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From: aet@felix.ee.mu.OZ.AU (Albert Edward THOMPSON)
Newsgroups: sci.math,sci.math.symbolic,comp.theory,comp.lang.functional
Subject: do computers believe in real numbers?
Message-ID: <7197@munnari.oz.au>
Date: 28 Mar 91 03:51:56 GMT
Article-I.D.: munnari.7197
Sender: news@cs.mu.oz.au
Reply-To: aet@felix.ee.mu.OZ.AU (Albert Edward THOMPSON)
Organization: Dept. of Electrical Engineering, University of Melbourne
Lines: 38


	I am trying to implement real numbers in a normal-order functional language--
NOT floating point numbers or any other approximation.
	This is my problem:
		I don't know what real numbers are 
		or even whether they can exist in any pragmatic (i.e. computational) sense.

	I looked at Bishop's 'Constructive Analysis' but could work out how to 
construct only rational reals. 
	I've heard about some things called the computable reals.
(some sort of countable subset of the reals?)
What are they? Are there reals which are not computable?
If so, in what way do they "exist"?

Concretely, my picture of real arithmetic on computer is something like this:
user>    sqrt 2
machine>   answer1    ;machine gives answer in unforced form.
user> answer1 5       ;user asks to see answer in explicit form accurate to 5 decimal places.
machine> 1.41421      ;machine forces answer as asked for.
user> log pi
machine> answer2
user> answer2 8
machine> 1.14472988

	It is not important whether the machine rounds off (which it probably must do anyway)
	or gives the "true" digits in the decimal expansion.
	(although this is theoretically interesting -- comments welcome!)
	What is important is that the machine should do much work only when digits are 
	explicitly requested and that the machine ideally should give _any_ accuracy.

	Perhaps some symbolic maths language uses reals rather than approximations?

Sorry if I'm being nebulous or ill-informed:
I still don't know exactly what is required in this problem.

		Preemptive thanks for any suggestions,

												Bert.