SOLUTION: What is the reasoning behind the fact that it is not possible for the solution of an irrational number multiplied by a whole number (other than 0) to be a rational number? I know t

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Question 342086: What is the reasoning behind the fact that it is not possible for the solution of an irrational number multiplied by a whole number (other than 0) to be a rational number? I know that it is not possible, which is the first step of the question, I just don't know the exact reasoning behind it.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let w = any whole number. We can write 'w' as w%2F1


Also, let i = any irrational number


Now assume that multiplying 'w' and 'i' will give you a rational number 'r'. We know it's false, but let's just hypothetically say that this is the case. We're basically looking for a contradiction which arises because of this assumption.


We can represent 'r' by r=a%2Fb. So this means that wi=r and that wi=a%2Fb.


Now solve for 'i' to get i=a%2F%28bw%29. Since a%2F%28bw%29 is clearly a rational number, this means that 'i' is a rational number. But wait, we clearly stated that 'i' is an irrational number and it cannot be both. So we have a contradiction.


So this means that the product of 'w' and 'i' is NOT rational. So it must be irrational (as it's the only other option)


Therefore, the product of a whole number and an irrational number is an irrational number.


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Jim