# Complex versus real numbers

Started by August 25, 2009
```"brent" <bulegoge@columbus.rr.com> wrote in message
> a complex exponential is like an atom and a sinusoid is like a
> molecule.  It takes two atoms (complex exponentials) to make a
> molecule (sinusoid).  It is better to work with and understand the
> atoms (complex exponentials) than to only work with molecules
> (sinusoids).

That's a very good way of thinking about it and with your
permission, I'd suggest a slight variation.

Use atoms for the sinusoids and electrons and nuclei for the
complex numbers, because atoms are something real that we can
hold in our hand; whilst the internals of atoms tend not to be.

It suggests the concept that inside a real number is a collection
of complex numbers.

Which is reminiscent of, ...

"Inside every fat girl there is a thin girl trying to get out"

(Which has a corollary, ...

"Outside every thin girl there is a fat man trying to get in")

And something we learnt at school, ...

"Two little atoms,
Walking home from school,
Bumped into each other,

```
```Gary <invalid@invalid.invalid> wrote:
< "brent" <bulegoge@columbus.rr.com> wrote in message
<> a complex exponential is like an atom and a sinusoid is like a
<> molecule.  It takes two atoms (complex exponentials) to make a
<> molecule (sinusoid).  It is better to work with and understand the
<> atoms (complex exponentials) than to only work with molecules
<> (sinusoids).

< That's a very good way of thinking about it and with your
< permission, I'd suggest a slight variation.

< Use atoms for the sinusoids and electrons and nuclei for the
< complex numbers, because atoms are something real that we can
< hold in our hand; whilst the internals of atoms tend not to be.

< It suggests the concept that inside a real number is a collection
< of complex numbers.

I am not so convinced.

In optics, for example, the sine/cosine basis (linear polarization)
is as valid as the circular polarizing basis.  It happens that
for many physical cases one basis works better, but overall they
are just two different orthogonal basis sets.

There are some nice demonstrations of quantum mechanics that
can be done with a few sheets of polaroid filters.

There are also many cases in physics where the states we are
given are not the eigenstates of the system.  For the neutral
Kaon (K meson), there are two forms that enter into reactions,
the K0 and anti-K0 (or K0 with a bar over it).  The eigenstates
are (K0+K0bar)/sqrt(2) and (K0-K0bar)/sqrt(2).  But which one
is the sine/cosine basis and which the exp(ikx)/exp(-ikx) basis?

-- glen
```
```"glen herrmannsfeldt" <gah@ugcs.caltech.edu> wrote in message
news:h75iea\$s5r\$1@naig.caltech.edu...
>
> There are also many cases in physics where the states we are
> given are not the eigenstates of the system.  For the neutral
> Kaon (K meson), there are two forms that enter into reactions,
> the K0 and anti-K0 (or K0 with a bar over it).  The eigenstates
> are (K0+K0bar)/sqrt(2) and (K0-K0bar)/sqrt(2).  But which one
> is the sine/cosine basis and which the exp(ikx)/exp(-ikx) basis?

I'm sorry, but I haven't a clue what you're talking about. But it does
seem to be some form of advanced physics; advanced to the point
well past that at which some novice is attempting to understand the
need or convenience of complex numbers.

```
```Gary <invalid@invalid.invalid> wrote:
> "glen herrmannsfeldt" <gah@ugcs.caltech.edu> wrote in message
> news:h75iea\$s5r\$1@naig.caltech.edu...

>> There are also many cases in physics where the states we are
>> given are not the eigenstates of the system.  For the neutral
>> Kaon (K meson), there are two forms that enter into reactions,
>> the K0 and anti-K0 (or K0 with a bar over it).  The eigenstates
>> are (K0+K0bar)/sqrt(2) and (K0-K0bar)/sqrt(2).  But which one
>> is the sine/cosine basis and which the exp(ikx)/exp(-ikx) basis?

> I'm sorry, but I haven't a clue what you're talking about. But it does
> seem to be some form of advanced physics; advanced to the point
> well past that at which some novice is attempting to understand the
> need or convenience of complex numbers.

At some level, it is advanced physics, but at another it is
just another example of a system that can be represented in
different sets of orthogonal basis functions.

Maybe you like the optical case better, where you can use
the (x,y) basis for linear polarization, the (x+y, x-y)/sqrt(2)
basis linearly polarized with the axis rotated 45 degrees, or the
(x+iy, x-iy)/sqrt(2) basis for right/left circular polarization.

The optical cases are exactly the same as those for radio
frequency or microwave polariation using wire antennas.

-- glen
```
```On Aug 27, 4:39=A0am, "Gary" <inva...@invalid.invalid> wrote:
> "brent" <buleg...@columbus.rr.com> wrote in message
>

>
> "Two little atoms,
> Walking home from school,
> Bumped into each other,

I like your thinking.  If we can somehow get some kind of pornographic
theme going with complex numbers mating together or something, maybe
we can get more people to understand them.

Well, come to think of it, we do use the word "conjugating" when
dealing with complex numbers, maybe we should change that to something
else.

Lets see: two complex numbers conjugate to form a real number.

How about: Two complex numbers  __ck around to form a real number?

That should get Chris Bore's clients attention.

