Re: .999... ?= 1

From: *** T. Winter (***.Winter_at_cwi.nl)
Date: 06/09/04 

Date: Wed, 9 Jun 2004 23:49:59 GMT

In article <40C6B2E7.4020108@et.uni-magdeburg.de> Eckard Blumschein <blumschein@et.uni-magdeburg.de> writes:
> *** 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?

No, this leads nowhere, I think. The reals do not provide an order for
counting and the complex numbers do not provide a measure.

> > 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?

In a sense it is, but that is not how the inverse of a function is
defined. Worse, there are functions that do not have an inverse
(like y = x^2).

> 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 do not see what the problem is.

> 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.

Think about what the Fourier transform is.

> > 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)

I am not talking about the relation > at all. I am talking about three
properties of numbers, if a, b and c are numbers:
1. a*(b + c) = a * b + a * c
2. For any a there is a b such that a + b = 0
3. Let's have a + a = a, and let a + b = 0, then:
    a = b + a + a = b + a = 0.
(BTW, it can also be done without the additive inverse:
2'. 1/0 = oo = 2/0
3'. 0.oo = 1 and 0.oo = 2 (by the definition of inverse), so 1 = 2.

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

Where do I want the property a > b back? I am not talking about that at
all. I am talking about basic properties of number systems, whether they
are ordered or not.

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

Adding a single infinity to the real numbers does not result in a number
system (in some sense). You need more infinities (hyperreals, surreals,
or whatever) to get a working number system again. But in *none* of them
is 1/0 a valid operation.

> > (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.

I am *not* talking about a>b. What I wrote above is also valid when you
consider complex numbers (where the > relation does not exist).

> > 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.

Ah, so you fit in the group of cranks that think the number of integers is
finite. Fine.

> > 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.

In that case he was not an expert.

> Also I was told that zero is not necessarily a number,

Not by an expert.

> and if I recall
> correctly, I found that zero is the only infinitesimal within the
> hyperreal numbers.

Your recollection is incorrect. 

-- 
*** t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~***/