Thinking about complex numbers

Greetings Earthlings,

   It's been a bit quiet here on comp.dsp for the 
last few days, so I thought I'd post something 
with the sole purpose of rattling your cages.

Regarding the complex-valued j-operator, where 
j = sqrt(-1), I was thinking about the interesting 
notion that j raised to the j-th power, j^j, is a 
real-valued number.

To increase my limited knowledge of complex numbers,
I wondered about other examples of raising a number 
to the j-th power.  Isn't it interesting that:

     0.001867442731708^j = 1^j = 1

[-Rick-]

On Saturday, January 31, 2015 at 7:44:08 PM UTC-6, Rick Lyons wrote:

> To increase my limited knowledge of complex numbers,
> I wondered about other examples of raising a number 
> to the j-th power.  Isn't it interesting that:
> 
>      0.001867442731708^j = 1^j = 1

I had never thought about it before, so I found some good info here: http://www.milefoot.com/math/complex/exponentofi.htm

A complex number raised to a complex exponent turns out to be periodic. That's not really surprising, given the periodicity in the special case of Euler's Identity.

Rick Lyons <R.Lyons@_bogus_ieee.org> wrote:
 
>   It's been a bit quiet here on comp.dsp for the 
> last few days, so I thought I'd post something 
> with the sole purpose of rattling your cages.
 
> Regarding the complex-valued j-operator, where 
> j = sqrt(-1), I was thinking about the interesting 
> notion that j raised to the j-th power, j^j, is a 
> real-valued number.

When I was in high school, I was trying out some PL/I
programs using complex numbers.  When Fortran only
had (complex)**(integer), PL/I had (complex)**(complex),
so I wrote a program to read in two numbers and write
out X**Y. 

My first try was 1I,1I and I was surprised to see a
real number.  Then I tried ones like 2I,1I and 1I,2I
finding that they were not real. 

But my favorite explanation is in "Feynman: Lectures on Physics".

In one chapter, he explains the origin of logarithms, starting
with 10 successive square roots of 10. (The is, 10**0.5, 10**0.25,
10**0.125, ..., 10**(1/1024).

Then with some multiplication, you can fill in the table, such
as 10**0.75 and 10**0.375, and finally learn that linear
interpolation is plenty close enough. 

Once you have a table of powers of 10, invert the table to
make a table of logs base 10, again with interpolation.
Also, he notes that for small x, 10**x is 1+2.3052x
(five digits might  have been about right for early log tables).

Now he makes the not so obvious, but reasonable assumption, that
10**x is still about 1+2.3052x for small complex x, such as i/1024,
so 10**(i/1024) is 1+2.3052i/1024. 

Next, using ordinary complex multiplication he computes powers
of 10**(i/1024), to generate a graph of 10**ix as a function of x,
both real and imaginary parts.

Finally, noticing that they are sinusoidal, but with periods 
not 2*pi, he calculates the base raised to the i*x power that
generates sinusoids with period 2*pi, and finds e.

By the way, anyone notice how many calculators that claim to
do complex arithmetic won't do things like log(-1) but
only ln(-1).  Also, won't do asin(2) or sin(i).

-- glen

On 01.02.2015 02:43, Rick Lyons wrote:

> Regarding the complex-valued j-operator, where
> j = sqrt(-1), I was thinking about the interesting
> notion that j raised to the j-th power, j^j, is a
> real-valued number.

Actually, if you say only that, I would probably respond that this isn't 
quite true. Unlike addition or multiplication, the power function is not 
a well-defined function on the complex-plane. It is a multi-valued 
function (actually, this is a contradiction in terms), or rather, it is 
defined on some multi-sheeded overlay of the complex plane with itself 
that depends on the basis. So for example,

z^{1/2}

can be either be defined as a discontinuous singly-valued function, or 
as a function on the double-overlay of C with itself where it is 
analytic outside 0. (aka "multivalued function").


> To increase my limited knowledge of complex numbers,
> I wondered about other examples of raising a number
> to the j-th power.  Isn't it interesting that:
>
>       0.001867442731708^j = 1^j = 1

Actually, not so much. It's all the same sport, and comes from the fact 
that e^{2 \pi i} = 1 or rather that the log is a "multivalued function".

Thus, the "values" the function a^b can have, once projected down from 
its natural domain (overlay of many copies of C) onto C are given by

a^b e^{2 b \pi i N} =  e^{b (\log a + 2 \pi i N)}

where "N" enumerates the overlays. As you can see, strange and wonderful 
things can happen if b is irrational and the geometry of the objects can 
be rather complicated.

