Xref: utzoo comp.theory:1756 comp.lang.functional:715 Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!swrinde!ucsd!ucbvax!agate!bionet!parc!boehm From: boehm@parc.xerox.com (Hans Boehm) Newsgroups: comp.theory,comp.lang.functional Subject: Re: do computers believe in real numbers? Message-ID: Date: 2 Apr 91 20:48:59 GMT References: <7197@munnari.oz.au> <91090.154205NN1@awiwuw11.wu-wien.ac.at> <6878@rex.cs.tulane.edu> Sender: news@parc.xerox.com Organization: Xerox PARC Lines: 15 Aberth makes the observation however that if instead of asking for the first n 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 ... Hans (boehm@xerox.com)