Proof of Transcendentals Uncountable

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In summary, the conversation discusses the countability of the set of algebraic numbers and the uncountability of the set of transcendentals. The speaker has already proven the countability of algebraics but is unsure how to prove the uncountability of transcendentals. They suggest that the complement of algebraics in R is the set of transcendentals, but they are not certain. Another person suggests that transcendentals are defined as any number that is not algebraic and that it is a general theorem that R - A is uncountable for any countable subset A. The speaker believes this confirms their proof.
  • #1
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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  • #2
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.
 

1. What does "Proof of Transcendentals Uncountable" mean?

"Proof of Transcendentals Uncountable" refers to a mathematical proof that demonstrates that the set of transcendental numbers is uncountable. This means that there are infinitely many transcendental numbers and they cannot be listed or counted in a finite way.

2. What are transcendental numbers?

Transcendental numbers are real numbers that cannot be expressed as a finite or repeating decimal. Examples include pi and e. They are considered "transcendental" because they cannot be obtained through the operations of addition, subtraction, multiplication, and division from integers or other rational numbers.

3. Why is it important to prove that transcendental numbers are uncountable?

Proving that transcendental numbers are uncountable has significant implications in mathematics. It helps us understand the nature of real numbers and their relationship to rational numbers. It also has practical applications in fields such as computer science, where transcendental numbers are used in calculations and algorithms.

4. How is the proof of transcendentals uncountable achieved?

The proof of transcendentals uncountable is typically achieved through a proof by contradiction. This means assuming that the set of transcendental numbers is countable and then showing that this assumption leads to a contradiction. This contradiction then proves that the original assumption was false, and therefore, the set of transcendental numbers must be uncountable.

5. Are there any other types of uncountable sets?

Yes, there are many other types of uncountable sets in mathematics, including the set of real numbers, the set of irrational numbers, and the set of all functions from one set to another. The proof of transcendentals uncountable is just one example of a proof that demonstrates the uncountability of a specific set.

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