Looks good to me. Now you can conclude that that there are uncountably many transcendental numbers!Hello everyone! Once again there are no answers in the book so this is just another check just to make sure. Thank you very much in advance.
Question: "Are the set of all algaebraic numbers countable? Justify your respose"
I will attempt to prove that the algaebraic numbers are countable. Once again for my sake let be the set of all algaebraic numbers.
Answer: Let be an algaebraic number. Then by definition there exists a polynomial , such that . Therefore let us define the number by the n-tuple . So now that we have shown that each algaebraic number may be expressed as a n-tuple we may state that where . So now since we have shown that the set of all algaebraic numbers may expressed as a set of n-tuples with each element of the n-tuple being an element of the integers, a countable set, we may conclude that the algaebraic numbers are countable
I know its relatively simple, but I just want to sure I am doing this right.