Greetings,
	Thomas

On Saturday, January 31, 2015 at 8:44:08 PM UTC-5, Rick Lyons wrote:
> Greetings Earthlings,
> 
>    It's been a bit quiet here on comp.dsp for the 
> last few days, so I thought I'd post something 
> with the sole purpose of rattling your cages.
> 
> Regarding the complex-valued j-operator, where 
> j = sqrt(-1), I was thinking about the interesting 
> notion that j raised to the j-th power, j^j, is a 
> real-valued number.
> 
> To increase my limited knowledge of complex numbers,
> I wondered about other examples of raising a number 
> to the j-th power.  Isn't it interesting that:
> 
>      0.001867442731708^j = 1^j = 1
> 
> [-Rick-]

I too find identies surrounding sqrt(-1) fascinating.
In this case there's an easy way to see through it:

 0.001867442731708^j = 1^j = 1

ln(0.001867442731708) = -2*pi
(ln(0.001867442731708))^j = (-2*pi)^j
exp((ln(0.001867442731708))^j) = exp(-j*2*pi) = 1

Cool.

Lito

On Sat, 31 Jan 2015 18:12:06 -0800 (PST), Greg Berchin
<gjberchin@charter.net> wrote:

>On Saturday, January 31, 2015 at 7:44:08 PM UTC-6, Rick Lyons wrote:
>
>> To increase my limited knowledge of complex numbers,
>> I wondered about other examples of raising a number 
>> to the j-th power.  Isn't it interesting that:
>> 
>>      0.001867442731708^j = 1^j = 1
>
>I had never thought about it before, so I found some good info here: http://www.milefoot.com/math/complex/exponentofi.htm

Hi Greg,
   I had not seen that web page before.  But I 
have encountered that web page's 'next to the 
last' equation before.  (In a different, but 
equivalent, form.)

I've always said, "Learning signal processing is 
not something you accomplish, it's a journey 
you take."

So, in my quest to understand complex numbers my 
journey has caused me to fall into a cave dominated 
by a four-headed Hydra, ...a beast that is the web 
page's 'next to the last' equation.  (The beast has 
one flayling, fanged, and fluctuating head for each 
of the four variables 'a', 'b', 'c', and 'd'.)

A few days ago, cutting off three of the 
Hydra's heads, I spent time studying what is  
equivalent to that web page's 'next to the last' 
equation under the restrictions that b = c = 0, 
and d = 1.  That is, I studied the simple equation:

    a^j = cos(ln(a)) + jsin(ln(a))       [1]

Under those restrictions, Eq. [1] is a complex 
number whose magnitude is always unity, but whose 
real and imaginary parts fluctuate in amplitude 
in a most interesting way.

Try plotting Eq. [1]'s real and imaginary parts, 
over some range of the real-valued independent 
variable 'a', using software.  (Variable 'a' is 
on the x-axis of your plots.) You'll see what 
I mean by "fluctuate in amplitude in a most 
interesting way."  

After playing around with your plots, change those 
plots so that the x-axis is not linear, but rather  
logarithmic.  Try it, see what happens.

Back to the web page that you posted Greg:
as for that web page's last (final) equation, 
I'm suspicious.  I'm not sure that it is correct.

>A complex number raised to a complex exponent turns out to be periodic.

As Rocky Balboa would say, "That is very true."

> That's not really surprising, given the periodicity in the special case of Euler's Identity.

[-Rick-]

On Sun, 1 Feb 2015 07:42:28 +0000 (UTC), glen herrmannsfeldt
<gah@ugcs.caltech.edu> wrote:

>Rick Lyons <R.Lyons@_bogus_ieee.org> wrote:
> 
>>   It's been a bit quiet here on comp.dsp for the 
>> last few days, so I thought I'd post something 
>> with the sole purpose of rattling your cages.
> 
>> Regarding the complex-valued j-operator, where 
>> j = sqrt(-1), I was thinking about the interesting 
>> notion that j raised to the j-th power, j^j, is a 
>> real-valued number.
>
>When I was in high school, I was trying out some PL/I
>programs using complex numbers.  When Fortran only
>had (complex)**(integer), PL/I had (complex)**(complex),
>so I wrote a program to read in two numbers and write
>out X**Y. 
>
>My first try was 1I,1I and I was surprised to see a
>real number.  Then I tried ones like 2I,1I and 1I,2I
>finding that they were not real. 
>
>But my favorite explanation is in "Feynman: Lectures on Physics".
>
>In one chapter, he explains the origin of logarithms, starting
>with 10 successive square roots of 10. (The is, 10**0.5, 10**0.25,
>10**0.125, ..., 10**(1/1024).
>
>Then with some multiplication, you can fill in the table, such
>as 10**0.75 and 10**0.375, and finally learn that linear
>interpolation is plenty close enough. 
>
>Once you have a table of powers of 10, invert the table to
>make a table of logs base 10, again with interpolation.
>Also, he notes that for small x, 10**x is 1+2.3052x
>(five digits might  have been about right for early log tables).
>
>Now he makes the not so obvious, but reasonable assumption, that
>10**x is still about 1+2.3052x for small complex x, such as i/1024,
>so 10**(i/1024) is 1+2.3052i/1024. 
>
>Next, using ordinary complex multiplication he computes powers
>of 10**(i/1024), to generate a graph of 10**ix as a function of x,
>both real and imaginary parts.
>
>Finally, noticing that they are sinusoidal, but with periods 
>not 2*pi, he calculates the base raised to the i*x power that
>generates sinusoids with period 2*pi, and finds e.
>
>By the way, anyone notice how many calculators that claim to
>do complex arithmetic won't do things like log(-1) but
>only ln(-1).  Also, won't do asin(2) or sin(i).
>
>-- glen

