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MathGroup Archive 2004

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Re: Re: Zero divided by a number...

  • To: mathgroup at smc.vnet.net
  • Subject: [mg52040] Re: [mg52030] Re: Zero divided by a number...
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Mon, 8 Nov 2004 03:13:21 -0500 (EST)
  • References: <20041103190554.052$Ay@newsreader.com> <cmcn2q$ihh$1@smc.vnet.net> <200411070604.BAA18132@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

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

It is to say the least controversial if "numbers" are what complex 
analysts deal with. 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. A point on the Riemann sphere is 
not a "number". 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, given that this 
is the only "number" about which Mathematica does not even know if it 
is equal to itself:


ComplexInfinity==ComplexInfinity

ComplexInfinity==ComplexInfinity

But then perhaps analysts mean something different by "arithmetic" and 
"algebra" from the rest of us ;-)


Andrzej Kozlowski
Chiba, Japan
http://www.akikoz.net/~andrzej/
http://www.mimuw.edu.pl/~akoz/


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