Question

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

Answer 1

Given: Number `(5 + sqrt(3))`

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

Proof:

1. Assume the contrary: Suppose `(5 + sqrt(3))` is rational.

Let `5 + sqrt(3) = a/b`, where a, b are integers and b ≠ 0.

2. Rearrange to isolate `sqrt(3)`:

`sqrt(3) = a/b - 5`

`sqrt(3) = (a - 5b)/b`

3. Analyze the right-hand side:

Since a, b are integers, a – 5b is an integer as well.

Therefore, `(a - 5b)/b` is a rational number. 

4. Contradiction:

This implies `sqrt(3)` is rational.

However, it is well-known and proven that `sqrt(3)` is irrational.

Our assumption is false.

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