# Rationalizing irrational number

In this chapter we will learn to rationalize irrational number which is present in denominator of the fraction.

## Irrational number in denominator

We know that irrational number cannot be represented in the form of fraction p / q.

But there are irrational numbers which are present in the denominator of the fraction.

Since the value of irrational number are non repeating and non terminating in nature, it gets very difficult to find the exact value of fraction.

For example,
Consider the number \mathtt{\ \frac{1}{\sqrt{2}}}

Here \mathtt{\sqrt{2}} is an irrational number whose value is given as;

\mathtt{\sqrt{2} \ =\ 1.41428\ .\ .\ .\ }

So the fraction \mathtt{\ \frac{1}{\sqrt{2}}} can be represented as;

\mathtt{\frac{1}{\mathtt{\ 1.41428\ .\ .}}}

Imagine how difficult is to find the exact value of above fraction.

So, in order to simplify the fraction with irrational number, we will use the rationalization technique.

In simple words, using rationalization, we are trying to bring the irrational number in the numerator for better calculation.

## Rationalization of Irrational number

Consider the fraction with irrational number \mathtt{\frac{1}{\sqrt{2}}}

To simplify the fraction, simply multiply numerator and denominator by same irrational number and do the further calculation.

\mathtt{\Longrightarrow \ \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\sqrt{2}}{2}}

Hence, the fraction \mathtt{\frac{1}{\sqrt{2}}} is rationalized to \mathtt{\frac{\sqrt{2}}{2}}

I hope you understood the above process. Given below are some examples for better clarity

Example 01
Simplify the fraction \mathtt{\frac{1}{\sqrt{13}}}

Solution
Multiply numerator and denominator by \mathtt{\sqrt{13}}

\mathtt{\Longrightarrow \ \frac{1}{\sqrt{13}} \times \frac{\sqrt{13}}{\sqrt{13}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\sqrt{13}}{\left(\sqrt{13}\right)^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\sqrt{13}}{13}}

Hence, the fraction is simplified to \mathtt{\frac{\sqrt{13}}{13}}

Example 02
Simplify the fraction \mathtt{\frac{1}{\sqrt{55}}}

Solution
Multiply numerator and denominator by \mathtt{\sqrt{55}}

\mathtt{\Longrightarrow \ \frac{1}{\sqrt{55}} \times \frac{\sqrt{55}}{\sqrt{55}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\sqrt{55}}{\left(\sqrt{55}\right)^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\sqrt{55}}{55}}

Hence, \mathtt{\frac{\sqrt{55}}{55}} is the required fraction.

## Rationalizing irrational number in subtraction / addition form

In some type of fractions, denominator contains irrational number added / subtracted with other number.

Common examples of such fractions are \mathtt{\frac{1}{a+\sqrt{x}} \ \ or\ \frac{1}{a-\sqrt{x}}} .

To simplify such fractions, we have to multiply the given number with its conjugate.

\mathtt{a+\sqrt{x}} conjugate is \mathtt{a-\sqrt{x}} .

\mathtt{a-\sqrt{x}} conjugate is \mathtt{a+\sqrt{x}} .

Given below are some examples for further clarity.

Example 01
Simplify \mathtt{\frac{1}{2+\sqrt{5}}}

Solution
Conjugate of \mathtt{2+\sqrt{5}} is \mathtt{2-\sqrt{5}} .

Multiply numerator and denominator by conjugate.

\mathtt{\Longrightarrow \ \frac{1}{2+\sqrt{5}} \times \frac{2-\sqrt{5}}{2-\sqrt{5}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2-\sqrt{5}}{\left( 2+\sqrt{5}\right)\left( 2-\sqrt{5}\right)}}

Referring the formula;

\mathtt{( a+b)( a-b) =a^{2} -b^{2}}

Using the formula, we get;

\mathtt{\Longrightarrow \ \frac{2-\sqrt{5}}{2^{2} -\left(\sqrt{5}\right)^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2-\sqrt{5}}{4-5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2-\sqrt{5}}{-1}}\\\ \\ \mathtt{\Longrightarrow \ -2+\sqrt{5}}

Hence, the irrational number is transferred from denominator to numerator.

Example 02
Rationalize \mathtt{\frac{1}{6-\sqrt{11}}}

Solution
The conjugate of \mathtt{6-\sqrt{11}} is \mathtt{6+\sqrt{11}} .

Multiply numerator and denominator by conjugate.

\mathtt{\Longrightarrow \ \frac{1}{6-\sqrt{11}} \times \frac{6+\sqrt{11}}{6+\sqrt{11}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6+\sqrt{11}}{\left( 6-\sqrt{11}\right)\left( 6+\sqrt{11}\right)}}

Referring the formula;

\mathtt{( a+b)( a-b) =a^{2} -b^{2}}

Using the formula;

\mathtt{\Longrightarrow \ \frac{6+\sqrt{11}}{6^{2} -\left(\sqrt{11}\right)^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6+\sqrt{11}}{36-11}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6+\sqrt{11}}{25}}

Hence, the fraction is simplified to \mathtt{\frac{6+\sqrt{11}}{25}}

Example 03
Rationalize \mathtt{\frac{1}{\sqrt{2} +\sqrt{3}}}

Solution
Conjugate of \mathtt{\sqrt{2} +\sqrt{3}} is \mathtt{\sqrt{2} -\sqrt{3}}

Multiplying numerator and denominator by conjugate.

\mathtt{\Longrightarrow \ \frac{1}{\sqrt{2} +\sqrt{3}} \times \frac{\sqrt{2} -\sqrt{3}}{\sqrt{2} -\sqrt{3}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\sqrt{2} -\sqrt{3}}{\left(\sqrt{2} +\sqrt{3}\right)\left(\sqrt{2} -\sqrt{3}\right)}}

Referring the formula;

\mathtt{( a+b)( a-b) =a^{2} -b^{2}}

Using the formula, we get;

\mathtt{\Longrightarrow \ \frac{\sqrt{2} -\sqrt{3}}{\left(\sqrt{2}\right)^{2} -\left(\sqrt{3}\right)^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\sqrt{2} -\sqrt{3}}{2-3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\sqrt{2} -\sqrt{3}}{-1}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt{3} -\sqrt{2}}

Hence, the fraction is rationalized to \mathtt{\sqrt{3} -\sqrt{2}}

Next chapter : Questions on irrational numbers