Question

Prove that 2 − 3√5 is an irrational number, given that √5 is irrational.

Answer 1

Step-by-step explanation:

Let x=2−3

5

be a rational number.

3

5

=2−x

5

=

3

2−x

Since x is rational, 2-x is rational and hence

3

2−x

is also rational number

⇒

5

is a rational numbers, which is a contradiction.

Hence 2−3

5

must be an irrational number.

Answer 2

Solution

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let, 2 - 3√5 is a rational number which is equal to x

$: \longrightarrow \sf 2 - 3 \sqrt5 = x$

$: \longrightarrow \sf - 3 \sqrt{5} = x - 2$

$: \longrightarrow \sf \sqrt{5} = \frac{2 - x}{3}$

as x is rational number,

( 2 - x )/3 is also a rational number. But √5 is an irrational number which is equal to ( 2 - x )/3

hence, x is not a rational number.

$\bf hence,$

$\: \: \: \: \: \: \: \: \: \: \: \: \bold{x} \: \: is \: \: not \: \: a \: \: rational \: \: number$

$\green{ \therefore \: \: \bf 2 - 3 \sqrt{5} \: \: is \: \: an \: \: irrational \: \: number} \\ \\ \bf [ \:Proved\:]$

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