Working with the set of real numbers
Steven D'Aprano
steve at pearwood.info
Thu Feb 13 04:31:09 CET 2014
On Wed, 12 Feb 2014 21:07:04 +1100, Ben Finney wrote:
> Chris Angelico <rosuav at gmail.com> writes:
>
>> On Wed, Feb 12, 2014 at 7:56 PM, Ben Finney
>> <ben+python at benfinney.id.au> wrote:
>> > So, if I understand you right, you want to say that you've not found
>> > a computer that works with the *complete* set of real numbers. Yes?
>>
>> Correct. […] My point is that computers *do not* work with real
>> numbers, but only ever with some subset thereof […]
>
> You've done it again: by saying that “computers *do not* work with real
> numbers”, that if I find a real number – e.g. the number 4 – your
> position is that, since it's a real number, computers don't work with
> that number.
That answer relies on the assumption that "computers do not work with X"
implies:
for each element x in X:
it is true that "computers do not work with x"
that is to say, a single counter-example of computers working with an
element of X, even if it is a fluke, is enough to disprove the rule.
To give a real-world, non-programming example:
"The former South African apartheid government did not respect the
Universal Human Rights of blacks."
Under your strict interpretation, we would have to say that even a single
example of the apartheid government respecting even a single human rights
of a single black person would be sufficient to disprove the claim.
But there's another interpretation available to us, one which is more
suited to natural language statements as made by Chris: we interpret
"computers do not work with X" as meaning:
there is at least one element x, such that it is true that
"computers do not work with x"
In the case of real numbers, there is an *uncountably infinite* number of
such elements x. In fact, we can pick any two distinct numbers, no matter
how close together, say:
1
1.000000000001
and be sure that there are an uncountably infinite number of real numbers
which computers do not work with between those two values.
For the record, "uncountable infinite" is not just me emphasising that
infinity is too big to count. It's a technical term from mathematics. In
a nutshell it means that not only are there too many elements to count,
but even in an infinite amount of time you couldn't count them all, not
even if you counted infinitely fast.
In fact, it isn't just that there are *specific* real numbers which
computers cannot represent (say, irrationals like pi or e, really tiny
numbers like 1/(googleplex**googleplex**googleplex), or really huge ones
like Graham's Number), but that the fundamental mathematical laws of the
reals are violated by computers.
For example, it is not true that for every number x, 1/1(x)) == x.
py> 1/(1/93.0) == 93.0
False
Nor is it always true that a*(b+c) equals a*b + a*c, or that a+b+c is
necessarily equal to b+c+a.
So it isn't even that floats are merely a subset of reals. They're
actually not reals at all, since the fundamental properties of real
numbers do not always apply to floating point calculations.
--
Steven
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