Question
Classify the following number as rational or irrational:$\big(2+\sqrt{5}\big)$
✓
Answer
Let $3+\sqrt2$ be rational.
Hence, $2+\sqrt5$ and $\sqrt5$ are rational.
$\therefore\big(2+\sqrt{5}\big)-2=2+\sqrt5-2=\sqrt{5}=$ rational $[\because$ Difference of two rational is rational$]$
This contradicts the fact that $\sqrt5$ is irrational.
The contradiction arises by assuming $2-\sqrt5$ is rational.
Hence, $2-\sqrt5$ is irrational.
Hence, $2+\sqrt5$ and $\sqrt5$ are rational.
$\therefore\big(2+\sqrt{5}\big)-2=2+\sqrt5-2=\sqrt{5}=$ rational $[\because$ Difference of two rational is rational$]$
This contradicts the fact that $\sqrt5$ is irrational.
The contradiction arises by assuming $2-\sqrt5$ is rational.
Hence, $2-\sqrt5$ is irrational.
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