On 20/07/2013 12:02 AM, Kevin Wampler wrote:
> I'm trying to think of an answer which won't make you regret asking that
> question :)

In that case I'll leave it 'till the morning to read it. As I said 
earlier. "I've had a long day" and I've crossed a timezone. The wrong 
way. :-(


-- 
Regards
     Stephen

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On 20/07/2013 12:07 AM, Nekar Xenos wrote:
> On Sat, 20 Jul 2013 00:02:11 +0200, Kevin Wampler <nob### [at] nowherenet>
> wrote:
>
>> On 7/19/2013 2:32 PM, Stephen wrote:
>
>> 2*w = "count in forever in multiple of two"
>> w*2 = "count forever, then when you're done, do it again."
>>
>
> I see a light bulb! ;)
>
>

Eureka! :-D

But from what you quoted.
Don’t they have a BODMAS? (well that is what I was taught the order of 
operations, was called). And the meanings.


-- 
Regards
     Stephen

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On 7/19/2013 2:38 PM, Nekar Xenos wrote:
> 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)

What you mean be "density" is a whole 'nother issue which is 
surprisingly complex when you get into it.  When talking about questions 
like "Infinity+Infinity" it's best not to worry about things like "density".

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

That's basically correct.  More precisely, for the definition of 
infinity which turns out to be most useful for the reals 
Infinity+Infinity=Infinity.  For the definition of Infinity which turns 
out to be the most useful for the complex numbers Infinity+Infinity is 
undefined.

People can (and occasionally do) use different definitions of Infinity 
for the reals/complexes than these, depending on what's useful for the 
problem they want to solve.  But you've got the right idea for the most 
common choices.

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On 7/19/2013 3:07 PM, Nekar Xenos wrote:
> On Sat, 20 Jul 2013 00:02:11 +0200, Kevin Wampler <nob### [at] nowherenet>
> wrote:
>
>> On 7/19/2013 2:32 PM, Stephen wrote:
>
>> 2*w = "count in forever in multiple of two"
>> w*2 = "count forever, then when you're done, do it again."
>>
>
> I see a light bulb! ;)
>


Excellent!  I should point out that this is an entirely different 
definition of "number" than Le_Forgeron/Orchid have been using.  They 
were treating numbers as reasoning about quantities, whereas which I 
wrote treats numbers as reasoning about sequences/orders.  Both are 
equally valid, but they give different answers to your question.

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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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On 7/19/2013 4:06 PM, clipka wrote:
>
> 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>
>

This is indeed what I meant by "complex infinity".  As far as I'm aware, 
it's not decomposable into the form a+bi where "a" and "b" are real, 
since doing so would require a "real infinity" which is distinct from 
the complex infinity (I mean, you probably *could* define such a thing, 
but I've never seen it done).

Good catch that Nekar was using "complex infinity" in a different way 
than I was thinking he was.

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On 19/07/2013 11:02 PM, Kevin Wampler wrote:
> So while counting forever by multiples of two is basically the same as
> counting forever normally, counting forever and then doing it again is
> another thing entirely.
>
> Provided you're willing to accept that any of this makes sense at all,
> that's what the difference is (for that definition of "number" of
> course, which surprisingly is a relatively common one).

That all made sense and no new ideas. Knowing the syntax and putting it 
in context, helps. Thanks.

It was well explained, so do you lecture in mathematics or is it just 
part of your job?

-- 
Regards
     Stephen

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On 7/20/2013 3:34 AM, Stephen wrote:
>
> It was well explained, so do you lecture in mathematics or is it just
> part of your job?
>

Thanks!  I'm glad to hear that you liked the explanation.  I don't 
lecture in mathematics or use this sort of math for my job.  I just 
enjoy mathematics and have studied various bits of it more or less for fun.

I do use a lot of math in my job though, but it's just applied math and 
not set theory stuff like this.

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Nekar Xenos <nek### [at] gmailcom> wrote:
> I was thinking of adding any particular infinity to itself.

In a sense, adding infinity to itself doesn't make any sense because
infinity is not a number. It's a concept used to describe an abstraction.

There are certain situations where you can use mathematical operators
on infinities as a kind of special notation.

Calculating limits is one situation where infinity-as-a-notation can be
useful.

For example, if lim(x->a) f(x) is infinity, and lim(x->a) g(x) is
infinity, then you can use the notation that lim(x->a) f(x)+g(x) =
infinity + infinity = infinity. In other words, since both f(x) and
g(x) approach infinity when x approaches a, you know that their sum
will also do so.

This doesn't mean that "infinity" is some kind of "special number" that
you can add or do special operations. It's not a number. As said, this
is just a notation you can use to make such calculations easier.

The same is true for multiplication. lim(x->a) f(x)*g(x) is also infinity.

This is also the reason why it's said that eg. "infinity - infinity" is
indetermined. It basically means that the notation system fails and
cannot be used to determine the correct result.

In other words, if it were lim(x->a) f(x)-g(x), the result depends on
what f(x) and g(x) are. You cannot know the answer only from knowing
that their individual limits go to infinity.

-- 
                                                          - Warp

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Nekar Xenos <nek### [at] gmailcom> wrote:
> 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)

You can uniquely map each complex number to each real number on a
one-to-one basis, which means that the set of complex numbers is as
big as the set of reals.

Likewise you can map each rational number to each natural number on
a one-to-one basis, which means that the set of rational numbers is as
big as the set of natural numbers. (There's actually a rather easy way
to do this enumeration. See if you can figure it out.)

Perhaps a bit surprisingly, there is no one-to-one mapping between
rational numbers and real numbers, which means that the set of real
numbers is larger than the set of rational numbers.

This last thing is quite curious given that, for example, between any
two real numbers there is an infinite amount of rational numbers, and
vice-versa. It feels like there should be the same amount...

-- 
                                                          - Warp

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