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Being transcendental implies necessarily being irrational?

Michael Hardy
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dot dot
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    The contrapositive of your statement (rational implies not transcendental) is trivially true. – William 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-Álvarez Apr 06 '12 at 03:03
  • wow, looking again at this question now haha. Back then I was starting to read math books. I guess I wasn't familiar with taking the contrapositive of an statement and simple proof strategies. – dot dot Aug 13 '23 at 10:50

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

Aryabhata
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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 – davidlowryduda Mar 14 '12 at 05:48
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    @mixedmath: Nope, not offended. Here is one more for your collection: http://math.stackexchange.com/questions/2284/irrationality-of-powers-of-pi/2285#2285 :-) – Aryabhata Mar 14 '12 at 05:59
  • 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. – Marcos Jun 07 '13 at 00:24
  • @MarkJ: Zero is a root of $x=0$. I don't get your point. – Aryabhata Jun 07 '13 at 00:58
  • 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. – Marcos Jun 07 '13 at 01:36
  • @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. – Aryabhata Jun 07 '13 at 02:48
  • "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?" – Marcos Jun 07 '13 at 02:53