Re: CHAT: mathematics

Carlos Thompson <carlos_thompson@...> wrote:
> Yown Ha Ley wabbe:
>
> > On Fri, 17 Nov 2000, Carlos Thompson wrote:
> >
> > >   duolrino :  real numbers (from rino S - oil)
> >
> > ... could you enlighten me as to how "oil" suggests real numbers? ...
    What you write below is basically correct, but your terminology is a bit
strange, and you make a minor mistake or two:

> Well, kind of joke.  Well, the history comes this way: rational
> numbers are not continuum, topologically speaking.  This means that
> you can have a series that seems to converge analyzing the series by
> itself:
>   Given the series {a_n}; given any positive real epsilon, there is a
> natural N shuch as any naturals n > N and m > N, | a_n - a_m | <
> epsilon.
    This sort of sequence (not "series") is a Cauchy sequence.

>   However such a series could not converge in Q.
    Close.  This sort of sequence does not necessarily converge in Q.  There
are Cauchy sequences which do converge in Q, such as {1/2, 1/3, 1/4, 1/5,
1/6, ...}, which converges to 0, (or, for that matter, {1, 1, 1, 1, ...},
which converges to 1).  The fact that there are Cauchy sequences which don't
converge, however, (such as {2, 2.7, 2.71, 2.718, 2.7182, ...} (the decimal
expansion of e)), is described by saying "Q is not complete" (yet another
meaning of "complete" in mathematics).  The reals, in fact, can be defined
in terms of equivalence classes of Cauchy sequences of rationals.  Because of
this, we say that the reals are the "completion of the rationals".

> This means that Q is full of wholes.
    That's "holes".

 ===========================================================================

             Dennis Paul Himes    <>    dennis@himes.connix.com
                  http://www.connix.com/~dennis/dennis.htm

Disclaimer: "True, I talk of dreams; which are the children of an idle
brain, begot of nothing but vain fantasy; which is as thin of substance as
the air."                      - Romeo & Juliet, Act I Scene iv Verse 96-99