Question 124041
We are given A is a whole number and B is irrational.  Let's assume that the product A*B is rational.  That means that there are two integers p and q such that {{{AB=p/q}}}.


Dividing both sides of the equation by A results in:


{{{B=p/(Aq)}}}.


p was chosen as an integer, and since A is a whole number and q is an integer, Aq must be an integer.  Therefore {{{p/(Aq)}}} is a rational number.


But that means B must be rational, contradicting the given condition that B be irrational.  Therefore the assumption that A*B is rational must be false and A*B must be irrational.