actually what I am asking for is "how many ways do there exist to create a real number out of a sequence of coefficients?" there is the solution of polynomials through radicals, some polynomials can be solved in terms of hypergeometric functions, then there are the roots of unity (which basically are algebraic numbers corresponding to polynomial solutions P to sin(x*n)=P_n(sin(x)) and the same for cos -- always forget how those polynomials are called). are there any other solutions for polynomials expressed in formulas using specially introduced functions? I suspect some algebraic numbers can be represented as some kind of integrals, i.e. just like those hypergeometric functions. however, just defining a function which maps coefficients onto an infinite sequence of digits for a float doesn't count here: my question is if one would create such a function, how many other functions must be included into it to generally solve all polynomial equations generally as the output of a polynomial in those functions. i.e roots are binary functions, roots of unity are unary functions, hypergeometric functions are functions taking many numbers as input. do these 3 suffice to express all algebraic numbers as a polynomial in these with input being again polynomial in these functions and so on? or does one need some other functions too? what research is being done in this respect?
my goal is to represent algebraic numbers in the computer in order to replace the float datatype by a datatype of infinite precision. any work being done in that direction?
Probably the easiest way to do that is to represent an algebraic number via its minimal polynomial along with enough information to pick out which root of the minimal polynomial it is. (e.g. a floating point approximation, or alternatively intervals of rational numbers that bound the real and imaginary parts such that only one root of the polynomial satisfies these bounds)