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Re: Re: Re: Zero divided by a number...
You might as well argue that there is no distinction between projective
spaces and affine spaces, manifolds and euclidean spaces, schemes and
rings and so on. Obviously you will can can always call each of the
concepts an extension of the other and then just decide that, you might
as well call them both by the same name. It still won't make them the
same.
In any case, this has completely no relevance to how this discussion
started and what it was originally about.
I think before one joins a thread it would be a good idea to start by
looking at the original posting that began it.
Andrzej Kozlowski
Chiba, Japan
On 15 Nov 2004, at 17:17, Garry Helzer wrote:
> *This message was transferred with a trial version of CommuniGate(tm)
> Pro*
> Historically "complex numbers" or sometimes "hypercomplex numbers"
> referred to any extension of the reals--field or no. I. M. Yaglom
> wrote a nice little book called something like " Complex numbers and
> Geometry". In it he describes three systems of "complex numbers" and
> and uses each system to prove a theorem in plane geometry (one
> Euclidean and two non-Euclidean theorems). As I recall (I no longer
> have the book and cannot find any reference to it on the web) he first
> shows that, up to ring isomorphism, there are exactly three rings of
> the of the form R[x]/<p(x)> where p(x) is a quadratic polynomial. The
> three distinct types are obtained by taking p(x) to be x^2+1, x^2, and
> x^2-1. He then extends each system of complex numbers by adding
> infinite elements--a single infinity for x^2+1 and an infinity of
> infinities in (both?) of the other two cases. He then states and proves
> the same plane geometry theorem in each of the three contexts using the
> arithmetic of the three number systems.
>
> Discussions as to whether these systems are "really" numbers--or
> complex numbers--are more suited to a discussion group for lawyers than
> a discussion group for mathematicians.
>
> --Garry Helzer
>
> On Nov 8, 2004, at 10:37 PM, David W. Cantrell wrote:
>
>> Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
>>> On 7 Nov 2004, at 15:04, David W. Cantrell wrote:
>>>
>>>> BTW, concerning whether an improper element of an extended number
>>>> system should be called a "number" or not, it might be noted that:
>>>> Floating-point arithmetic is surely the most widely used number
>>>> system
>>>> in the world, in terms of the number of computations performed per
>>>> day.
>>>> There is an internationally accepted standard for that arithmetic.
>>>> The
>>>> standard clearly distinguishes between those floating-point objects
>>>> which are numbers and those which aren't (the NaNs). According to
>>>> the standard, -Infinity and +Infinity are numbers (while things such
>>>> as 0*Infinity yield NaN).
>>>
>>> According to Mathematica:
>>>
>>> NumericQ[ComplexInfinity]
>>>
>>> False
>>>
>>> NumericQ[Infinity]
>>>
>>> False
>>
>> I was either unaware or had forgotten that Mathematica considers
>> ComplexInfinity and Infinity to be nonnumeric, so thanks for pointing
>> that
>> out. And knowing that can indeed be important when programming in
>> Mathematica. But as far as _mathematics itself_ is concerned, I had
>> already
>> said
>>
>> "Seriously, both 0 and ComplexInfinity are quite peculiar, no doubt.
>> And if
>> you don't want to call ComplexInfinity a number, that's just fine. But
>> it's essentially irrelevant whether we call it a 'number' or not. It's
>> an
>> element of C*, and what's important is knowing what you can (and
>> can't) do
>> with it."
>>
>> On second reading, perhaps it was not clear that I was talking about
>> mathematics itself, rather than Mathematica, but C* was what I had in
>> mind
>> when I wrote that.
>>
>>> It is to say the least controversial if "numbers" are what complex
>>> analysts deal with.
>>
>> It would be absurd to suggest that "numbers" be _restricted_ to what's
>> used
>> in complex analysis. But I gather that that is not your point.
>>
>>> ComplexInfintiy and Infinity do not belong to any
>>> family of numbers known to number theorists (who ought to be the
>>> people
>>> who know best what numbers are), e.g. algebraic numbers,
>>> transcendental
>>> numbers, or even computable numbers.
