Let x be given to be a rational number and y be given to
be an irrational number.

We want to show that x+y is irrational.

There exist integers p and q such that x = p/q

For contradiction, assume x+y is rational

Then there exist integers r and s such that x+y = r/s.

Then p/q + y = r/s 
           y = r/s - p/q
           y = (rq-ps)/(sq)

But that says that y is rational.  But it was give to
be irrational.  So we have reached a contradiction, 
and therefore the assumption that x+y is rational is
false.  Therefore x+y is irrational.

Edwin