular fraction.
Here, A = 768, B = 768123, $10^p = 10^4$
So, $0.0768\overline{123} = \frac{768123-768}{(10^3-1)(10^4)} = \frac{767355}{9990000} = \frac{51157}{222000}$

Irrational Numbers: The numbers which cannot be written in the form of 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….

Real Numbers: All rational & irrational numbers constitute Real numbers.
Real numbers can be represented on a number line.

So, what is a number line?
A number line can be defined as a straight with numbers placed at equal or segments along its length with ‘zero’ as the reference point, positive numbers to the right of ‘zero’ and negative numbers to the left of ‘zero’.
A number line can be extended infinitely in any direction and is usually represented horizontally.

Number Line