Question
Classify the following number as rational or irrational:$\big(2-\sqrt{3}\big)$
✓
Answer
Let $2-\sqrt3$ be rational.
Hence, 2 and $2-\sqrt3$ are rational.
$\therefore\big(2-2+\sqrt{3}\big)=\sqrt3=$ rational $[\because$ Difference of two rational is rational$]$
This contradicts the fact that $\sqrt3$ is irrational.
The contradiction arises by assuming $2-\sqrt3$ is rational.
Hence, $2-\sqrt3$ is irrational.
Hence, 2 and $2-\sqrt3$ are rational.
$\therefore\big(2-2+\sqrt{3}\big)=\sqrt3=$ rational $[\because$ Difference of two rational is rational$]$
This contradicts the fact that $\sqrt3$ is irrational.
The contradiction arises by assuming $2-\sqrt3$ is rational.
Hence, $2-\sqrt3$ is irrational.
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