Using fractions instead of floats

[mensanator at aol.com]

On Oct 2, 1:12 am, Robert Kern <robert.k... at gmail.com> wrote:
> mensana... at aol.com wrote:
> > On Oct 1, 8:17?pm, richy... at gmail.com wrote:
> >> On Oct 1, 9:03 pm, "mensana... at aol.com" <mensana... at aol.com> wrote:
>
> >>> On Oct 1, 7:20 pm, richy... at gmail.com wrote:
> >>>> On Oct 1, 8:30 am, Nick Craig-Wood <n... at craig-wood.com> wrote:
> >>>>>   >>> mpq(1,3)+0.6
> >>>>>   mpq(14,15)
> >>>> Golly!  That's quite impressive.  And more than a little bit magic as
> >>>> well, since 0.6 is definitely not the same as 3/5.  
> >>> It's not? Since when?
> >> The 0.6 above is a floating point number, mathematically very close to
> >> 0.6 but definitely not equal to it, since 0.6 can't be represented
> >> exactly as a float.
>
> > Oh, you mean something like this, right?
>
> >>>> import gmpy
> >>>> a = 0.6
> >>>> a
> > 0.59999999999999998
>
> > So, the rational should have 59999999999999998
> > in the neumerator and 100000000000000000 in the
> > denominator?
>
> Actually (5404319552844595 / 2**53) would be best.
>
> > But it doesn't
>
> >>>> b = gmpy.mpq(a)
> >>>> b
> > mpq(3,5)
>
> > Why do you suppose that is?
>
> For the same reason that str() does. See below.
>
> > For that matter, why
> > does
>
> >>>> str(a)
> > '0.6'
>
> > give me an EXACT representation?
>
> It doesn't. It just rounds at a lower number of decimal places than is necessary
> to faithfully represent the number. str() is intended to give friendly results,
> not strictly correct ones. In this case it happens that float(str(0.6)) == 0.6,
> but this is not guaranteed. For most numbers that a user or programmer might
> enter, there will only be a relatively small amount of precision necessary, and
> float(str(x)) == x will tend to hold. That's why it does this.
>
> > Didn't you just
> > say it couldn't be represented exactly?
>
> Yup.
>
> > Which is correct,
>
> >>>> str(a)
> > '0.6'
>
> > or
>
> >>>> repr(a)
> > '0.59999999999999998'
>
> > ?
>
> The latter.
>
> > How does gmpy make the conversion from float to rational?
>
> gmpy has a configurable transformation between floats and the internal
> representation. I believe the default goes through str().

How do you do that? Is it configurable at run time or something that
has to be done when compiled?

But it is still wrong to say "0.6 is definitely not the same as 3/5".
One can say 0.6 doesn't have an exact float representation and that
inexact representation is not the same as 3/5. And I suppose one can
be surprised that when this inexact representation is coerced to a
rational the result is now exact.

Would that be a fairer way of putting it?

>
> --
> Robert Kern
>
> "I have come to believe that the whole world is an enigma, a harmless enigma
>  that is made terrible by our own mad attempt to interpret it as though it had
>  an underlying truth."
>   -- Umberto Eco