A real number x ER is algebraic if -0akxk 0 for some n E N and...

 

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A real number x ER is algebraic if Σ-0akxk 0 for some n E N and integers ao,..., an. A real number x ER is transcendental if it is not algebraic. Is the set of algebraic real numbers countable or uncountable? What about the set of transcendental real numbers? (Remember to justify each of your answers with a proof.) For this problem, you may use the fact that for a fixed n E N and a fixed choice of integers ao,..., an, there are (at most) finitely many values of x ER with Σkoakxk = 0. (This follows from the fundamental theorem of algebra.) A real number x ER is algebraic if Σ-0akxk 0 for some n E N and integers ao,..., an. A real number x ER is transcendental if it is not algebraic. Is the set of algebraic real numbers countable or uncountable? What about the set of transcendental real numbers? (Remember to justify each of your answers with a proof.) For this problem, you may use the fact that for a fixed n E N and a fixed choice of integers ao,..., an, there are (at most) finitely many values of x ER with Σkoakxk = 0. (This follows from the fundamental theorem of algebra.)

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