CBSE 10th Mathematics | Irrational Numbers and Solved Examples

IRRATIONAL NUMBERS

All real numbers which are not rational are called irrational numbers. $\displaystyle \sqrt{2}$ , $\displaystyle \sqrt[3]{3}$ , $\displaystyle -\sqrt{5}$ are some examples of irrational numbers.

There are decimals which are non-terminating and non-recurring decimal.

Example: 0.303003000300003…

Hence, we can conclude that

An irrational number is a non-terminating and non-recurring decimal and cannot be put in the form $\displaystyle \frac{p}{q}$ where p and q are both co-prime integers and q ¹ 0.

Solved Examples Based on Irrational Numbers

Question:

Prove that $\displaystyle \sqrt{2}$ is not a rational number.

Solution:

Let $\displaystyle \sqrt{2}$ is a rational number

\ $\displaystyle \sqrt{2}=\frac{p}{q}$ [p and q are co-prime and q ¹ 0]

Squaring both sides

$\displaystyle 2=\frac{{{p}^{2}}}{{{q}^{2}}}$

Þ $\displaystyle 2{{q}^{2}}={{p}^{2}}$

Þ $\displaystyle {{p}^{2}}$ is even or p is even.

Let p = 2r

Þ $\displaystyle 2{{q}^{2}}={{(2r)}^{2}}=4{{r}^{2}}$

Þ $\displaystyle {{q}^{2}}=2{{r}^{2}}$

Þ $\displaystyle {{q}^{2}}$ is even so q is even.

Hence, p and q have 2 as a common factor or p and q are not co-prime.

So, our supposition is wrong.

\ $\displaystyle \sqrt{2}$ is not a rational number.

Question:

Prove that $\displaystyle \sqrt{3}+\sqrt{5}$ is an irrational number.

Solution:

Suppose $\displaystyle \sqrt{3}+\sqrt{5}$ is a rational number and can be taken as $\displaystyle \frac{a}{b}$ , b ¹ 0 and a, b are co-prime.

Þ $\displaystyle \sqrt{3}+\sqrt{5}=\frac{a}{b}$ [Rational]

Squaring both sides

$\displaystyle 3+5+2\sqrt{3}\sqrt{5}=\frac{{{a}^{2}}}{{{b}^{2}}}$

Þ $\displaystyle 8+2\sqrt{15}=\frac{{{a}^{2}}}{{{b}^{2}}}$

Þ $\displaystyle \sqrt{15}=\frac{{{a}^{2}}-8{{b}^{2}}}{2{{b}^{2}}}$

LHS is $\displaystyle \sqrt{15}$ which is irrational while RHS is rational.

So, our supposition is wrong.

Hence, $\displaystyle \sqrt{3}+\sqrt{5}$ is not a rational number.

Question:

Show that there is no positive integer n for which $\displaystyle \sqrt{n-1}+\sqrt{n+1}$ is rational and can be expressed in the form $\displaystyle \frac{a}{b}$ , b ¹ 0 and a & b are co-prime.