Re: .999... ?= 1

From: David W. Cantrell (DWCantrell_at_sigmaxi.org)
Date: 06/09/04 

Date: 09 Jun 2004 01:43:21 GMT

"*** T. Winter" <***.Winter@cwi.nl> wrote:
> In article <40C5F9DD.8070703@et.uni-magdeburg.de> Eckard Blumschein
> <blumschein@et.uni-magdeburg.de> writes:
> > Robin Chapman wrote:
> > > Your handling of zero is ceertinly uncertain.
> > >
> > > Is "infinity" actually a number?
> >
> > I know that there is seemingly no need for treating it like a number
> > while this status is commonly attributed to its reciprocal.
>
> Depends on what you define as a number (I have not seen a definition in
> this thread yet), and how you define inverse/reciprocal. Commonly in
> the definition-process, there are only two starting operations:
> addition and multiplication, and how they hang together in a ring.
> At some point the reciprocal of a number a is defined as the number
> ra such that a * ra = 1 (the unity of the multiplication). It is
> only after that that division is defined as a shorthand for
> multiplication by the reciprocal. Similarly, negative numbers are
> defined as numbers that add up to the original, such that their sum is 0
> (the unity of the addition). Only after that subtraction is defined.
>
> Now let's see how that works in a ring where 0 (the unity of the
> addition) has an inverse, say oo. So 1/0 = oo and 0.oo = 1. Now what is
> 2/0? If it is also oo, we have:
> 1. oo + oo = 1/0 + 1/0 = 2/0 = oo (using the distributive property)
> 2. oo + oo = oo -> oo = 0 (using the property of the additive inverse)
> So that will not work. We need more infinities to make it work.
> (Unless you throw away one of the two properties used above.)
>
> But the strange thing is, it does not matter how many infinities we add,
> we can not get something consistent (David Cantrell argues otherwise,
> yes, I know).

What?! Are you claiming that I argue "otherwise" specifically in relation
to the statement "it does not matter how many infinities we add, we can not
get something consistent"? For the record, I have _never_ argued such.
I invite anyone interested to read what I have written, rather than what
someone else says that I have written. And I say that with all due respect
to you, ***. I certainly cannot imagine that you intentionally distorted
anything I wrote. I'm not sure where you think I espoused such an untenable
position, but perhaps it was in my article "How I Divided by Zero, and
Lived to Tell About It!" at
<http://mathforum.org/discuss/sci.math/a/m/338231/547380>.

Regards,
David W. Cantrell

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