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.



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

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