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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: comp.lang.c
Subject: Re: More complex numbers than reals?
Date: Tue, 9 Jul 2024 12:00:29 -0700
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On 7/9/2024 1:47 AM, Kaz Kylheku wrote:
> On 2024-07-09, Chris M. Thomasson <chris.m.thomasson.1@gmail.com> wrote:
>> On 7/8/2024 3:59 PM, Ben Bacarisse wrote:
>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>
>>>> Are there "more" complex numbers than reals?
>>>
>>> If you ask this in an appropriate group (sci.math?) I'll answer.  Can
>>> you really think this is topical in comp.lang.c?
>>>
>>
>> Ahhhh shit! this was meant for sci.math! Damn it! Cursing, ..., ..., .....
> 
> Anyway, a complex number is a + ib where a and b are real.
> 
> We can take any two reals (wlog, in the range [0, 1)):
> 
>    a = 0 . a0 a1 a2 a3 a4 ....  (a0 a1 ... are decimal digits of a)
> 
>    b = 0 . b0 b1 b2 b3 b4 ....
> 
> and intertwine the digits to make a new real number:
> 
>    c = 0. a0 b0 a1 b1 a2 b2 ...
> 
> That new number c is still among the reals.
> 
> The intertwining is undoable: you can recover the original pair
> of numbers by taking the even or odd digits.
> 
> Thus, any complex number can be encoded as a real number,
> which implies that there can't be more of them than reals.

Makes sense to me. By the way, did you take a look at my "fun" 
experiment wrt storing data in the roots of complex numbers?

https://groups.google.com/g/comp.lang.c++/c/05XwgswUnDg/m/s_RNcUHCBQAJ

The fun part is that we can store data in fractals... :^)