More complex numbers than reals?

[Chris M. Thomasson]

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?

[Ben Bacarisse]

Post by Chris M. Thomasson
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?

[James Kuyper]

Post by Ben Bacarisse

Post by Chris M. Thomasson
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?

I haven't seen more of Chris's message than what you've quoted. In the
context of C, it's a easy (even trivial) question to answer.

"Each complex type has the same representation and alignment
requirements as an array type containing exactly two elements of the
corresponding real type; the first element is equal to the real part,
and the second element to the imaginary part, of the complex number."
(6.2.5p17).

Therefore, the number of different complex numbers that can be
represented is therefore the square of the number of different numbers
that can be represented in the corresponding real type.

The corresponding question suitable for sci.math is much trickier to answer.

[Tim Rentsch]

Post by Chris M. Thomasson
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?

I haven't seen more of Chris's message than what you've quoted. In the
context of C, it's a easy (even trivial) question to answer.
"Each complex type has the same representation and alignment
requirements as an array type containing exactly two elements of the
corresponding real type; the first element is equal to the real part,
and the second element to the imaginary part, of the complex number."
(6.2.5p17).
Therefore, the number of different complex numbers that can be
represented is therefore the square of the number of different numbers
that can be represented in the corresponding real type.

The answer is still no, because the question is about complex
numbers and real numbers, not representable values.

[Kaz Kylheku]

Post by Ben Bacarisse

Post by Chris M. Thomasson
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?

It can be. There are clearly more complex doubles than
there are doubles.

[Chris M. Thomasson]

Post by Kaz Kylheku

Post by Ben Bacarisse

Post by Chris M. Thomasson
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?

It can be. There are clearly more complex doubles than
there are doubles.

Bingo! I think... :^)

[Ben Bacarisse]

Post by Kaz Kylheku

Post by Ben Bacarisse

Post by Chris M. Thomasson
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?

It can be. There are clearly more complex doubles than
there are doubles.

Oh come on. If the question was about C representations, why did Chris
put "more" in scare quotes? It's disingenuous to answer as if that was
the intent without first asking what Christ meant by "more" (rather than
just more).

[Richard Harnden]

On 09/07/2024 1

Post by Ben Bacarisse
put "more" in scare quotes? It's disingenuous to answer as if that was
the intent without first asking what Christ meant by "more" (rather than
just more).

Fish? Bread? Maybe Wine?

[Ben Bacarisse]

Post by Richard Harnden
On 09/07/2024 1

Post by Ben Bacarisse
put "more" in scare quotes? It's disingenuous to answer as if that was
the intent without first asking what Christ meant by "more" (rather than
just more).

Fish? Bread? Maybe Wine?

Ha! An unusual typo for me. Given the way I see words, I had to read
it quite a few times before I saw it, even with your comment.

[James Kuyper]

Post by Kaz Kylheku

Post by Chris M. Thomasson
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?

It can be. There are clearly more complex doubles than
there are doubles.

Oh come on. If the question was about C representations, why did Chris
put "more" in scare quotes? It's disingenuous to answer as if that was
the intent without first asking what Christ meant by "more" (rather than
just more).

It may be disingenuous, but deliberately misinterpreting his question as
if it had been one that would be on-topic in this newgroup is an
entirely appropriate way of reminding him that his question was off-topic.

[Chris M. Thomasson]

Post by Ben Bacarisse

Post by Chris M. Thomasson
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, ..., ..., .....

Damn it. Sorry!

[Kaz Kylheku]

Post by Chris M. Thomasson

Post by Ben Bacarisse

Post by Chris M. Thomasson
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.

[Chris M. Thomasson]

Post by Kaz Kylheku

Post by Chris M. Thomasson

Post by Ben Bacarisse

Post by Chris M. Thomasson
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.
a = 0 . a0 a1 a2 a3 a4 .... (a0 a1 ... are decimal digits of a)
b = 0 . b0 b1 b2 b3 b4 ....
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... :^)

[Lawrence D'Oliveiro]

Post by Chris M. Thomasson
By the way, did you take a look at my "fun"
experiment wrt storing data in the roots of complex numbers?

You can encode messages in anything.