>>
>> Number theory is a specific branch of mathematics, as you know. There
>> are
>> types of objects used in mathematics which are often called numbers
>> but
>> which are not normally considered in number theory itself. Cantor's
>> cardinal and ordinal numbers come to mind, for example. DrBob has
>> objected
>> previously in this thread that ComplexInfinity doesn't behave like a
>> number; I'd say, rather, that it doesn't behave like a _finite_
>> number.
>> Transfinite numbers do not behave like finite numbers; that's to be
>> expected, of course. The mathematicians' drinking song "Aleph_nought
>> bottles of beer on the wall" never ends (well, at least, in theory; in
>> practice, we'd get too drunk to continue singing, I suppose). And 1 +
>> omega
>> is not the same as omega + 1.
>>
>> Earlier in this thread, you said "The word 'number' is ambiguous..."
>> That's
>> certainly true. There is no generally accepted definition of number.
>> There
>> are, of course, generally accepted definitions for _specific_ number
>> systems.
>>
>>> A point on the Riemann sphere is not a "number".
>>
>> Are you being pedantic? If so, I agree with you. We may say that the
>> Riemann sphere is merely a way to visualize C*, just as the number
>> line is
>> a way to visualize R. And no point on the line or the sphere _is_ a
>> number.
>> But there is an obvious _correspondence_ between the points of the
>> line and
>> the elements of R and between the points of the sphere and the
>> elements of
>> C*.
>>
>> Each point on the Riemann sphere, except its "North Pole", corresponds
>> with
>> an element of C, that is, with a complex number. The North Pole
>> corresponds
>> with oo, an element of C*. Whether we wish to call that element a
>> "number"
>> or not is essentially irrelevant, as I've said before. Please note
>> that I
>> have not said anywhere in this thread that I think it _should_ be
>> called a
>> number. It's adequate, as far as I'm concerned, merely to say that oo
>> is an
>> element of C*.
>>
>>> But I found this reply of yours to Bob particulalry incredible :
>>>
>>>>> ComplexInfinity isn't a number if you can't do
>>>>> arithmetic with it,
>>>>
>>>> But you can do arithmetic with it.
>>>
>>> Really? And presumably algebra too? I am curious how,
>>
>> Really! (And I can't imagine why you thought that was "particulalry
>> incredible".) In C*, we have oo + 1 = oo and 1/oo = 0, for example.
>> That's
>> doing arithmetic involving oo. Correspondingly, in Mathematica, we
>> have
>>
>> In[1]:= ComplexInfinity + 1
>>
>> Out[1]= ComplexInfinity
>>
>> In[2]:= 1/ComplexInfinity
>>
>> Out[2]= 0
>>
>>> given that this
>>> is the only "number" about which Mathematica does not even know if it
>>> is equal to itself:
>>
>> First, note that I never said it was a number. And as you pointed out
>> above, Mathematica considers it to be nonnumeric.
>>
>>> ComplexInfinity==ComplexInfinity
>>>
>>> ComplexInfinity==ComplexInfinity
>>
>> In C*, oo = oo is true, of course. The fact that Mathematica cannot at
>> present decide about ComplexInfinity==ComplexInfinity is, in my
>> opinion, a
>> deficiency. But of course, the designers may have had a good reason,
>> unknown to me, for leaving ComplexInfinity==ComplexInfinity
>> unevaluated.
>>
>>> But then perhaps analysts mean something different by "arithmetic"
>>> and
>>> "algebra" from the rest of us ;-)
>>
>> Not that I'm aware of.
>>
>> Perhaps it would be helpful if I discuss the construction of C*. I'll
>> also
>> construct a more flexible system, much like the one that Mathematica
>> uses.
>> Of course, to keep this post to a reasonable length, this must be a
>> discussion "in a nutshell", just a bare-bones outline. I hope that
>> readers
>> will be able to follow it if they're familiar with the constructions
>> of the
>> reals from the rationals via Dedekind cuts and via equivalence classes
>> of
>> Cauchy sequences. (BTW, the constructions outlined will not be
>> aesthetically optimal IMO, but rather optimal for speed of
>> presentation
>> here.)
