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Proof of transcendentals uncountable

  • #1
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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
 

Answers and Replies

  • #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?
 
  • #3
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Thanks, dick. So this would work as an answer? Simply proving the countability of the algebraic numbers?
 
  • #4
Dick
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Thanks, dick. So this would work as an answer? Simply proving the countability of the algebraic numbers?
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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