Re: More complex numbers than reals?

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.