std.complex

[Joseph Rushton Wakeling]

On 26/11/13 22:11, David Nadlinger wrote:
> 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, …

Well, it has been 12+ years ... :-P  Still, it's very, very annoying when one 
misplaces fundamental stuff like that.  Thank you for reminding me :-)

Now I need to dust off my copy of "What is Mathematics?" ...

> 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.

I'm sure you've heard that old anecdote of the professor back in the 1950s, or 
was it the 1920s, who, on hearing a student say "infinity", said: "I won't have 
bad language in class!" :-)