Question

How do you find the sum of 2 irrational numbers?

Answer 1

Choose two irrational numbers xx and yy randomly, say independently and each according to the Lebesgue measure on (0,1)(0,1). Then x+yx+y is irrational with full probability. (Proof: For every real number zz, P[x+y=z]=0P[x+y=z]=0, hence P[x+y∈Q]=∑z∈QP[x+y=z]=0P[x+y∈Q]=∑z∈QP[x+y=z]=0, QED.)

In this sense, the sum of "almost every" pair of irrational numbers is irrational. Note that, to begin with, "almost every" real number is irrational...