std.complex

[David Nadlinger]

On Tuesday, 26 November 2013 at 16:30:30 UTC, Joseph Rushton 
Wakeling wrote:
> On 26/11/13 17:11, David Nadlinger wrote:
>> On Monday, 25 November 2013 at 08:18:43 UTC, Joseph Rushton 
>> Wakeling wrote:
>>> But if you just think of 0 as a number, then
>>>
>>> 0 * lim{x --> inf} x
>>>             = lim{x --> inf} (0 * x)
>>
>> This is where your argument falls apart, as mathematically, 
>> you can't do that
>> unless lim{x --> inf} x is well-defined.
>
> I was using a very lazy shorthand there, I'm glad someone 
> thought to call me on it.  Can we take it as read that I was 
> basically thinking of a sequence {x_n} such that for every K 
> there is an N such that for n > N, x_n > K ... ? :-)
>
> And then you have
>
>     0 * lim{n --> inf} x_n = ... etc.

x_n = n actually fulfils that property, and I think most people 
would understand the limit notation for real numbers exactly the 
way you intended.

But that was not my point. To be able to write "lim{n --> inf} 
x_n" in a meaningful way (and consequently »pull in« the 
multiplication), a (or as it turns out, the) limit must exist in 
the metric space you are working in.

If your metric space contains ∞, and it has the property that 0 . 
∞ = 0, then your argument is correct. But such a symbol ∞ does 
not exist in the real numbers.

I guess it might help to think back to your first 
university-level analysis courses, where I'm sure these 
distinctions were discussed many times in proofs concerning the 
existence of limits, e.g. integrability of certain functions, …

>> See also: http://en.wikipedia.org/wiki/Riemann_sphere
>
> I don't recall ever actually studying the Riemann sphere, which 
> really seems to me like a gap in my education :-\

Well, the reason I bring this up is that what the »right« 
behavior is all comes down to the definition of your numerical 
system.

IEEE 754 includes infinity as an actual value, contrary to the 
usual definition of real numbers in mathematics. However, it also 
distinguishes between +∞ and -∞, so it can't model the Riemann 
sphere, which is one of the most straightforward ways to perform 
the extension of the complex plane with a concept infinity in 
mathematics.

David