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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: More complex numbers than reals?
Date: Tue, 9 Jul 2024 12:14:47 -0700
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On 7/9/2024 12:08 PM, Chris M. Thomasson wrote:
> On 7/9/2024 3:02 AM, FromTheRafters wrote:
>> FromTheRafters pretended :
>>> Chris M. Thomasson explained on 7/9/2024 :
>>>> Are there "more" complex numbers than reals? It seems so, every real 
>>>> has its y, or imaginary, component set to zero. Therefore for each 
>>>> real there is an infinity of infinite embedding's for it wrt any 
>>>> real with a non-zero y axis? Fair enough, or really dumb? A little 
>>>> stupid? What do you think?
>>
>> Corrected.
>>
>>> In a sense there are 'more' since the reals are all on the x axis 
>>> line whereas the 2D R x R space is filled with complex numbers. R is 
>>> contained in C. In another sense they are the same size set, C being 
>>> basically R by R in the same sense as Q being Z by Z).
>>>
>>> Are there any other sizes of sets between countable Q and uncountable 
>>> R? How about between uncountable R and uncountable C?
> 
> Seems to boil down to:
> 
> Is uncountable infinity the same "size", as any other uncountable 
> infinity? Say reals vs. complex numbers...

This is where I like to ponder on so-called, density of infinity. The 
reals are "denser" than the naturals... Agreed? The complex numbers seem 
denser than the reals. Oh well, that is in my mind for some damn reason.

:^)