On Thu, 28 Aug 2014 20:21:08 -0400, Randy Yates wrote:

> Tim Wescott <seemywebsite@myfooter.really> writes:
>> [...]
>> Your confusion confuses me.  Rather, I'm not quite sure where your
>> confusion lies.
>>
>> Quaternions are, as close as you can come, the next higher dimensional
>> "complex numbers".  They have a complete algebra that is more than 4-
>> vectors.  They're like complex numbers in that they have imaginary
>> parts, but unlike complex numbers in that they have three imaginary
>> parts.  You can add them, subtract them, and multiply them. 
>> Multiplication is not commutative -- a * b does not necessarily equal b
>> * a, in quaternion-land.
>>
>> They're exceedingly useful as a compact way of representing rotations
>> is 3D space, however there are so many rules and exceptions that you
>> have to remember that they'll make your brain twitch.
> 
> I did a little reading on quaternions and division rings. What a rich
> set of mathematics this algebra! I surely wish I had the time and money
> to study this area of mathematics intead of writing my umpteenth
> thousandth line of C/C++ code...

Rich and weird.

Actually, if you do an "ordinary" vector dot or cross product, you're just 
doing part of a quaternion multiply: if a and b are quaternions 
representing positions in 3D space (meaning their real parts are zero), 
then the real part of a * b is the negative of the dot product of the 
corresponding 3D vectors, and the imaginary part of a * b is either the 
vector cross product or the negative thereof.

Basically, people extracted an easier to work with subset out of 
quaternions, called it 3D vectors, and proceeded to do useful (and easier 
to understand) mathematics with it.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Tim Wescott <seemywebsite@myfooter.really> writes:
> [...]
> Your confusion confuses me.  Rather, I'm not quite sure where your 
> confusion lies.
>
> Quaternions are, as close as you can come, the next higher dimensional 
> "complex numbers".  They have a complete algebra that is more than 4-
> vectors.  They're like complex numbers in that they have imaginary parts, 
> but unlike complex numbers in that they have three imaginary parts.  You 
> can add them, subtract them, and multiply them.  Multiplication is not 
> commutative -- a * b does not necessarily equal b * a, in quaternion-land.
>
> They're exceedingly useful as a compact way of representing rotations is 
> 3D space, however there are so many rules and exceptions that you have to 
> remember that they'll make your brain twitch.

I did a little reading on quaternions and division rings. What a rich
set of mathematics this algebra! I surely wish I had the time and money
to study this area of mathematics intead of writing my umpteenth
thousandth line of C/C++ code...
-- 
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com

On Thu, 28 Aug 2014 18:04:40 -0400, robert bristow-johnson wrote:

> On 8/28/14 1:55 PM, Tim Wescott wrote:
>> On Thu, 28 Aug 2014 12:24:56 -0400, robert bristow-johnson wrote:
>>
>>> On 8/27/14 5:07 AM, gyansorova@gmail.com wrote:
>>>> On Wednesday, August 27, 2014 5:19:13 AM UTC+12, mako...@yahoo.com
>>>> wrote:
>>>>>
>>>>> regarding the physicality of complex numbers....  if you replace the
>>>>> word  "complex" with "2 dimensional"  then it doesn't seem so
>>>>> mysterious..
>>>>>
>>>>>
>>> but complex numbers are *not* merely ordered pairs of real numbers.
>>> there is more to them than that.
>>>
>>>>> In fact I do have an oscope that will display a "complex" or 2
>>>>> dimensional signal,, I simply connect the 2 wires to ch1 and ch2 and
>>>>> put the ocsope in XY mode.
>>>
>>> same issue.
>>>
>>>
>>>> Why stop at 2? Why not use quaternions
>>>>
>>>> http://en.wikipedia.org/wiki/Quaternion
>>>
>>> hey, why not N-dimensional "complex numbers".  better define how
>>> operations like multiplication get done.  and, if one of those 4
>>> dimensions is the "real axis", then the rules of those operations
>>> should default to the existing real operations when the other 3
>>> dimensions have a value of 0.
>>
>> I'm not sure what you're trying to get at, Robert.  You seem to start
>> by asking why not an arbitrary number of dimensions for "complex
>> numbers", then you finish off the sentence talking about "those four
>> dimensions".
>>
>> At any rate, as to your stipulation for the real part, real numbers,
>> complex numbers, quaternions and octonions all follow that rule.
>>
>> As for N-dimensional "complex numbers", if you stipulate that they must
>> be a certain kind of algebra or group (I can't remember which -- I
>> think they must be a Lie algebra), then there are exactly four kinds
>> possible: reals,
>> complex, quaternions, and octonions.  Hamilton, in fact, was stuck for
>> years trying to extend the notion of complex numbers to three
>> dimensions, and only got unstuck when he realized that he must
>> represent the operations with four components, not three.
>>
>>
> Tim, i dunno much about quaternions (like what they're good for), but my
> main issue is that while a complex number is fully described as a pair
> of real numbers, the set of complex numbers and its algebra are more
> than that of 2-vectors.  C is not the same as R^2 .

