Prove that the number sqrt{5}+1 is irrational | vedclass

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(B) Assume that $\sqrt{5}+1$ is a rational number.Then,we can write $\sqrt{5}+1 = \frac{p}{q}$,where $p$ and $q$ are integers and $q 
eq 0$.Rearrangi

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Irrational

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(B) Assume that $\sqrt{5}+1$ is a rational number.Then,we can write $\sqrt{5}+1 = \frac{p}{q}$,where $p$ and $q$ are integers and $q 
eq 0$.Rearranging the equation,we get $\sqrt{5} = \frac{p}{q} - 1$.This simplifies to $\sqrt{5} = \frac{p-q}{q}$.Since $p$ and $q$ are integers,$\frac{p-q}{q}$ must be a rational number.This implies that $\sqrt{5}$ is a rational number.However,this contradicts the known fact that $\sqrt{5}$ is an irrational number.Therefore,our initial assumption is false,and $\sqrt{5}+1$ must be an irrational number.

Prove that the number $\sqrt{5}+1$ is irrational.

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