Show that 2-3 is an irrational number. - Vidyadip

Question

Show that $2-\sqrt{3}$ is an irrational number.

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Answer

Let us assume that $2-\sqrt{3}$ is rational.
Then, there exist positive co primes a and b such that,
$2-\sqrt{3}=\frac{\text{a}}{\text{b}}$
$\sqrt{3}=2-\frac{\text{a}}{\text{b}}$
This implies, $\sqrt{33}$ is a rational number, which is a contradiction.
Hence, $2-\sqrt{3}$ is irrational number

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