Question

show that 1 +root 5 is an irrational number​

Answer 1

$\large\bf\underline\red{Correct \: Question \: :- }$

$\small\sf{Show \: that \: \frac{1}{\sqrt{5}} \: is \: an \: irrational \: number.}$

$\large\bf\underline\red{Answer \: :- }$

$\small\sf{Let \: us \: assume \: \frac{1}{ \sqrt{5} } \: is \: rational \: number.}$

then it will be of the form a/b, where a and b are co prime

$\sf{and \: b \neq 0 }$

Now,

$\sf{ \frac{a}{b} = \frac{1}{\sqrt{5}}}$

$:\sf\implies{ \frac{a}{b} = \frac{1 \times \sqrt{5} }{ \sqrt{5} \times \sqrt{5} } } \\ \\:\sf\implies{ \frac{5a}{b} = \sqrt{5} } \: \: \: \: \: \: \: \: \:$

Since, 5a is an integer and b is also an integer.

So 5a/b is a rational number.

$\sf\implies{ \sqrt{5} \: is \: a \: rational \: number. }$

But this contradicts to the fact that √5 is an irrational number.

Therefore, our assumption is wrong.

Hence, 1√5 is an irrational number.

Hence Proved

_______________________________________

Answer 2

Answer:

It is irrational number.

Step-by-step explanation:

Let, 1 + √5 is rational number.

so, 1 + √5 = p/q

suppose, p and q have a common factor other than 1 .

so, 1+ √5 = a/b, where a and b are co-prime.

1+ √5 = a/b

√5 = a/b - 1

√5 = a - b /b

Because , √5 is irrational number.

Therefore, 1 + √5 is irrational number.

Hence proved