Question

Prove that the following number is irrational.

`sqrt(3) - sqrt(2)`

Answer 1

We need to prove that `sqrt(3) - sqrt(2)` is irrational.

Step 1: Recall the property of irrational numbers

If x and y are irrational numbers, then in most cases x ± y is also irrational (with some exceptions, like `sqrt(2) + (2 - sqrt(2)) = 2`, which is rational).

So, to prove rigorously, we use proof by contradiction.

Step 2: Assume the contrary

Suppose `sqrt(3) - sqrt(2)` is rational

That means `sqrt(3) - sqrt(2) = p/q, p, q ∈ ℤ, q ≠ 0`

Step 3: Rearrange

`sqrt(3) = p/q + sqrt(2)`

Since `p/q` is rational and `sqrt(2)` is irrational, their sum must be irrational.

But the left-hand side is `sqrt(3)`, which is known to be irrational.

Step 4: Contradiction

This leads to a contradiction because we assumed `sqrt(3) - sqrt(2)` was rational.

Step 5: Conclusion

Therefore, our assumption is false, and hence `sqrt(3) - sqrt(2)` is irrational.