Question 342086
Let w = any whole number. We can write 'w' as {{{w/1}}}



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/b}}}. So this means that {{{wi=r}}} and that {{{wi=a/b}}}. 



Now solve for 'i' to get {{{i=a/(bw)}}}. Since {{{a/(bw)}}} 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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