Re: .999... ?= 1

From: Eckard Blumschein (blumschein_at_et.uni-magdeburg.de)
Date: 06/09/04 

Date: Wed, 09 Jun 2004 07:49:11 +0100

*** T. Winter wrote:

> Depends on what you define as a number (I have not seen a definition in
> this thread yet),

Being a layman, I looked for such definition and was disappointed, too.
So I came to the idae a number has at least two applications: Providing
an order for counting and providing a measure. Does this lead on the
right track?

and how you define inverse/reciprocal.

I naively imagine the reciplocal of a number might be related to but not
the same as the reciprocal of a function or a graph. Isn't ln(x) the
inverse of exp(x) in the sense of symmetry with respect to y=x?
I wonder why y=exp(x) within IR can be one-to-one translated into
y=ln(x) within IR+ while changing multiplication of x values into
addition, etc of belonging y values.
I also wonder why Fourier transform of a continuous periodic function
(aleph2) consists of dicrete values (aleph1) and vice versa no matter
whether or not it is compex-valued.

> 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

This looks reasonable to me.

and 0.oo = 1.

I know, this is not allowed. Nonetheless it might be pointless while
also reasonable.

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

I see this correct because the relation a>b fails at oo.

-> oo = 0 (using the property of the additive inverse)

Once having lost the a>b property you must not expect getting it back
with a return.

> So that will not work. We need more infinities to make it work.

???

> (Unless you throw away one of the two properties used above.)

I am a miser who does not throw away anything. Loss of a>b is the price
for crossing the border between finite and infinite.

> 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).

So called counterintuitive strangeness is not strange to me at all.
I am an engineer. To my understanding "no matter how _many_" implies a
finite number.

>>I vaguely recall: Only a few experts do not consider zero a number. Is this correct?
>
> Not as far as I know, I have never met an expert who did consider 0 not
> a number.

I merely found this within at least one book on Cauchy(?) written as or
more likely based on a thesis by an outsider. Don't ask me which one.
Also I was told that zero is not necessarily a number, and if I recall
correctly, I found that zero is the only infinitesimal within the
hyperreal numbers.

Eckard Blumschein