🚀 go-pugleaf

On 30 Mrz., 16:21, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
> MoeBlee says...
>
>
>
> >On Mar 29, 3:58=A0pm, Transfer Principle <lwal...@lausd.net> wrote:
>
> >> for each natural number n, let (n,0) be the
> >> positive integer +n, and (0,n) be the negative integer -n.
>
> >> let (a,b), with b>0, gcd(a,b)=1, be
> >> the rational a/b.
>
> >Have you checked that you can get that all to work with operations of
> >addition, multiplication, subtraction, division, and the standard
> >ordering? Especially, arithmetic on integers requires taking them also
> >as differences reckoned not just with 0, and arithmetic on rationals
> >ordinarily requires taking them also as quotients not necessarily in
> >the gcd form you stipulate.
>
> I don't think that's true. The advantage to defining integers
> as pairs <x,y> is that addition and multiplication of integers
> can be defined equationally in terms of addition and multiplication
> of naturals:
>
> <x,y> + <z,w> = <x+z,y+w>
> <x,y> * <z,w> = <x*z+y*w, y*z+x*w>
>
> In contrast, letting the integers be a disjoint union
> of pairs of the form <0,y> and <x,0> requires a more
> complicated definition by cases for addition:
>
> <0,y> + <0,z> = <0,y+z>
> if y<=z, then <0,y> + <z,0> = <z-y,0>
> if y>z, then <0,y> + <z,0> = <0,y-z>
>
> On the other hand, the rule for multiplication is still
> pretty simple:
> <0,y> * <0,z> = <y*z,0>
> <0,y> * <z,0> = <0,y*z>
>
> Addition and multiplication of rationals is pretty
> straight-forward in either case, as well.
>
> Where the difference between using equivalence classes
> and using unique representations makes a big difference
> is in operations involving REALS.
>
> Here are two ways to think about computability for
> reals: (Off the top of my head, this is probably not
> the way that computability theorists would think of it).
>
> (1) If you view reals as equivalence classes of rationals,
> then you can define a function f from reals to reals to be
> computable if there is a corresponding computable function
> F from rationals to rationals such that whenever
> limit of q_0, q_1, ... = r
> then
> limit of F(q_0), F(q_1), ... = f(r)
>
> (2) If you view reals as unique decimal representations,
> then we can say that a function f from reals to reals is
> computable if there is computable function F from finite
> decimal approximations of r to finite decimal approximations
> to f(r), and every digit of f(r) is computed by F on a
> sufficiently accurate approximation of r.
>
> By this definition, multiplication and addition
> are not computable: For example, let f(r) = 3*r,
> and consider a computation on r = 0.333...
> No amount of accuracy in the input is sufficient
> to determine whether the output starts 0.xxx or
> 1.xxx
>

But nobody would compute 3*0.333... in that way and nobody have to
compute it in that way. But in the case of irrational numbers a lot of
computation have to be done in that way and there is no other way to
do the job. That's the essential difference between rational and
irrational numbers. Decimal or other adic expansions are only crutches
for talking about irrationals. No adic expansion defines an irrational
number and no adic expansion is able to define an irrational number.

The view of the reals as unique decimal representations is a Fata
Morgana. E.g. nobody can be sure what is meant with the expression
"3.1415926535897932384626433832795..." .

Best
Albrecht