>>
>> Assume that the positive rationals have already been constructed. We
>> may
>> then construct the nonnegative extended reals, [0, +oo], from the
>> positive
>> rationals either by cuts or by equivalence classes of appropriate
>> rational
>> sequences. Bertrand Russell used the former construction with the cut
>> having lower class {q | q in Q, q > 0} being "the real number
>> infinity", to
>> use his own words. For the other method of construction, we proceed as
>> usual except that we also include an equivalence class which consists
>> of
>> all rational sequences which increase without bound. That equivalence
>> class
>> is +oo. [Those reading closely may protest that the sequences in that
>> equivalence aren't Cauchy. But in fact, if desired, they can easily be
>> made
>> Cauchy simply by using an appropriate metric.] Note that, regardless
>> of the
>> method of construction chosen, +oo is the same type of object as the
>> nonnegative reals which were constructed along with it. In that light,
>> it
>> would certainly not be unreasonable to call +oo a number.
>>
>> The (signed) reals, R, may then be constructed easily from the
>> nonnegative
>> reals.
>>
>> An extended complex number system may then be constructed in which the
>> elements are equivalence classes of ordered pairs (r, theta) with r in
>> [0, +oo] and theta in R, with ordered pairs (r, theta_1) and (r,
>> theta_2)
>> being in the same equivalence class whenever theta_1 = theta_2 mod(2
>> Pi).
>> This system should in essence be the same as Mathematica's system of
>> complex numbers together with _directed infinities_. (The only web
>> reference I know to such a system is "A family of compactifications
>> accounting for all arguments of infinity" by Gingold and Gingold at
>> <http://www.slackworks.com/~yotam/gammasphere/metric/
>> gammasphere.pdf>.)
>> OTOH, if we lump all of the ordered pairs having r = +oo into a single
>> equivalence class (without regard to theta), we get C*, which should
>> in
>> essence be Mathematica's system of complex numbers together with
>> ComplexInfinity. [Note of course that, in C*, the undirected infinity
>> oo is
>> defined as a specific equivalence class, and so there is no doubt that
>> oo = oo is true. That's why I'm surprised that Mathematica does not
>> assert
>> that ComplexInfinity==ComplexInfinity is true.]
>>
>> The system which Mathematica actually uses is equivalent neither to C
>> with
>> just directed infinities nor to C with just the undirected infinity.
>> Rather, Mathematica uses a "hybrid" system having both the directed
>> infinities and the undirected infinity available. This is laudable
>> IMO,
>> allowing a specific directed infinity to be given whenever feasible
>> (and
>> thereby retaining as much information as possible), but also allowing
>> the
>> undirected infinity to be given in cases when no direction can be
>> specified.
>>
>> In this brief outline, I haven't yet mentioned the definitions of the
>> arithmetic operations, but they can be defined in fairly obvious ways.
>> Let's go back to the system [0, +oo], for example, and consider, say,
>> 1/(+oo). To compute that, take a representative sequence <a_n> of
>> nonnegative rationals in the equivalence class 1 and a representative
>> sequence <b_n> of nonnegative rationals in the equivalence class +oo.
>> The
>> desired quotient is then the equivalence class containing the sequence
>> <a_n/b_n>. Since a_n converges to 1 and b_n increases without bound,
>> a_n/b_n must converge to 0. Thus, 1/(+oo) = 0 in the system [0, +oo].
>> But
>> what if we similarly attempt to compute, say, 0/0. We must consider
>> two
>> representative sequences, <a_n> and <b_n>, in the equivalence class 0,
>> and
>> look at <a_n/b_n>. Alas, that sequence may very well not reside in
>> _any_ of
>> the equivalence classes constituting [0, +oo], and so we must say that
>> 0/0
>> is undefined in this system. In floating-point arithmetic, when this
>> situation arises, the result is called NaN (Not a Number); in
>> Mathematica,
>> it's called Indeterminate.
>>
>> Someone had asked me, in a private email concerning this thread, if
>> such
>> systems aren't fields, then what are they? Well, for a system like C*
>> with
>> another element (like NaN or Indeterminate) adjoined to handle cases
>> such
>> as 0/0, the term is "wheel" (and that extra element is often called
>> "bottom"). The best source (known to me) of information on wheels is
>> Jesper
>> Carlstrom's thesis "Wheels -- On Division by Zero", available at
>> <http://www.matematik.su.se/~jesper/research/wheels/>.
>>
>> Sorry that I didn't have time to go into more detail.
>>
>> David W. Cantrell
>>
>>
> Garry Helzer
> gah at math.umd.edu
>
>
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