Working with the set of real numbers

Gregory Ewing greg.ewing at canterbury.ac.nz
Wed Feb 12 23:24:51 CET 2014


Chris Angelico wrote:
> Sure, but nobody said the text file had to be _stored_ anywhere :)
> Computers are quite capable of working with streams of incoming data
> that are potentially infinite in size.

However, they *can't* work with arbitrary real numbers in an
exact way, even if they are represented by infinitely long
digit streams, and you're willing to run the program for
an infinitely long time to get the result.

Consider adding two of these numbers, for example. You have
to do it starting at the big end, because the small end is
infinitely far away. And you only have a limited amount of
buffer space, so you need to start writing out result
digits before you've seen all the input digits.

But you can't do that, because it's possible that some
pair of input digits you haven't seen yet will cause a
carry-over that ripples back and affects something you've
already written out.

This is where schemes such as computable reals get into
trouble. Doing arithmetic with them gets extremely messy.

-- 
Greg



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