Prove that if x Q and y is irrational then x y is irrationa

Prove that, if x Q and y is irrational, then x + y is irrational. Is x^2 + y^2 irrational?

Solution

Given x is rational

x=p/q and y is irrational

x+y =p/q+y=(p+qy)/y

Since y is irrational qy is also irrational,then

p+qy is irrational

And hence p+qy/y is rational i e x+y is irrational

2) if x=2,y = (3)^1/3

x^2+y^2=4+3^2/3 whuch is irrational.i.e x^2+y^2 is irrational only if irrational y is not a squar root .