Question

Prove that `3 + 2sqrt(5)` is an irrational number.

Answer 1

Given: A number `3 + 2sqrt(5)` where `sqrt(5)` is known to be irrational.

To Prove: The number `3 + 2sqrt(5)` is irrational.

Proof:

Step 1: Assume, on the contrary, that `3 + 2sqrt(5)` is rational.

Step 2: Let `3 + 2sqrt(5) = r` where r is a rational number.

Step 3: Subtract 3 from both sides `2sqrt(5) = r - 3`.

Since r and 3 are rational numbers, their difference r – 3 is also rational.

Thus, `2sqrt(5)` = rational. 

Step 4: Divide both sides by 2: `sqrt(5) = (r - 3)/2`

The right side is a rational number because it is a ratio of two rational numbers.

Step 5: But `sqrt(5)` is known to be irrational proved by contradiction in the classical way, assuming `sqrt(5) = a/b` rational leads to a and b having a common factor 5, contradicting the assumption.

Step 6: Therefore, the assumption that `3 + 2sqrt(5)` is rational leads to a contradiction, since it implies `sqrt(5)` is rational.

Hence, `3 + 2sqrt(5)` is irrational.