Hi glen,
   Thanks for your detailed post.  It deserves 
careful and thorough examination. 

Your point about hand calculators is well-taken.
My really nice Casio scientific calculator claims 
to work with complex numbers, and it does.  It'll 
add, multiply, and divide complex numbers.  But when 
I raise a number to a complex power, my calculator 
responds with the message "Math Error."  Ha ha.

[-Rick-]
 

On Sun, 01 Feb 2015 09:52:30 +0100, Thomas Richter
<thor@math.tu-berlin.de> wrote:

>On 01.02.2015 02:43, Rick Lyons wrote:
>
>> Regarding the complex-valued j-operator, where
>> j = sqrt(-1), I was thinking about the interesting
>> notion that j raised to the j-th power, j^j, is a
>> real-valued number.
>
>Actually, if you say only that, I would probably respond that this isn't 
>quite true. 

Hi Thomas,
   are you saying j^j is not equal to a 
real-valued number?

>Unlike addition or multiplication, the power function is not 
>a well-defined function on the complex-plane. 

You're not jokin'.  Willy-nilly use of the 
Exponent Law:

     (a^p)^q

can lead to paradoxical contradictions. 

>It is a multi-valued 
>function (actually, this is a contradiction in terms), or rather, it is 
>defined on some multi-sheeded overlay of the complex plane with itself 
>that depends on the basis. So for example,
>
>z^{1/2}
>
>can be either be defined as a discontinuous singly-valued function, or 
>as a function on the double-overlay of C with itself where it is 
>analytic outside 0. (aka "multivalued function").

It sure looks like you know a lot more about 
all of this than I do.  Darn, I don't understand what 
"double-overlay of C with itself" means.  I don't 
even know what your mysterious 'C' represents.

>> To increase my limited knowledge of complex numbers,
>> I wondered about other examples of raising a number
>> to the j-th power.  Isn't it interesting that:
>>
>>       0.001867442731708^j = 1^j = 1
>
>Actually, not so much. It's all the same sport, and comes from the fact 
>that e^{2 \pi i} = 1 or rather that the log is a "multivalued function".
>
>Thus, the "values" the function a^b can have, once projected down from 
>its natural domain (overlay of many copies of C) onto C are given by
>
>a^b e^{2 b \pi i N} =  e^{b (\log a + 2 \pi i N)}
>
>where "N" enumerates the overlays. As you can see, strange and wonderful 
>things can happen if b is irrational and the geometry of the objects can 
>be rather complicated.
>
>Greetings,
>	Thomas

"Strange" and "wonderful" are certainly appropriate 
words to use.  Thanks for your thoughts Thomas.

[-Rick-]

On Sun, 1 Feb 2015 11:44:03 -0800 (PST), lito844@gmail.com wrote:

   [Snipped by Lyons]
>
>I too find identies surrounding sqrt(-1) fascinating.
>In this case there's an easy way to see through it:
>
> 0.001867442731708^j = 1^j = 1
>
>ln(0.001867442731708) = -2*pi
>(ln(0.001867442731708))^j = (-2*pi)^j
>exp((ln(0.001867442731708))^j) = exp(-j*2*pi) = 1
>
>Cool.
>
>Lito

Hi Lito,
  My response to your post is a single 
word used in 18th-century England: 

   Correctamundo!

Good job Lito.

[-Rick-]

On 1/31/2015 8:43 PM, Rick Lyons wrote:
>
> Greetings Earthlings,
>
>     It's been a bit quiet here on comp.dsp for the
> last few days, so I thought I'd post something
> with the sole purpose of rattling your cages.
>
> Regarding the complex-valued j-operator, where
> j = sqrt(-1), I was thinking about the interesting
> notion that j raised to the j-th power, j^j, is a
> real-valued number.
>
> To increase my limited knowledge of complex numbers,
> I wondered about other examples of raising a number
> to the j-th power.  Isn't it interesting that:
>
>       0.001867442731708^j = 1^j = 1

Hell, anyone can *think* about complex numbers!  When is anyone going to 
*DO* anything about them?

-- 

Rick