Working with the set of real numbers (was: Finding size of Variable)

[Rustom Mody]

On Tuesday, March 4, 2014 9:16:25 AM UTC+5:30, Chris Angelico wrote:
> On Tue, Mar 4, 2014 at 2:13 PM, Rustom Mody  wrote:
> >> But it's a far cry from "all real numbers". Even allowing for
> >> continued fractions adds only some more; I don't think you can
> >> represent surds that way.
> > See
> > http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html#sqrts

> That's neat, didn't know that. Is there an efficient way to figure
> out, for any integer N, what its sqrt's CF sequence is? And what about
> the square roots of non-integers - can you represent √π that way? I
> suspect, though I can't prove, that there will be numbers that can't
> be represented even with an infinite series - or at least numbers
> whose series can't be easily calculated.

You are now asking questions that are really (real-ly?) outside my capacities.

What I know (which may be quite off the mark :-) )

Just as all real numbers almost by definition have a decimal form (may
be infinite eg 1/3 becomes 0.33333...) all real numbers likewise have a CF form

For some mathematical (aka arcane) reasons the CF form is actually better.

Furthermore:

1. Transcendental numbers like e and pi have non-repeating infinite CF forms
2. Algebraic numbers (aka surds) have repeating maybe finite(?) forms
3. For some numbers its not known whether they are transcendental or not
(vague recollection pi^sqrt(pi) is one such)
4 Since e^ipi is very much an integer, above question is surprisingly non-trivial