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Prove that of the Numbers `2 + Sqrt (5)` is Irrational: - Mathematics

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Prove that of the numbers  `2 + sqrt (5)` is irrational:

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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.

Concept: Concept of Irrational Numbers
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