Proof of Transcendentals Uncountable

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SUMMARY

The discussion centers on proving that the set of algebraic numbers is countable and the set of transcendental numbers is uncountable. The user has successfully demonstrated the countability of algebraic numbers and seeks clarification on whether the complement of algebraic numbers in the real numbers (R) constitutes the set of transcendental numbers. A participant confirms that transcendental numbers are indeed defined as those numbers that are not algebraic, and asserts that the uncountability of R minus a countable subset is a well-established theorem, thus concluding the proof.

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  • Understanding of algebraic numbers and their properties
  • Familiarity with transcendental numbers and their definitions
  • Knowledge of set theory, specifically countability and uncountability
  • Basic comprehension of real numbers (R) and subsets
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  • Study the properties of algebraic and transcendental numbers in detail
  • Learn about set theory, focusing on countable vs. uncountable sets
  • Explore the implications of Cantor's theorem on real numbers
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Mathematicians, students of mathematics, and anyone interested in number theory and the foundations of real analysis.

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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Isn't a trascendental number by definition any number which is not algebraic?
And that R - A is uncountable for any countable (or finite, of course) subset A is a general theorem.
So I think you are done.
 

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