Question

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Prove that 2 + 3√5 is an irrational number.




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Answer 1

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Assume that 2+3root 5 is rational. 2+3root5=p/q,where p and q are coprimes and q not equals to 0.

3root 5=p/q-2

root 5=p/q-2/3.

A rational number never equals to an

irrational number.

We assume that p and q are coprimes and q not equals to 0.

so, our assumption is wrong.

2+3root 5 is an irrational number.

ᴀʏᴀɴ Bʜᴀɪʏᴀ ɪɴʙX ᴋʀᴅᴏ ᴩʟᴢᴢᴢ

Answer 2

$\bf Let \: x = 2 - 3 \sqrt{5} \: be \: a \: rational \: number.$

$\bf 3 \sqrt{5} = 2 - x$

$\bf \sqrt{5} = \frac{2 \: - \: x}{3}$

$\bf Since \: x \: is \: rational, \: 2 - x \: is \: rational \: and \: hence \: is \: also \: rational \: number.$

$\bf \implies \sqrt{5} \: is \: a \: rational \: numbers, \: which \: is \: a \: contradiction.$

$\bf Hence \: \: 2 - 3 \sqrt{5} \: must \: be \: an \: irrational \: number.$