Re: More complex numbers than reals?

On 7/9/2024 1:24 AM, Chris M. Thomasson wrote:

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?

Not stupid, but also not correct.
Galileo asked a similar question.
https://en.wikipedia.org/wiki/Galileo%27s_paradox
The natural numbers can be mapped 1.to.1 to
  the natural.number squares.
  In an obvious way.
There is room for all the numbers in the squares.
Thus, there are at least as many squares as numbers.
There aren't more squares than numbers; they fit.
But there are "more" numbers than squares.
  Also obviously.
Galileo's resolution was to stop asking that question.
Less obviously,
there are 1.to.1 maps from the complex plane
to the real line,
and not more plane than line.
And also "more" plane than line.
Here's a handwave of one way to do that.
I'm sure there are loose threads dangling,
but I'm also sure they can be tucked in.
Define the stretch.operator 🗚x which interleaves 0s
with the decimal digits of real number x
For x = x₀.x₁x₂x₃...
🗚x = x₀.0x₁0x₂0x₃0...
x+iy ⟼ 10⋅🗚x+🗚y = x₀y₀.x₁y₁x₂y₂x₃y₃...
There isn't "more" plane than line. Not.obviously.