A real number r is rational if and only if there exists two integer a and b such that ( b is not equal to zero) Let us assume that x is an irrational number and y is a rational number . Suppose for the sake of contradiction that is not irrational since s is irrational , x cannot be written as the ratio of two integers Since in not irrational, is rational . By the definition of rational, there exists integers c and d not equal to zero such that
Since a,b,c,d are integers bc - ad and bd are also integers. Moreover bd is zero as b and d are both non zero. However , this then implies that x is a rational number (as it is written as the ratio of tow integers). which is in contradiction with the fact that with the fact that x is an irrational number. Thus our supposition that is not irrational is false. Which means that is irrational .