The continued fraction form is particularly nice. It seems it is just a nice way to write a completion. Consider the approximation for sqrt(x):
Pick a, b such that x^2 > a and b=x^2 - a. Then there is a continued fraction of the form:
x = a + b/(2a + b/(2a + \dots)).
The other way to find the sqrt is with Newton's method.
The rationals given by Newton appear in the list generated by the partial fraction (convergents).
The first step in the completion is going to involve a tangent space, which I think looks a bit like the first step in Newton. I think the idea with storing information in a "continued fraction" form could then be generalised a bit. It probably applies to more things than just irrationals.
The fact there are multiple continued fractions for real numbers is okay because it is a quotient, you have picked a representative.
Pick a, b such that x^2 > a and b=x^2 - a. Then there is a continued fraction of the form:
x = a + b/(2a + b/(2a + \dots)).
The other way to find the sqrt is with Newton's method. The rationals given by Newton appear in the list generated by the partial fraction (convergents).
The first step in the completion is going to involve a tangent space, which I think looks a bit like the first step in Newton. I think the idea with storing information in a "continued fraction" form could then be generalised a bit. It probably applies to more things than just irrationals.
The fact there are multiple continued fractions for real numbers is okay because it is a quotient, you have picked a representative.