Name: Rationalisation of Surds
Uploaded: 2019-07-19
Duration: 5 min 28 s
Description: When the denominator of an expression is a surd that can be reduced to an expression with a rational denominator, this process is called the rationalisation of a surd.

Question

Prove that $\sqrt{5}$ is an irrational number. Hence, show that $- 3 + 2 \sqrt{5}$ is an irrational number.

Accepted Answer

Let $\sqrt{5}$ is an rational number, then

$\sqrt{5}$ = $\frac{p}{q}$ , where $p, q$ are integers, $q \neq 0$ and $p, q$ have no common factors (Except $1$ )

$\Rightarrow 5 = \frac{p^{2}}{q^{2}}$

$5 q^{2} = p^{2} . . . . . . . (i)$

As $5$ divides $5 q^{2}$ , So $5$ divides $q^{2}$ and $5$ is prime $\Rightarrow 5$ divides $q$ .

Let $p = 5 k$ where $k$ is some integer.

Substitute this value of $p$ in $E q (i)$ we have:

${(5 k)}^{2} = 5 q^{2}$

$25 k^{2} = 5 q^{2}$

$q^{2} = 5 k^{2}$

As $5$ divides $5 k^{2}$ , so $5$ divides $p^{2}$ $\Rightarrow 5$ is a prime.

$\Rightarrow 5$ divides $p$

Thus, $p$ and $q$ have a common factor $5$ . This contradicts that $p$ and $q$ have no common fact (Except $1$ ) .

Therefore, $\sqrt{5}$ is not a rational number. It is irrational.

Let $- 3 + 2 \sqrt{5}$ is rational number equal to $r$ .

$\Rightarrow - 3 + 2 \sqrt{5} = r$

$\Rightarrow \frac{r + 3}{2} = \sqrt{5}$

If $3, 2, r$ are rational numbers, then $\sqrt{5}$ is also rational number. This contradicts the fact that $\sqrt{5}$

is irrational.

Thus, we get $- 3 + 2 \sqrt{5}$ is irrational.

Hence, it is proved that $\sqrt{5} and - 3 + 2 \sqrt{5}$ are irrational.

Prove that $\sqrt{5}$ is an irrational number. Hence, show that $- 3 + 2 \sqrt{5}$ is an irrational number.

Important Questions on Rational and Irrational Numbers

Learn and Prepare for any exam you want !

Prove that 5 is an irrational number. Hence, show that -3+25 is an irrational number.

Important Questions on Rational and Irrational Numbers

Learn and Prepare for any exam you want !

Prove that $\sqrt{5}$ is an irrational number. Hence, show that $- 3 + 2 \sqrt{5}$ is an irrational number.