Subject: Successive Square Roots of i                              [S/MIME]
Date: Sun, 25 May 1997 01:45:03 -0400                                Signed
From: "David C. Manchester" 
Organization: David Manchester
To: "David C. Manchester" 

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Article 31 of 36

Subject:      Successive Square Roots of i
From:         Vead@q.continuum.net (D. Manchester)
Date:         1996/02/15
Message-Id:   <4g0b7l$3gp@news.continuum.net>
Newsgroups:   sci.math,sci.physics,sci.fractals
[More Headers]

 A number of years ago (Nov. 1988) I lived in NC and my good friend
Chris was about to graduate from RIT way up in NY.  As we are both
from SE CT, I was concerned about losing touch, but happily that
hasn't happened.  At that time Chris wrote me a letter on a subject
of mutual interest, the square root of -1. (He was, after all, a
Math major, and was aware of my interest in building a mathematical
Pulsor (discontinuously pulsating Twistor) model of Space-Time.

As fortune would have it, I promptly lost the letter. :(

But a few years later, it turned-up in my canvas "bag 'o math" :)

Chris has graciously given his ok for me to post it (so long as I
do the data entry work).

Any errors in transcription are mine. -dcm (Vead@q.continuum.net)
[my comment]
===================================================================

                                                    [11-88]

Hey Dave!

 How's it sliding?  I am here in Rochester, basking in the
 hedonistic glow of vacation, and have had some Free time to
 diddle with mathematics.  Being thus vacated and diddling,
 I happened upon Some Identities that caused me to grin
 uncontrollably.  I thought I'd share them with you, in the
 hopes you might incorporate them into a new theory of
 consciousness-time-space, and that I might subsequently
 steal all the credit.

 The First identity is based on the Familiar

            2    2
           a  - b  = (a+b)(a-b).

 We don't really _need_ a difference of _squares_, however,
 we could just as well write

  a - b = (sqr(a) + sqr(b)) (sqr(a) - sqr(b))

     [in the letter, Chris used radical signs...prettier-dcm]

 We can repeat the process and Factor the BLUE term [on right above]
 similarly:

  a - b =
(sqr(a)+sqr(b)) (sqr(sqr(a))+(sqr(sqr(b))) (sqr(sqr(a)) - (sqr(sqr(b)))

 We will always have a difference term left, so we can repeat this
 process indefinitely.  After taking the limit and throwing around
 a lot of calculus, we get:
                                        1             1
                                     ( ___ )      (  ____ )
  I.                     infinity       n             n
                          _____        2             2
    a - b = (ln(a) - ln(b) | |   (   a      +     b           )
                           | |       ---------------
                          n=1               2

            [this look better in longhand.
              the exponent above should be 1/(2**n)..dcm]

  This is what we get if we carry on the Factorization
  infinitely many times.  I'll spare you the proof.
  (I can't find it!)

  Notice the 'log  ' popped up.
                 e

  Now consider a special case of I. where b=1:


                  infinity     (1/2**n)
      a - 1       _______    (a        ) + 1
     --------  =    | |     ------------------
      ln(a)         | |             2
                    i=1

            [sorry about the notation switch...this goes on...dcm)

   or,

  II.

      ln(a) =

              2                  2                    2
 (a-1)(-----------------)(-----------------)(-----------------) ...
            1/2                1/4               1/8
           a    + 1           a    + 1          a    + 1

  A new expression For log , aside from the usual Taylor Series!
                          e

  Now, the _natural_ thing to take the log of is -1, since ln(1)=i*pi.
                                          e

  So let's plug '-1' in for 'a':

   ln(-1)=i*pi=

                    2               2                2
         (-2)(--------------)(--------------)(--------------) ...
                  1/2               1/4           1/8
               (-1)    + 1      (-1)    + 1    (-1)    + 1

  or,

 III.
                      2               2                2
    i*pi = (-2)(--------------)(--------------)(--------------) ...

                                     1/2             1/4
                   i + 1           (i   +1)        (i   +1)

  But what the hell is sqr(i)?  Or sqr(sqr(i)) ?  And so on?

  This question brings me to my prettiest set of equations,
  the successive square roots of i...

