- #1
gottfried
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Homework Statement
Prove that there is no rational x such that x2=3
2. The attempt at a solution
Suppose that there is a rational x=[itex]\frac{a}{b}[/itex]=[itex]\sqrt{3}[/itex] and that the fraction is fully simplified. (ie. a and b have no common factor)
Then a2/b2=3 which means a2=b2.3 and it follows that a is a factor of 3 and be written a=3.k (k is an integer). Therefore a2/b2 = 9.k2/b2=3. Therefore b2=3.k2.
This means both a and b are multiples of 3 and this contradicts our original assumption that the fraction is fully simplified.
3. Question
As far as I can tell the proof is sound but if you replace 3 with 4 the same logic holds and this means there is no rational number x such that x2=4 which is obviously wrong since 2 does. So what is wrong with this proof.