Question

Prove /5+1 is irrational

Answer 1

GIVEN:

• √5 + 1

TO FIND:

• Prove that √5 + 1 is an irrational number.

SOLUTION:

Let √5 + 1 be a rational number,

it can be written in the form of p/q (q ≠ 0), where p and q are coprimes

$\rm{\dashrightarrow \sqrt{5} + 1 = \dfrac{p}{q} }$

$\rm{\dashrightarrow \sqrt{5} = \dfrac{p}{q} - 1 }$

$\rm{\dashrightarrow \sqrt{5} = \dfrac{p - q}{q} }$

since p and q are integers, we get p –q/q is rational, and so √5 is rational.

But this contradicts the fact that √5 is not a rational number.

So, we conclude that √5 is an irrational number.

Hence, √5 + 1 is an irrational number.

______________________