Re: JSH: Critique means slow, and thorough

Jesse F. Hughes wrote:
> Will Twentyman <wtwentyman@xxxxxxxxxxx> writes:
>
> > jstevh@xxxxxxx wrote:
> >> Again, for those who wonder what the latest discussion is about,
if you
> >> append 1/2 to the ring of integers, you get the reals--if you
allow
> >> infinite sums.
> >>
> >
> > James, in this post you have claimed the following:
> >
> > 1) Z[1/2] is in the rationals.
>
> He didn't claim that, did he?
>

My position is that adjoining fractions like 1/2 to the ring of
integers gives you reals, as you end up with numbers that result from
infinite sums, whether you want them or not.

> > 2) Z[1/2] = reals
> > 3) pi is not rational.
> >
> > The only conclusion that can be drawn is that pi is not a real
number.

Nope.

The proper conclusion is that rationals are a description and not a
true field.

The description of a rational then is a number a/b where a is an
integer and b is a non-zero integer (rational integer for those who are
picky).

My point is that the "field of rationals" is in fact, not a field, as
that's a misnomer.

Actually, the act of adjoining any fraction like 1/2 or 1/3 to a ring,
where 1 and -1 are the only rational units is the field of reals, and
if i is in the ring, then you have the field of complex numbers.

> > Are you sure that's what you want to claim?
>
> Here's my guess.
>
> Z is a subset of Q.
>
> Q contains 1/2.
>
> Z[1/2] is a superset of Q.
>
> Therefore, Z[1/2] is not the minimal ring containing 1/2 and a
> superset of Z.
>
> If you try to add 1/2 to Z, you automatically overshoot Q. Now, if
> you *start* with Q and "add" 1/2 to Q, probably you stay with Q. So
> the process of adjoining elements to rings is non-monotonic (and a
> touch unpredictable).
>
> Anyway, that's my guess.
>

The entire idea of the "field" of rationals is that you have numbers
a/b where a is an integer and b is a non-zero integers, but you DO NOT
ALLOW INFINITE SUMS.

However, mathematically, *saying* you do not allow infinite sums does
not stop them from entering anyway, when the ring is infinite in size.

An infinite sized ring will allow convergent infinite sums.


James Harris

.

Relevant Pages

  • Re: JSH: Keep it simple
    ... R is closed under convergent infinite sums, ... First note that since R is a ring, ... So we only need to show that R is dense in the reals. ...
    (sci.math)
  • Re: JSH: Critique means slow, and thorough
    ... >> Adjoining such an element gives you the field of reals. ... >> something as simple as adjoining 1/2 to the ring of integers will ... as you cannot block the infinite sums. ...
    (sci.math)
  • Re: JSH: Critique means slow, and thorough
    ... > My point is that the "field of rationals" is in fact, not a field, as ... > if i is in the ring, then you have the field of complex numbers. ... > However, mathematically, *saying* you do not allow infinite sums does ...
    (sci.math)
  • Re: JSH: Critique means slow, and thorough
    ... My point is that the "field of rationals" is in fact, not a field, as ... if i is in the ring, then you have the field of complex numbers. ... An infinite sized ring will allow convergent infinite sums. ... Blah blah blah. ...
    (sci.math)
  • Re: JSH: Critique means slow, and thorough
    ... Zis in the rationals. ... integers gives you reals, as you end up with numbers that result from infinite sums, whether you want them or not. ... claim they do not satisfy one of the field axioms. ...
    (sci.math)
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