On Mon, Apr 22, 2013 at 08:06:29PM +0200, Telmo Menezes wrote: > > > On 22 avr. 2013, at 19:14, Craig Weinberg <whatsons...@gmail.com> wrote: > > > A quote from someone on Facebook. Any comments? > > > > "Computers can only do computations for rational numbers, not for real > > numbers. Every number in a computer is represented as rational. No computer > > can represent pi or any other real number... So even when consciousness can > > be explained by computations, no computer can actually simulate it." > > Of course it can, the same way it represents the letter A, as some sequence > of bits. And it can perform symbolic computations with it. It can calculate > pi/2 + pi/2 = pi and so on. > >
To expand a bit on Telmo's comment, the computer represents pi, e, sqrt(2) and so on as a set of properties, or algorithms. Computers can happily compute exactly with any computable number (which are of measure zero in the reals). They cannot represent nondescribable numbers, and cannot compute with noncomputable numbers (such as Chaitin's Omega). Also, computers do not compute with rational numbers, they compute with integers (often of fixed word size, but that restriction can easily be lifted, at the cost of performance). Rational numbers can obviously be represented as a pair of integers. What are called "real" numbers in some computer languages, or more accurately "float" numbers in other computer languages, are actually integers that have been mapped in a non-uniform way onto subsets of the real number line. Their properties are such that they efficiently generate adequate approximations to continuous mathematical models. There is a whole branch of mathematics devoted to determining what "adequate" means in this context. Cheers -- ---------------------------------------------------------------------------- Prof Russell Standish Phone 0425 253119 (mobile) Principal, High Performance Coders Visiting Professor of Mathematics hpco...@hpcoders.com.au University of New South Wales http://www.hpcoders.com.au ---------------------------------------------------------------------------- -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list?hl=en. For more options, visit https://groups.google.com/groups/opt_out.
