Prove that 3 + 2√5 is irrational.

1.	Prove that 3 + 2√5 is irrational.
Answer» 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 that 3 + 2√5 is a rational number is wrong. Hence, 3 + 2√5 is irrational.

Answer»

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 that 3 + 2√5 is a rational number is wrong.

Hence, 3 + 2√5 is irrational.

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