Xref: utzoo comp.theory:1781 comp.lang.functional:722 Path: utzoo!utgpu!news-server.csri.toronto.edu!rpi!zaphod.mps.ohio-state.edu!usc!snorkelwacker.mit.edu!bloom-beacon!eru!hagbard!sunic!mcsun!ukc!stl!tom@nw.stl.stc.co.uk From: tom@nw.stl.stc.co.uk Newsgroups: comp.theory,comp.lang.functional Subject: Re: do computers believe in real numbers? Message-ID: <4186@stl.stc.co.uk> Date: 5 Apr 91 15:37:00 GMT References: <7197@munnari.oz.au> <91090.154205NN1@awiwuw11.wu-wien.ac.at> <6878@rex.cs.tulane.edu> Sender: news@stl.stc.co.uk Reply-To: tom@nw.stl.stc.co.uk Lines: 23 digits in the infinite expansion, I ask for the best rounded n digit approximation (where I'm counting digits to the right of the decimal point), and I occasionally allow the machine to give me the best n+1 digit rounded approximation instead, then the result is computable, provided I can approximate the number to an arbitrarily small tolerance, i.e. I'm starting with a recursive real number. I simply evaluate the number to a tolerance of less than one in the n+2nd digit. If the n+1st and n+2nd digits are 5 and 0, then I print the first n digits followed by a 5. In all other cases, I can safely round to n digits. (This is of course not optimal.) I can also ask for the first n truncated digits if I use a redundant notation, e.g. if I allow digits -9 through -1 as well. Of course that gets hard to read ... ====== Stuff and nonsense! How do you know when you are close enough? A recursive real number is simply a function f:N->R satisfying a convergence constraint (and of course being recursive too) [R=2 * N * (N-{0}), ie the rationals]. I don't see how you can use the convergence constraint to know when your approximation is close enough to be right to so many decimal places (suppose f is identically zero on the first twenty milliion naturals, and 1 thereafter: how do you know after looking at 10 million terms that you still don't have the first digit? how do you know after looking at the first 50 million terms that you do now h the first digit?)