Your confusion confuses me.  Rather, I'm not quite sure where your 
confusion lies.

Quaternions are, as close as you can come, the next higher dimensional 
"complex numbers".  They have a complete algebra that is more than 4-
vectors.  They're like complex numbers in that they have imaginary parts, 
but unlike complex numbers in that they have three imaginary parts.  You 
can add them, subtract them, and multiply them.  Multiplication is not 
commutative -- a * b does not necessarily equal b * a, in quaternion-land.

They're exceedingly useful as a compact way of representing rotations is 
3D space, however there are so many rules and exceptions that you have to 
remember that they'll make your brain twitch.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

On 8/28/14 1:55 PM, Tim Wescott wrote:
> On Thu, 28 Aug 2014 12:24:56 -0400, robert bristow-johnson wrote:
>
>> On 8/27/14 5:07 AM, gyansorova@gmail.com wrote:
>>> On Wednesday, August 27, 2014 5:19:13 AM UTC+12, mako...@yahoo.com
>>> wrote:
>>>>
>>>> regarding the physicality of complex numbers....  if you replace the
>>>> word  "complex" with "2 dimensional"  then it doesn't seem so
>>>> mysterious..
>>>>
>>>>
>> but complex numbers are *not* merely ordered pairs of real numbers.
>> there is more to them than that.
>>
>>>> In fact I do have an oscope that will display a "complex" or 2
>>>> dimensional signal,, I simply connect the 2 wires to ch1 and ch2 and
>>>> put the ocsope in XY mode.
>>
>> same issue.
>>
>>
>>> Why stop at 2? Why not use quaternions
>>>
>>> http://en.wikipedia.org/wiki/Quaternion
>>
>> hey, why not N-dimensional "complex numbers".  better define how
>> operations like multiplication get done.  and, if one of those 4
>> dimensions is the "real axis", then the rules of those operations should
>> default to the existing real operations when the other 3 dimensions have
>> a value of 0.
>
> I'm not sure what you're trying to get at, Robert.  You seem to start by
> asking why not an arbitrary number of dimensions for "complex numbers",
> then you finish off the sentence talking about "those four dimensions".
>
> At any rate, as to your stipulation for the real part, real numbers,
> complex numbers, quaternions and octonions all follow that rule.
>
> As for N-dimensional "complex numbers", if you stipulate that they must be
> a certain kind of algebra or group (I can't remember which -- I think they
> must be a Lie algebra), then there are exactly four kinds possible: reals,
> complex, quaternions, and octonions.  Hamilton, in fact, was stuck for
> years trying to extend the notion of complex numbers to three dimensions,
> and only got unstuck when he realized that he must represent the
> operations with four components, not three.
>

Tim, i dunno much about quaternions (like what they're good for), but my 
main issue is that while a complex number is fully described as a pair 
of real numbers, the set of complex numbers and its algebra are more 
than that of 2-vectors.  C is not the same as R^2 .


-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

On Thu, 28 Aug 2014 12:24:56 -0400, robert bristow-johnson wrote:

> On 8/27/14 5:07 AM, gyansorova@gmail.com wrote:
>> On Wednesday, August 27, 2014 5:19:13 AM UTC+12, mako...@yahoo.com
>> wrote:
>>>
>>> regarding the physicality of complex numbers....  if you replace the
>>> word  "complex" with "2 dimensional"  then it doesn't seem so
>>> mysterious..
>>>
>>>
> but complex numbers are *not* merely ordered pairs of real numbers.
> there is more to them than that.
> 
>>> In fact I do have an oscope that will display a "complex" or 2
>>> dimensional signal,, I simply connect the 2 wires to ch1 and ch2 and
>>> put the ocsope in XY mode.
> 
> same issue.
> 
> 
>> Why stop at 2? Why not use quaternions
>>
>> http://en.wikipedia.org/wiki/Quaternion
> 
> hey, why not N-dimensional "complex numbers".  better define how
> operations like multiplication get done.  and, if one of those 4
> dimensions is the "real axis", then the rules of those operations should
> default to the existing real operations when the other 3 dimensions have
> a value of 0.

I'm not sure what you're trying to get at, Robert.  You seem to start by 
asking why not an arbitrary number of dimensions for "complex numbers", 
then you finish off the sentence talking about "those four dimensions".

At any rate, as to your stipulation for the real part, real numbers, 
complex numbers, quaternions and octonions all follow that rule.

As for N-dimensional "complex numbers", if you stipulate that they must be 
a certain kind of algebra or group (I can't remember which -- I think they 
must be a Lie algebra), then there are exactly four kinds possible: reals, 
complex, quaternions, and octonions.  Hamilton, in fact, was stuck for 
years trying to extend the notion of complex numbers to three dimensions, 
and only got unstuck when he realized that he must represent the 
operations with four components, not three.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

On 8/27/14 5:07 AM, gyansorova@gmail.com wrote:
> On Wednesday, August 27, 2014 5:19:13 AM UTC+12, mako...@yahoo.com wrote:
>>
>> regarding the physicality of complex numbers....  if you replace the word  "complex" with "2 dimensional"  then it doesn't seem so mysterious..
>>

but complex numbers are *not* merely ordered pairs of real numbers. 
there is more to them than that.

