Question 994509

 "The product of a non-zero rational number and an irrational number is irrational."

Indirect Proof (Proof by Contradiction) of the statement:

Assume the opposite of what you want to prove, and show it leads to a contradiction of a known fact.
   
assume {{{x}}} is an irrational number, and the product of {{{x}}} and a rational {{{a/b}}} is rational {{{c/d}}}, where {{{a}}},{{{b}}},{{{c}}}, and {{{d}}} are integers ({{{a<>0}}},{{{b<>0}}},{{{d<>0}}}).

Then {{{x(a/b)=c/d}}}.
By division, {{{x=(c/d)/(a/b)}}}=>{{{x=(cb)/(da)}}}.
Since integers are closed under multiplication, {{{cb}}} and {{{da}}} are integers, making {{{ (cb)/(da)}}} a rational number by definition. This is a contradiction to the given fact that {{{x}}} is an irrational number. The assumption is wrong. The product of a non-zero rational number and an irrational number is an irrational number.