Question

Prove the following as irrational root 5+1

Answer 1

1 + √5 is irrational number

__________ [TO PROVE]

Let us assume that, 1 + √5 is a rational number

=> 1 + √5 = $\dfrac{a}{b}$

Here, a and b are co-prime numbers.

=> √5 = $\dfrac{a}{b}$ - 1

=> √5 = $\dfrac{a\:-\:b}{b}$

Here..

$\dfrac{a\:-\:b}{b}$ is rational number.

So, √5 is also a rational number.

But we know that √5 is irrational number.

So, our assumption is wrong.

1 + √5 is irrational number.

___________ [HENCE PROVED]