Proof of transcendentals uncountable

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Homework Help Overview

The discussion revolves around proving the countability of algebraic numbers and the uncountability of transcendental numbers within the context of real numbers. The original poster seeks clarification on whether the complement of algebraic numbers in the real numbers constitutes the set of transcendental numbers.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster considers the relationship between algebraic and transcendental numbers and questions the definitions involved. Some participants affirm the definition of real transcendental numbers as the complement of real algebraic numbers and suggest that proving the countability of algebraic numbers could lead to a contradiction regarding the uncountability of reals.

Discussion Status

The discussion is active, with participants providing guidance on definitions and implications of countability. There is an exploration of the relationship between algebraic and transcendental numbers, but no consensus has been reached on the next steps for the proof.

Contextual Notes

The original poster has indicated a willingness to share their proof of the countability of algebraic numbers if requested, suggesting that the discussion may involve reviewing or validating that proof.

mynameisfunk
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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 believe 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
 
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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?
 
Thanks, dick. So this would work as an answer? Simply proving the countability of the algebraic numbers?
 
mynameisfunk said:
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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