What are Irrational Numbers?

We saw in the previous section that the numbers of the form $\dfrac{a}{b}$, where a and b are Integers, and b ≠ 0 are called Rational numbers.
These numbers can be expressed as a ratio of two numbers.
e.g. $\dfrac{3}{-4}$, $\dfrac{14}{55}$, $\dfrac{-21}{46}$ are all Rational Numbers.

There are numbers which cannot be expressed as a ratio of two integers. They are called Irrational numbers.
Some of the famous Irrational numbers are $\pi$, $e$ and $\phi$
$\pi$ stands for Pi and it value is $\frac{22}{7}$ which can be approximated to 3.14159.
$e$ is used to denote Euler's Number.
$\phi$ stands for Golden Ratio (called 'phi').


A number which can neither be expressed as a terminating decimal nor as a repeating decimal is called an Irrational number.
Let us see a few examples

(i) A non terminating and a non repeating decimal.
For example, numbers such as 0.91991999199991 , 0.12112111211112

(ii) Square root of a positive integer which is not a perfect square is irrational
For example, numbers such as $\sqrt{2}$, $\sqrt{3}$, $\sqrt{11}$

(iii) Cube root of a positive integer which is not a perfect cube is irrational
For example, numbers such as $\sqrt[3]{2}$, $\sqrt[3]{3}$, $\sqrt[3]{11}$

Properties of Irrational Numbers

(i) The sum of two irrational numbers need not result in an irrational number
For example (5 + $\sqrt{3}$) and (6 - $\sqrt{3}$) are two irrational numbers.
But their sum (5 + $\sqrt{3}$) + (6 - $\sqrt{3}$) = 5 + 6 = 11 is a rational number

Due to the same logic, the difference of two irrational numbers need not be an irrational number.

(ii) The sum of a rational number and an irrational number always results in an irrational number.
For example (5 + $\sqrt{3}$) is an irrational number and 6 is a rational number.
Their sum (5 + $\sqrt{3}$) + 6 = (11 + $\sqrt{3}$) is an irrational number.

(iii) The product of two irrational numbers need not result in an irrational number
For example 5$\sqrt{3}$ and 4$\sqrt{3}$ are two irrational numbers.
But their product 5$\sqrt{3}$ * 4$\sqrt{3}$ = 20 * 3 = 60 is a rational number

(iv) The product of an irrational number with any nonzero rational number gives an irrational number.
For example (5 + $\sqrt{3}$) is an irrational number and 10 is a rational number.
Their product (5 + $\sqrt{3}$) * 10 = (50 + 10$\sqrt{3}$) is an irrational number.

(v) The division of an irrational number by another irrational number need not always yield an irrational number.
For example 21$\sqrt{3}$ and 7$\sqrt{3}$ are two irrational numbers.
After division, the quotient is 3 which is a rational number.

Rationalisation

For a given number whose denominator is irrational, the process of converting its denominator to a rational number by multiplying its numerator and denominator by an appropriate number is called rationalisation.

For example

(i) If $\frac{4}{\sqrt{3}}$ is a given number, its denominator is irrational.
To convert it to a rational number, we can multiply its numerator and denominator by $\sqrt{3}$
$\frac{4}{\sqrt{3}} * \frac{\sqrt{3}}{\sqrt{3}}$ = $\frac{4\sqrt{3}}{3}$


(ii) If $\frac{4}{\sqrt[3]{7}}$ is a given number, its denominator is irrational.
To convert it to a rational number, we can multiply its numerator and denominator by $\sqrt[3]{7}^2$
$\frac{4}{\sqrt[3]{7}} * \frac{\sqrt[3]{7}^2}{\sqrt[3]{7}^2}$ = $\frac{4\sqrt[3]{7}^2}{7}$


(iii) If a denominator is of the form $(\sqrt{5} + \sqrt{7})$, rationalization of the same can be achieved by multiplying it by its conjugate.
$\begin{align*} & \frac{1}{(\sqrt{5} + \sqrt{7})} * \frac{(\sqrt{5} - \sqrt{7})}{(\sqrt{5} - \sqrt{7})} \\ & = \frac{(\sqrt{5} - \sqrt{7})}{(\sqrt{5}^2 - \sqrt{7}^2)} \\ & = \frac{(\sqrt{5} - \sqrt{7})}{-2} \\ \end{align*} $