Question

prove that 2+/3 is irritaional number
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Answer 1

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Answer in attachment:

Answer 2

$\bf To\ prove:\ \tt 2\ +\ \sqrt{3}\ is\ an\ irrational\ number.\\\\\\Let's\ assume\ that\ 2\ +\ \sqrt{3}\ is\ a\ rational\ number.\\\\\\2\ +\ \sqrt{3}\ =\ \dfrac{a}{b},\ where\ a\ and\ b\ are\ co-prime\ integers\ and\ b\ \ne\ 0.\\\\\\2\ +\ \sqrt{3}\ =\ \dfrac{a}{b}\\\\\\\sqrt{3}\ =\ \dfrac{a}{b}\ -\ 2\\\\\\\sqrt{3}\ =\ \dfrac{a\ -\ 2b}{b}\\\\\\Since\ a\ and\ b\ are\ integers,\ and\ \dfrac{a\ -\ 2b}{b}\ is\ a\ rational\ number,\\\\\\\implies\ \sqrt{3}\ is\ also\ rational.$

$\tt This\ contradicts\ the\ fact\ that\ \sqrt{3}\ is\ irrational.\\\\\\This\ contradiction\ has\ arisen\ due\ to\ our\ wrong\ assumption.\\\\\\Therefore,\ our\ assumption\ is\ wrong.\\\\\\2\ +\ \sqrt{3}\ is\ an\ irrational\ number.$