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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

Vedclass · Accepted Answer

Irrational

Vedclass · Answer

(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.

Similar Questions

$(\sqrt{3}+2)^{2} = \dots$

$\frac{47}{500}$ has $\ldots$ digits after the decimal point in its terminating decimal expansion.

Find the $\text{g.c.d.}$ of $867$ and $255$ using Euclid's division algorithm.

Write whether every positive integer can be of the form $4q + 2$,where $q$ is an integer. Justify your answer.

The rationalising factor of $5 \sqrt{2}$ is ..............

Vedclass Test Series

Exam Paper Generator

Online Exam Module