🚀 go-pugleaf

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

--
Daryl McCullough
Ithaca, NY