Question

Show that 3√5-1 is not a rational number

Answer 1

Answer:

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Answer 2

$\mathfrak{{\pmb{\underline{To~prove}}:}}$

• $\sf{3{\sqrt{5}}-1}$ is an irrational number (not a rational number)

$\mathfrak{{\pmb{\underline{Solution}}:}}$

Let us assume that, $\sf{3{\sqrt{5}}-1}$ is a rational number

$\sf\implies{3{\sqrt{5}}-1={\dfrac{p}{q}}}$

　Where p and q are integers and q ≠ 0

$\sf\implies{3{\sqrt{5}}={\dfrac{p}{q}}+1}$

$\sf\implies{3{\sqrt{5}}={\dfrac{p+1q}{q}}}$

$\sf\implies{3{\sqrt{5}}={\dfrac{p+q}{q}}}$

• p and q being integers, $\sf{\pmb{{\dfrac{p+q}{q}}}}$ represents a rational number ; but in LHS it is $\sf{\pmb{3{\sqrt{5}}}}$ which is an irrational number

$\sf\implies$ LHS ≠ RHS

$\sf\therefore$ our assumption is wrong

• $\sf{3{\sqrt{5}}-1}$ is not a rational number, it is an irrational number.