Question

Prove that `(4 + 3sqrt(5))` is irrational.

Answer 1

Given: Let `x = 4 + 3sqrt(5)`.

To Prove: x is irrational.

Proof [Step-wise]:

1. Suppose, for contradiction, that x is rational.

Then there exist integers a, b with b ≠ 0 and gcd(a, b) = 1 such that `x = a/b = 4 + 3sqrt(5)`.

2. Rearranging, `3sqrt(5) = a/b - 4`

= `(a - 4b)/b`

So, `sqrt(5) = (a - 4b)/(3b)`. 

The right-hand side is a ratio of integers, hence `sqrt(5)` would be rational.

3. But `sqrt(5)` is known to be irrational. A standard proof: assume `sqrt(5) = p/q` in lowest terms (p, q integers, q ≠ 0).

Squaring gives p² = 5q², so 5 divides p² and hence p. 

Write p = 5k; then 25k² = 5q², so q² = 5k² and 5 divides q, contradicting that p and q are coprime. 

Therefore, `sqrt(5)` is irrational.

4. The contradiction in step 2–3 shows the assumption that `x = 4 + 3sqrt(5)` is rational is false.

Therefore, `4 + 3sqrt(5)` is irrational.