Am 19.07.2013 23:38, schrieb Nekar Xenos:

>> As an aside, a notion of "complex Infinity" is actually extremely
>> useful in some areas mathematics.  Arguably much more useful than
>> "real Infinity" is.  The standard definition of complex Infinity does
>> not allow Infinity+Infinity though (it treats it as undefined, much
>> like 1/0 is commonly treated as undefined for the reals).
>>
> I imagined the complex set would have a larger "density" if you could
> call it that. (Real numbers would have a larger "density" than integers)

No, the set of complex numbers is just "as infinite" as the set of real 
numbers; you can use a digit-interleaving technique similar to that 
which gets you from the natural numbers to all of the integers.

> So if I have this right
> Infinity + Infinity =  Infinity
> is correct for real numbers and not for complex numbers.

If you mean

   Infinity := (number of elements in the set of complex numbers)

then no, Infinity + Infinity = Infinity still holds there.

I guess what he meant was that some

   ComplexInfinity := an infinite that behaves much like a
                      complex number, and possibly even has
                      a representation of the form (a + bi)
                      for some presumably weird choice of a,b

Gives you

   ComplexInfinity + ComplexInfinity = <undefined>

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