Question

If x be a non zero number and y is an irrational number prove that xy is irrational.

Answer 1

Let x be rational, y be rational and assume their product is rational.

xy=z (where z is rational)

Divide both sides by x:
y = z/x

z and x are rational, so z/x is rational, but this a contradiction since y is irrational. Therefore, our assumption that the product is rational is false; this means the product is irrational.

I hope this helped!