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Question
Prove that of the numbers `2 + sqrt (5)` is irrational:
Solution
Let `2 + sqrt (5)` be rational.
Hence, `2 + sqrt (5)` and `sqrt( 5 )` are rational.
∴ `(2 + sqrt (5))` – 2 = `2 + sqrt (5)`– `2 = sqrt (5)` = rational [∵Difference of two rational is rational]
This contradicts the fact that `sqrt (5)` is irrational.
The contradiction arises by assuming `2 - sqrt (5)` is rational.
Hence, `2 - sqrt (5)` is irrational.
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