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Prove that 3 + 2√5 is irrational. jaijishuysvvgscdxsd

 Prove that 3 + 2√5 is irrational.
 
jaijishuysvvgscdxsd

Grade:12

1 Answers

Harshit Singh
askIITians Faculty 5963 Points
3 years ago
Dear Student

Let 3 + 2√5 be a rational number.

Then the co-primes x and y of the given rational number where (y ≠ 0) is such that:

3 + 2√5 = x/y

Rearranging, we get,

2√5 = (x/y) – 3

√5 = 1/2[(x/y) – 3]

Since x and y are integers,
thus, 1/2[(x/y) – 3] is a rational number.

Therefore,√5 is also a rational number.
But this confronts the fact that√5 is irrational.

Thus,
our assumption that3 + 2√5 is a rational number is wrong.

Hence, 3 + 2√5 is irrational.

Thanks

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