can anyone tell me why the transcendental numbers are uncountable?
HallsofIvy said:Since a polynomial of degree n has exactly n coefficients, there are a countable number of such polynomials.
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).
murshid_islam said:i didn't understand this. could you elaborate a little?