🚀 go-pugleaf

On 31 Mrz., 08:19, Albrecht <albst...@gmx.de> wrote:
> On 30 Mrz., 16:21, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
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> > MoeBlee says...
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> > >On Mar 29, 3:58=A0pm, Transfer Principle <lwal...@lausd.net> wrote:
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> > >> 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:
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> > <x,y> + <z,w> = <x+z,y+w>
> > <x,y> * <z,w> = <x*z+y*w, y*z+x*w>
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> > 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:
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> > <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>
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> > 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>
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> > Addition and multiplication of rationals is pretty
> > straight-forward in either case, as well.
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> > Where the difference between using equivalence classes
> > and using unique representations makes a big difference
> > is in operations involving REALS.
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> > 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).
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> > (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)
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> > (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..." .
>

And you can't give a decimal expansion of pi so that anybody can be
sure about the exact number. There is no unique decimal representation
of pi or any other irrational number.

AS