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

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) (Show Source): You can put this solution on YOUR website!
Let w = any whole number. We can write 'w' as

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 . So this means that and that .

Now solve for 'i' to get . Since 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.

If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my tutoring website

Jim
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