Short Note

Prove that 3 +`sqrt 5` is an irrational number.

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#### Solution

Let us assume that `3 + sqrt 5` is a rational number.

`=> 3 + sqrt 5 = p /q`, where p and q are the integers and q ≠0.

`=> sqrt 5 = p/q - 3 = (p - 3q)/q`

Since p , q and 3 are integers. So, `(p - 3q)/q`is a rational number.

`=> sqrt 5` is also a rational number.

but this contradicts the fact that `sqrt 5` is an irrational number.

This contradiction has arisen due to the wrong assumption that `3 + sqrt 5` is a rational number.

Hence, `3 + sqrt 5` is an irrational number.

Concept: Concept of Rational Numbers

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