Question

27. Prove that 2 - /3 is irrational, given that /3 is irrational​

Answer 1

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Let 2 - √3 be a rational number  We can find co-prime a and b (b ≠ 0) Such that

$2 - \sqrt{3} = \frac{a}{b} \\ \frac{2-a}{b} = \sqrt{3} \\ \\ so \: we \: get \: \\ \frac{2a - b}{b} = \sqrt{3} \\ \\ Since \: a \: and \: b are \: integers, \\ we \: get \: \frac{2a - b}{b} \: is \: irrational \: \\ and \: so \: √3 \: is \: rational. \: \\ But \: √3 \: is \: an \: irrational \: number. \\ \\ Which \: contradicts \: our \: statement  \: \\ \: Therefore \: 2 - √3 \: is \: irrational.$