SOLUTION: Why is the product of a non zero rational number and an irrational number irrational ?

Algebra ->  Test -> SOLUTION: Why is the product of a non zero rational number and an irrational number irrational ?      Log On


   

Question 994509: Why is the product of a non zero rational number and an irrational number irrational ?
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

"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%2Fb is rational c%2Fd, where a,b,c, and d are integers (a%3C%3E0,b%3C%3E0,d%3C%3E0).
Then x%28a%2Fb%29=c%2Fd.
By division, x=%28c%2Fd%29%2F%28a%2Fb%29=>x=%28cb%29%2F%28da%29.
Since integers are closed under multiplication, cb and da are integers, making +%28cb%29%2F%28da%29 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.