Contradiction Proof:

x is a rational number and y is an irrational number. x and y are nonzero. The product of x and y is irrational. 

Proof: Suppose not, suppose a nonzero rational number x and an irrational number y such that x*y is rational. By def. of rational x = a/b and xy = c/d for some integers a,d,c and d. Where b,d are not zero. A is not zero because x is nonzero. 

		x*y = (a/b)y=c/d			Substitution 
		y = bc/ab				Algebra

bc and ad are intergers (by product of integers) and ad is not zero (by the zero product property). 

Thus, by def of rational y is rational.
CONTRADICTION of the suppsition which states that y is irrational. 

QED