Question

Prove that the following number is irrational.

`2 + sqrt(5)`

Answer 1

We need to prove that `2 + sqrt(5)` is irrational.

Step 1: Recall the definition

A number is rational if it can be expressed as `p/q`, where p, q ∈ ℤ (integers) and q ≠ 0.

A number is irrational if it cannot be written in this form.

Step 2: Assume the contrary

Suppose `2 + sqrt(5)` is rational.

That means we can write `2 + sqrt(5) = p/q`, where p, q ∈ ℤ, q ≠ 0

Step 3: Rearrange

`sqrt(5) = p/q - 2`

Since both `p/q` and 2 are rational, their difference is rational.

Thus, `sqrt(5)` would be rational.

Step 4: Contradiction

But it is well-known that `sqrt(5)` is irrational.

This contradicts our assumption.

Step 5: Conclusion

Therefore, the assumption that `2 + sqrt(5)` is rational is false.

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