Being transcendental implies necessarily being irrational?
Is a transcendental number necessarily irrational?
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elementary-number-theory
irrational-numbers
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3The contrapositive of your statement (rational implies not transcendental) is trivially true. – 2012-03-10
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0from 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. – 2012-03-17
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0Dear 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. – 2012-04-06
1 Answers
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Yes. If it were rational, then it would be the root of a degree one polynomial.
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1Wow. I like to keep track of those answers which get (I hope not to offend you) more votes than the mathematical content deserves. Cool – 2012-03-14
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1@mixedmath: Nope, not offended. Here is one more for your collection: http://math.stackexchange.com/questions/2284/irrationality-of-powers-of-pi/2285#2285 :-) – 2012-03-14
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0Zero 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. – 2013-06-07
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0@MarkJ: Zero is a root of $x=0$. I don't get your point. – 2013-06-07
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0That 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. – 2013-06-07
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0@MarkJ: Polynomials have precise definitions and based on that it is true and so is statement about transcendentals etc. Still don't get your point. – 2013-06-07
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0"Polynomials have precise definitions" is somewhat a matter of convention and habit. Beyond that there is only logical consistency. The question is whether these definitions stand in the light of new data. Please see the reference to geometry in the question "Is the diagonal of a square truly irrational?" – 2013-06-07