Question

Prove that if x is a rational number and y is an irrational number, then x + y is an irrational
number. If, in addition, x is not equal to 1, then show that xy is an irrational number.

Answer 1

x -rational
y - irrational
If z=x+y rational then y=z-x is rational too,
that contradicts assumptions.

The same is about z=xy and y=z/x. So, x+y
and xy are both irrational.