#!/usr/bin/python3

import sys

charset = ' .Hadefghilmnoprstwx'
modulo = 23
s = 1193321429126088671017703197607273471738040746714878246039040663141374779777444615292185709614467374016367591074170680369683264762098941
num = iter(range(2, 9999))
while s != 1 :
n = next(num)
if s % n == 0 :
sys.stdout.write("%s" % charset[(n - 1) % modulo])
s //= n
#end if
#end while
sys.stdout.write("\n")

[Chris M. Thomasson]

Post by Lawrence D'Oliveiro

Post by Chris M. Thomasson
By the way, did you take a look at my "fun"
experiment wrt storing data in the roots of complex numbers?

You can encode messages in anything.

Encoding data in fractals is fun. Also, one can gain some pretty renders
along the way... ;^)

[...]

[Chris M. Thomasson]

Post by Lawrence D'Oliveiro

Post by Chris M. Thomasson
By the way, did you take a look at my "fun"
experiment wrt storing data in the roots of complex numbers?

You can encode messages in anything.

[...]

https://groups.google.com/g/comp.lang.c++/c/bB1wA4wvoFc/m/ozDpUBlTAAAJ

[Richard Damon]

Post by Chris M. Thomasson
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?

In comp.lang.c the answer is yes, for if we assume the standard 8 byte
floating point numbers, there are approximatly 2^64 possible reals
(slightly less because the encoding is not totally exhaustive) but
approximately 2^128 possible complex numbers.

[Lawrence D'Oliveiro]

Post by Chris M. Thomasson
Are there "more" complex numbers than reals?

If you talking about the theoretical mathematical sets, then the answer is
no. They can be mapped 1:1.

An obvious way to do it is to interleave the digits of the real and
imaginary parts to create a real.

[Kaz Kylheku]

Post by Chris M. Thomasson
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?

The argument is not that simple. If we restrict to just integer complex
numbers like 4 + 5i, then no; there aren't more of these than integers.
Yet the same argument about axes and embeddings could be wrongly applied.

Integer complex numbers are countable: you can start at 0, and then go
in a spiral fashion: 1, 1 + i, i, -1 + i -1, ... thus they can be put
into correspondendce with the natural numbers.

[Chris M. Thomasson]

Post by Kaz Kylheku

Post by Chris M. Thomasson
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?

The argument is not that simple. If we restrict to just integer complex
numbers like 4 + 5i, then no; there aren't more of these than integers.
Yet the same argument about axes and embeddings could be wrongly applied.
Integer complex numbers are countable: you can start at 0, and then go
in a spiral fashion: 1, 1 + i, i, -1 + i -1, ... thus they can be put
into correspondendce with the natural numbers.

This was meant for sci.math... ARHG!!!! Well, shit. Sorry... Hummm...
its fun to think about complex as vectors wrt addition, subtraction, but
multiplication is different, division, ect...

Actually, here is some of my c++ code that might be relevant here wrt
precision of complex numbers.... float, double, ect...

https://groups.google.com/g/comp.lang.c++/c/05XwgswUnDg/m/Wku2Oym_BQAJ

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

More info where complex numbers and their precision is important:

https://paulbourke.org/fractals/multijulia/
(read all!)

:^)

[Chris M. Thomasson]

Post by Kaz Kylheku

Post by Chris M. Thomasson
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?

The argument is not that simple. If we restrict to just integer complex
numbers like 4 + 5i, then no; there aren't more of these than integers.
Yet the same argument about axes and embeddings could be wrongly applied.
Integer complex numbers are countable: you can start at 0, and then go
in a spiral fashion: 1, 1 + i, i, -1 + i -1, ... thus they can be put
into correspondendce with the natural numbers.

OOPS! I meant this link:

https://groups.google.com/g/comp.lang.c++/c/bB1wA4wvoFc/m/ozDpUBlTAAAJ

[Lawrence D'Oliveiro]

Integer complex numbers ...

Another fun thing about integer complex numbers is that some numbers that
are primes if you stay on the real line, become non-prime if complex
integer factors are allowed.

E.g.

5 = (2 + i)(2 - i)
29 = (5 + 2i)(5 - 2i)
101 = (10 + i)(10 - i)

[Blue-Maned_Hawk]

Post by Chris M. Thomasson
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?

No. You could draw a Hilbert curve (or any other space-filling curve) on
a square of the complex plane and then tile it around in a spiral to fill
up the rest of the plane. Then, you connect up all the ends of those
tiles together and pull on the ends of the curve to stretch it out to form
a line that is as infinite as the real number line.