```
```robert bristow-johnson wrote:
> On Aug 26, 10:58 pm, spop...@speedymail.org (Steve Pope) wrote:
>> To claim that DSP designers should not use complex numbers
>> in their algorithms is of course silly, but requiring that
>> they be separated into their real and imaginary parts
>> at some point in the design flow is not unreasonable.
>
> ultimately, that's what we do.  every piece of real equipment (like
> CPUs and DSPs and FPGAs) does arithmetic in reality on real numbers.
> ultimately our compact and concise equations involving complex
> quantities *is* separated into two sets, one equating the real parts
> on both sides of the = sign and the other equating the imaginary
> parts.
>
> i have still yet to apply a voltmeter to a voltage and measure 3 + j*4
> volts.
>
> scaling by a constant, the derivative (w.r.t. time), and the delay
> operation (w.r.t. time) applied to either sinusoids or exponentials
> will result in other sinusoids or exponentials respectively (with the
> same omega or alpha, respectively).  but we know it's more squirrelly
> with sinusoids.  exponentials make for very clean eigenfunctions with
> operations such as scaling, differentiation, and shifting (delay).
> fortunately Euler saved our sorry little asses with his most important
> formula that says that sinusoids *are* exponentials making
> differentiation, and shifting all come out to be equivalent to
> scaling.  hurray!!  but we can't cheer in the language of only real
> numbers.
>
> complex numbers and variable are not synonymous with complicated
> numbers and variables.

Some people seem to believe that mathematics is reality and the things
we touch are only Plato's shadows on the cave wall. Sometimes they are
the same people who ask how a computer distinguishes between text and
numbers if it's all only bits.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```Jerry Avins <jya@ieee.org> writes:
> [...]
> Some people seem to believe that mathematics is reality and the things
> we touch are only Plato's shadows on the cave wall.

Define "reality"...

No, really - try it. I have and never reached a final verdict.
--
Randy Yates                      % "With time with what you've learned,
Digital Signal Labs              %  they'll kiss the ground you walk
mailto://yates@ieee.org          %  upon."
http://www.digitalsignallabs.com % '21st Century Man', *Time*, ELO
```
```Randy Yates wrote:
> Jerry Avins <jya@ieee.org> writes:
>> [...]
>> Some people seem to believe that mathematics is reality and the things
>> we touch are only Plato's shadows on the cave wall.
>
> Define "reality"...
>
> No, really - try it. I have and never reached a final verdict.

I can't define it either. If we replace it here with "things that can be
seen or weighed", then there's nothing to say. Thanks for puzzling me.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```My first reply seems to have vanished.

brent wrote:
> On Aug 26, 11:15 pm, Jerry Avins <j...@ieee.org> wrote:
>> brent wrote:
>>
>>    ...
>>
>>> the only way to get repetitive motion explained mathematically is by
>>> using complex numbers.
>> Oh come now! Consider the elliptic repetitive motion defined by the
>> parametric equations
>>
>>   For 0<t<infinity; y = sin(at) and
>>   x = ((1 + sqrt(5))/2) * cos(t) (I think the golden mean is pretty.)
>>
>> Jerry
>> --
>> Engineering is the art of making what you want from things you can get.
>> &#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
> Well, you are employing circular reasoning ( ha).
>
> By employing cosines and sines you have employed complex numbers.

Do agree that a real number is a complex number with the imaginary part
equal to zero? Then I use complex numbers when I count my toes.

> a complex exponential is like an atom and a sinusoid is like a
> molecule.  It takes two atoms (complex exponentials) to make a
> molecule (sinusoid).  It is better to work with and understand the
> atoms (complex exponentials) than to only work with molecules
> (sinusoids).

You could as well claim that the sinusoids are atomic and the complex
exponentials are composite. Given sin(x) and cos(x),
exp(ix)=cos(x)+i*sin(x). Where is the imaginary part of
1 + x - x^3/3! + x^5/5! - x^7/7! + ... = sin(x) ?

> So If I say you cannot make steam without hydrogen and oxygen you
> don't get credit by saying that you can do it with water :-).

Try to make steam with hydrogen and oxygen and you'll probably make an
explosion instead. First make the water. Slowly.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```"Jerry Avins" <jya@ieee.org> wrote in message
>
> Do agree that a real number is a complex number with the imaginary part
> equal to zero? Then I use complex numbers when I count my toes.

I hesitate to use the term, "imaginary", for it has other connotations
as already discussed in this thread, but even the so-called "real" numbers
are just a figment of the imagination.

Ask someone to show you two pencils, then two apples, and then ask
them to show you "two". If they offer, "2" or "II", point out to them that
they are indeed symbols that represent two, but not two itself.

Two does not exist any more than does i*2 (or j*2 if you're electrical).

Real numbers are as imaginary as are imaginary numbers.

This is the standard response I give when tutoring those who claim that
they can never understand mathematics because it is abstract by pointing
out that they already understand the abstraction that is mathematics.

```