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The sequence is certainly not useless?? This is how you construct real numbers. It is what is called a completion. A real number is a sequence of rationals. You

The sequence is certainly not useless?? This is how you construct real numbers. It is what is called a completion. A real number is a sequence of rationals. You take the space of all sequences of rationals and then quotient out by an ideal of sequences which converge in a sense. I think about the limit object by thinking to some finite depth (truncation). The existence of a limit object is because there are arrows between the different finite steps. A continued fraction is just a nice representation, it’s the same thing.

Twisol 5y ago
I think their point is that if you have a convergent sequence of fractions, you can drop some finite number of them from the start and still converge to the same real number. In some sense, the rest of the sequence already contains all of the information, so early terms are redundant. With the decimal and continued fraction representations, that's not true. The later parts of the representations only describe refinements on the earlier parts; they don't themselves contain the earlier parts.
lupire 5y ago
You misquoted the slighly misphrased claim: > In a sense, every single one of the rational numbers in that sequence is useles. "Every single one" is not the same as "all". But it's not quite right, as you note. Better to say "every single one is mostly redundant information"
- cdsmith 5y ago
  As the author, I have just updated the article to try to make this clearer. Thanks.
  - fennecs 5y ago
    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.

Twisol 5y ago

I think their point is that if you have a convergent sequence of fractions, you can drop some finite number of them from the start and still converge to the same real number. In some sense, the rest of the sequence already contains all of the information, so early terms are redundant. With the decimal and continued fraction representations, that's not true. The later parts of the representations only describe refinements on the earlier parts; they don't themselves contain the earlier parts.

lupire 5y ago

You misquoted the slighly misphrased claim: > In a sense, every single one of the rational numbers in that sequence is useles. "Every single one" is not the same as "all". But it's not quite right, as you note. Better to say "every single one is mostly redundant information"

cdsmith 5y ago
As the author, I have just updated the article to try to make this clearer. Thanks.
- fennecs 5y ago
  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.