>> In fact I do have an oscope that will display a "complex" or 2 dimensional signal,, I simply connect the 2 wires to ch1 and ch2 and put the ocsope in XY mode.

same issue.

>
> Why stop at 2? Why not use quaternions
>
> http://en.wikipedia.org/wiki/Quaternion

hey, why not N-dimensional "complex numbers".  better define how 
operations like multiplication get done.  and, if one of those 4 
dimensions is the "real axis", then the rules of those operations should 
default to the existing real operations when the other 3 dimensions have 
a value of 0.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

On Wednesday, August 27, 2014 5:19:13 AM UTC+12, mako...@yahoo.com wrote:
> On Tuesday, August 26, 2014 12:32:12 PM UTC-4, Rob Gaddi wrote:
> 
> > On Mon, 25 Aug 2014 19:53:21 -0700
> 
> > 
> 
> > Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:
> 
> > 
> 
> > 
> 
> > 
> 
> > > On Mon, 25 Aug 2014 20:43:20 +0000 (UTC), glen herrmannsfeldt
> 
> > 
> 
> > > <gah@ugcs.caltech.edu> wrote:
> 
> > 
> 
> > > 
> 
> > 
> 
> > >    [Snipped by Lyons]
> 
> > 
> 
> > > >
> 
> > 
> 
> > > >When you touch something, it is the wave function of the electrons
> 
> > 
> 
> > > >in your finger interacting with the electrons in the object you
> 
> > 
> 
> > > >are touching.  Pauli exclusion keeps them from overlapping.
> 
> > 
> 
> > > >
> 
> > 
> 
> > > >Who believes that wave functions are complex, and who that it should
> 
> > 
> 
> > > >be two separate real functions?  Remember, unlike phasors and voltages,
> 
> > 
> 
> > > >it is the absolute square of the wave function that gives the
> 
> > 
> 
> > > >probability density.
> 
> > 
> 
> > > >
> 
> > 
> 
> > > >-- glen
> 
> > 
> 
> > > 
> 
> > 
> 
> > > Hi glen,
> 
> > 
> 
> > >    It's sounds like you may be a physicist, or at 
> 
> > 
> 
> > > least have a strong background in physics.
> 
> > 
> 
> > > 
> 
> > 
> 
> > > If so, I have a seemingly super-silly question 
> 
> > 
> 
> > > for you.  Do modern physicists know what is 
> 
> > 
> 
> > > an electron?
> 
> > 
> 
> > > 
> 
> > 
> 
> > > For example, if I tell my brother that electrons 
> 
> > 
> 
> > > from his car battery flow through the starter motor 
> 
> > 
> 
> > > and make the car engine start turning, and if he 
> 
> > 
> 
> > > asks me, "What are electrons?", what should I say?
> 
> > 
> 
> > > 
> 
> > 
> 
> > > Thanks,
> 
> > 
> 
> > > [-Rick-]
> 
> > 
> 
> > > 
> 
> > 
> 
> > 
> 
> > 
> 
> > Little pieces of electricity.  That is a statement that, while being
> 
> > 
> 
> > almost entirely wrong, conveys everything that you actually wanted to
> 
> > 
> 
> > say.
> 
> > 
> 
> > 
> 
> regarding the physicality of complex numbers....  if you replace the word  "complex" with "2 dimensional"  then it doesn't seem so mysterious..
> 
> 
> 
> In fact I do have an oscope that will display a "complex" or 2 dimensional signal,, I simply connect the 2 wires to ch1 and ch2 and put the ocsope in XY mode.
> 
> 
> 
> Mark

Why stop at 2? Why not use quaternions

http://en.wikipedia.org/wiki/Quaternion

Rick Lyons <R.Lyons@_bogus_ieee.org> wrote:

(snip)

>  thanks a lot for your detailed and cogent 
> explanation.  It sounds like a meaningful 
> description of an electron depends on what 
> you're doing with that electron.

> I was afraid you'd reply with, "Rick.  Electrons 
> are like women.  Nobody understands them."

When I was in grad. school, we had weekly group seminars. 

There was one subject that we were specifically not allowed
to talk about, as my advisor knew we didn't understand.

-- glen

On Tue, 26 Aug 2014 10:19:13 -0700 (PDT), makolber@yahoo.com wrote:

   [Snipped by Lyons]

>regarding the physicality of complex numbers....  if you replace the word  "complex" with "2 dimensional"  then it doesn't seem so mysterious..
>
>In fact I do have an oscope that will display a "complex" or 2 dimensional signal,, I simply connect the 2 wires to ch1 and ch2 and put the ocsope in XY mode.
>
>Mark

Hi Mark,
  right you are.  I've used those exact ideas 
in my writing about quadrature (I/Q) processing.

The great mathematician and electrical engineer 
Charles Steinmetz, who pioneered electrical power 
generation and distribution for General Electric 
(GE) in the late 1800s, did NOT like the term "complex 
numbers."  He preferred to call them "general numbers."

[-Rick-]