On Sun, Dec 29, 2019 at 2:44 PM David Mertz <me...@gnosis.cx> wrote:
>
> On Sat, Dec 28, 2019 at 10:31 PM Richard Damon <rich...@damon-family.org>
> wrote:
>>
>> Every value of the type float, except NaN and perhaps +inf and -inf
>> (depending on which version of the Real Number Line you use) IS actually
>> a representation of a Real Number (so I don't understand in what way you
>> can say they aren't "numbers"). And yes, they happen to also be Rationals.
>
>
> This is not subtle, and it is not something you do not understand. C'mon.
>
> Tell me three Real Numbers (or Rational Numbers) such that:
>
> (a + b) + c != a + (b + c)
>
> You will not succeed in that endeavor because of the nature of "numbers."
>
> ... and yes, because of the usual meaning given to the symbol '+', etc.
It's not about numbers, it's about the nature of addition in the real
numbers. Since floats are a subset of real numbers, it is entirely
possible for the result of any operation to be outside of the subset
representable as floats.
> OK... now I'll tell you three floating point "numbers" that follow that
> pattern:
>
> a = float("0.1")
> b = float("0.2")
> c = float("0.3")
>
> We can conveniently write the constructors for these in Python as `0.1` and
> so on. These objects that occupy computer memory are not "numbers" in the
> sense of obeying the axioms of arithmetic. But they kinda-sorta resemble
> numbers, and computers can do things with them quickly, so we pretend.
>
They really truly ARE numbers. They are, in fact, these numbers:
a = 3602879701896397 / 36028797018963968
b = 3602879701896397 / 18014398509481984
c = 5404319552844595 / 18014398509481984
When you perform addition on these, the result must repeatedly be
rounded to the available resolution. This is not about numbers, it is
about limited subsets of numbers.
ChrisA
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