        ***************************************
        *                                     *
        *  The Successive Square Roots of i:  *
        *                                     *
        ***************************************

  What happens when we take sqr(i)?  Do we get super-imaginary
  numbers?  Hallucinogenic numbers?  No, it turns out we just get
  ordinary complex numbers back.  Take a look at this:

       [lovely recursively drawn mutant smiley omitted ')...dcm]

         i  = sqr(-1).                                       [i**(1/1)]

     sqr(i) = 1/2 ( sqr(2) + sqr(2)i ).                      [i**(1/2)]

 sqr(sqr(i))= 1/2 ( sqr(2+sqr(2)) + sqr(2 - sqr(2))i ).      [i**(1/4)]

sqr(sqr(sqr(i))) =

   1/2 ( sqr(2+sqr(2+sqr(2))) + sqr(2 - sqr(2+sqr(2)))i ).   [i**(1/8)]


sqr(sqr(sqr(sqr(i)))) =                                      [i**(1/16)]

 1/2 ( sqr(2+(sqr(2+sqr(2+sqr(2)))) + sqr(2 - (sqr(2+sqr(2+sqr(2))))i ).

     ...
       ...
         ...

                          [as I said, the original scrawl was done   ]
                          [ using radical symbols...rewrite it using ]
                          [ them for a clearer picture of the        ]
                          [ recursive dynamic here....dcm            ]

 Again, I'll spare you the proof (I'll send it to you if you like.)
                                  ___________
 Interesting how the [radical]'  /             ' signs keep
                               \/  2 + (  )

 re-entering themselves;  G. Spencer-Brown would like it, I think.

              [G. Spencer Brown wrote THE LAWS OF FORM, a pre-
                mathematical treatment of symbolic logic...dcm]
      ----------------------------------------------------------------

 (Sharpened my pencil ... much better.)

 Now that we can handle sqr(i), sqr(sqr(i)) and the like,
 we can go back to equation III and write a new equation for
 Pi in terms of i ...
                2           4                      4
 i*Pi = (-2)(-------)(----------------)(----------------------------------)
              1 + i   2+sqr(2)+sqr(2)i  2+sqr(2+(sqr(2)))+sqr(2-(sqr(2)))i

   [continued...fifth factor below..dcm]

                                    4
 x (--------------------------------------------------------------------)
        2 + sqr(2+(sqr(2+sqr(2)))) + sqr(2 - (sqr(2+sqr(2))))

                                                       ... ... ...

 or, rearranged a little [God help me...],

*
      2       1-i   2+sqr(2)+sqr(2)i   2+sqr(2+sqr(2))+sqr(2-sqr(2))i
    ----- = (-----)(-----------------)(-------------------------------)
      Pi       2           4                         4

  [continued...fourth factor below..dcm]



      2 + sqr(2 + sqr(2 + sqr(2))) + sqr(2 - sqr(2 + sqr(2)))i
   x(----------------------------------------------------------------)...
                                   4

[ the 'x' above marked continuation in the original letter ]

  This equation connects Pi, i, and 2, while avoiding e.

  Oh, yea... I found the following identity completely by accident:

*
       Pi
  -  (-----) + (infinity)i =
        2

      (sqr(2)-sqr(2)i) (sqr(2 + sqr(2)) - sqr(2 - sqr(2))i)

 [continued...third factor below...x marks continuation..dcm]

   x ( sqr(2+(sqr(2+sqr(2)))) - sqr(2 - (sqr(2+sqr(2)))) i ) ...
                                                                 ...
                                                                  ...

 Well, that's all.  Hope you Find these Formulas amusing.
 I'm sending you a copy of a book (undeserving old goon that you be...)
 called "Cognizers: Neural Networks and Machines that Think".  It
 gives me an instant contact high.  Also, I'm sending you a Basic
 program that allows you to experiment with imaginary numbers,
 supporting +, *, /, -, e**x, ln(x), plus trig. Functions.

 Let me know how the job in Jersey works out, and specifically,
 which exit?

                            Unmistakeably yours,

                                        Chris Reiss.

 [ I hope this is of some use to someone...I found the forms of
   these equations quite aesthetic. I feel quite lucky to be blessed
   with such brilliant and witty friends.....dcm (Vead@q.continuum.net)]

=========================================================================
"Congress shall make no law respecting an establishment of religion,
or prohibiting the free exercise thereof; or abridging the freedom of
speech, or of the press; or the right of the people peaceably to assemble,
and to petition the Government for a redress of grievances."- Amendment I
to the Constitution of the United States.
D.Manchester   Vead@q.continuum.net
===========================================================================

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