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Being transcendental implies necessarily being irrational?

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The contrapositive of your statement (rational implies not transcendental) is trivially true. –  you Mar 10 '12 at 21:43
    
from me you get an upvote for this question. At least it is useful (otherwise i would not understand the high rating of the answer) and clear. –  miracle173 Mar 17 '12 at 16:11
    
Dear dot dot, we do not delete questions which already have answers (much less when the answers have been upvoted this much!) because at that point it would result in the work of the answered being deleted along with the question. –  Mariano Suárez-Alvarez Apr 6 '12 at 3:03

1 Answer 1

up vote 19 down vote accepted

Yes. If it were rational, then it would be the root of a degree one polynomial.

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Wow. I like to keep track of those answers which get (I hope not to offend you) more votes than the mathematical content deserves. Cool –  mixedmath Mar 14 '12 at 5:48
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@mixedmath: Nope, not offended. Here is one more for your collection: math.stackexchange.com/questions/2284/… :-) –  Aryabhata Mar 14 '12 at 5:59
    
Zero is not a root. Is it transcendental? I've started a little hullabaloo over at "Is the diagonal of a square truly irrational?" if you want to join in. –  Mark J Jun 7 '13 at 0:24
    
@MarkJ: Zero is a root of $x=0$. I don't get your point. –  Aryabhata Jun 7 '13 at 0:58
    
That doesn't look like a polynomial, only a "nomial". But this is where terminology becomes paramount. One can't argue on the grounds of reason here, because it involves definitions. –  Mark J Jun 7 '13 at 1:36

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