Working with the set of real numbers

[Chris Angelico]

On Wed, Feb 12, 2014 at 10:44 PM, Ben Finney <ben+python at benfinney.id.au> wrote:
> Chris Angelico <rosuav at gmail.com> writes:
>
>> On Wed, Feb 12, 2014 at 9:07 PM, Ben Finney <ben+python at benfinney.id.au> wrote:
>> > That's why I think you need to be clear that your point isn't
>> > “computers don't work with real numbers”, but rather “computers work
>> > only with a limited subset of real numbers”.
>>
>> Hmm, I'm not sure that my statement is false. If a computer can work
>> with "real numbers", then I would expect it to be able to work with
>> any real number.
>
> Likewise, if you claim that a computer *does not* work with real
> numbers, then I would expect that for any real number, the computer
> would fail to work with that number.
>
> Which is why neither of those is a good statement of your position, IMO,
> and you're better off saying the *limitations* you're describing.

I think we're using different words to say the same thing here :)

What I mean is that one cannot accurately say that a computer works
with real numbers, because it cannot work with them all. Of course a
computer can work with _some_ real numbers; but only some. (An awful
lot of them, of course. A ridiculously huge number of numbers. More
numbers than you could read in a lifetime! While the number is
extremely large, it still falls pitifully short of infinity.[1]) And
so we do have optimizations for some subset of reals: in approximate
order of performance, an arbitrary-precision integer type, a limited
precision floating point type, and two types that handle fractions
(vulgar and decimal). They're all, in a sense, optimizations. In pure
theory, we could have a single "real number" type and do everything
with that; all the other types are approximations to that.

[1] http://tools.ietf.org/search/rfc2795