Question

Proof that 4 + √5 is a irrational number ?

Answer 1

If possible, let 4+√5 be rational.
Then, (4+√5) ^2 is also rational.
Now,
(4+√5) ^2=16+5+8√5
=21+8√5, which is irrational.
Therefore, this contradicts the hypothesis that 4+√5 is rational.
Thus, 4+√5 is a irrational number.
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Answer 2

Assume that
$4 + \sqrt{5} \: \: \: is \: rational$
$4 + \sqrt{5 } = \frac{a}{b}$
where a and b are integers. and suppose that a and b are co-founder prime
✒√5 = a/b +4

✒√5 = a+4b
  _____
b



therefore it means that a and b have another Common Factor then one but a and b r coprime but this is not possible this contradiction is arising due to a wrong supposition that 4 + √5 is rational therefore 4 + √5 is irrational.

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