Main Question or Discussion Point
can anyone tell me why the transcendental numbers are uncountable?
i didn't understand this. could you elaborate a little?HallsofIvy said:Since a polynomial of degree n has exactly n coefficients, there are a countable number of such polynomials.
how??? can you explain further? i understood the rest.benorin said:You can prove A is countably infinite by:first defining the height of a polynomial as the sum of the absolute values of its coefficients and its degree, e.g., |P(z)|=|a0|+|a1|+...+|an|+n for the above polynomial.
Then prove that there are finitely many polynomials of a given height, and that each such polynomial has finitely many roots (use the fundamental theorem of algebra for the second part).
Do you know this theorem: If A and B are countable sets then AxB (Cartesian product: ordered pairs where first member is in A and second in B) is countable. To prove that, imagine that you make a table by listing all members of A horizontally (you can do that because A is countable) and listing all members of b vertically (you can do that because B is countable). The pair (a,b) is written in the column headed a along the top and the row headed b on the left. Now you can go through that table "diagonally"- you may have seen proofs that the set of all rational numbers is countable done that way.murshid_islam said:i didn't understand this. could you elaborate a little?