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

Question 994509: Why is the product of a non zero rational number and an irrational number irrational ?
Answer by MathLover1(20849) (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 is an irrational number, and the product of and a rational is rational , where ,,, and are integers (,,).
Then .
By division, =>.
Since integers are closed under multiplication, and are integers, making a rational number by definition. This is a contradiction to the given fact that is an irrational number. The assumption is wrong. The product of a non-zero rational number and an irrational number is an irrational number.

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