SOLUTION: Explain why the product of a nonzero rational number and an irrational number is irrational

Question 654623: Explain why the product of a nonzero rational number and an irrational number is irrational
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

Let

represent the set of irrational numbers.

Assume that for some

and

that

Then

by definition of rationals.

Is it possible then that the product of

and

is rational, that is

But that means

But since

are all integers and the integers are closed under multiplication,

must be rational since it is the quotient of integers, contradicting the original assumption that it was irrational.

Therefore, reductio ad absurdum, the product of a non-zero rational number and an irrational number cannot be rational. Q.E.D.

John

My calculator said it, I believe it, that settles it

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