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Homework Help: Proof of transcendentals uncountable

  1. Sep 8, 2010 #1
    Hi guys,
    My question is to prove that the set of algebraic numbers is countable, then also prove that the set of transcendentals are uncountable. I have already proved the countability of the algebraics but now i do not know how to proceed. I beleive it could be as simple as the complement of the algebraics in R is uncountable, but I am not sure if the complement of the algebraics numbers within R is the set of transcendentals or not. I was unable to find out if this is the case.. I saw that trascendtals could possibly be complex, but in any case, if transcendentals make up the rest of R, then I would be done.. Any help would be great.. If anyone needs to see my proof of algebraics being countable I will post if someone asks.
    Thanks
     
  2. jcsd
  3. Sep 8, 2010 #2

    Dick

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    Of course the real transcendentals are the complement of the real algebraic numbers in R. That's basically the DEFINITION of 'real transcendental'. What's your definition?
     
  4. Sep 8, 2010 #3
    Thanks, dick. So this would work as an answer? Simply proving the countability of the algebraic numbers?
     
  5. Sep 8, 2010 #4

    Dick

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    Sure. You know the reals are uncountable, right? If you've proved the algebraics are countable, then if the transcendentals were also countable, I think you'd have a contradiction.
     
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