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 -- https://mail.python.org/mailman/listinfo/python-list
