Rational sum of irrationals

Of course there are irrationals whose sum is rational. An example would be 2-√2 and √2.

If we limit ourselves to sums like √a + √b = q, where a, b, and q are nonnegative fractions but √a and √b are irrational, what are the solutions?

Assume that there exists a solution. Multiply by (√a - √b)/q (q ≠ 0 because otherwise √a = √b = 0 which is rational). We get (a - b)/q = √a - √b, a rational expression which we may call r. Knowing that both q and r are rational, we deduce that (q + r)/2 = √a is rational, but that's a contradiction, so there cannot be any solutions.

We've just investigated rational sums of irrational numbers. The other day, I heard about algebraic sums of transcendental numbers. Supposedly, it is not known whether e + π is algebraic or not. Maybe that's something to delve further into?