A. THE MEANING OF DIRECTION & SIGNAGE:

   We humans see ourselves in terms of position: vertical upright,
   horizontal on back, & horizontal on side.  These being our most
   common positions relative to Earth, they become the  3  basic
   dimensions that are seen as  3 linear axes orthogonal (at right
   angles) to each other. Furthermore, we think of measurable
   increments along each axis as being in the positive or negative
   direction depending upon their position relative to a zero point
   on the axis.   Such a concept gives us a 3-dimensional reference
   system that we see as absolute, albeit not necessarily so.    

   In mathematics unfortunately the plus and minus signs have two
   different meanings, depending upon the position relative to an
   operand. When touching an operand, it means the value of the
   operand is in the positive or negative direction along a linear axis.
   But it can also mean to add or subtract the operand in the absence
   of an explicit operator between two numbers.  And in the absence
   of a touching sign, the default is in the positive direction. When
   not touching an operand, it means addition or subtraction
   between two operands, ie,  it becomes an operator. So in an
   expression, there can be both directional signs and operational
   signs.

   As a side note & not relevant to this discussion, a minus sign in
   front of an exponent means to raise the reciprocal of the base to
   the power indicated by the exponent.

B. IMAGINARY NUMBERS:

   The imaginary number, i, is said to be the square root of -1 which
   is impossible, because according to current convention, there is no
   number multiplied by itself one time that yields a negative number.
   Lacking the ability to determine a numeric value, the square root
   of -1 is assigned the variable, “i” & complex numbers are
   mathematical expressions containing  “i”. A complex number is of
   the form (+or-)a +  (+or-)b * i, where “a” is the numerical offset,
   “b” is the numerical multiplicand, & “i” is the Multiplier. It is rare
   to see i * b, where “i”, as the Multiplier, precedes “b”, as the
   multiplicand. But that is going to change in this writing, as we
   shall soon see.

C. THE RULES OF MULTIPLICATION:

   The aforementioned indeterminate problem of not being able to
   evaluate “i” arises from the fact that mathematicians established
   long ago that A MINUS NUMBER TIMES A MINUS
   NUMBER SHOULD BE POSITIVE.  Furthermore, they
   established that THE PRODUCT OF TWO OPPOSITELY
   SIGNED VALUES SHOULD BE NEGATIVE. These
   conclusions arose due to the distributive law of mathematics.
   Let me state here my belief that when it comes to groupings
   via ( ..), the order of operations should dictate that expressions
   within a group should be evaluated first. But I will not quibble
   over the distributive law.

D. CURRENT MULTIPLY OPERATIONS IN USE:

   These current-day conventions affecting a change in value
   resulting from multiplication can be expressed as follows:

   Let:
            M = multiplier/operator

             m = multiplicand/operand

              “*” means times,
              (not to be confused with “**” which means exponent of)

             R = resulting product

   1. THE  PRODUCTS OF ACCUMULATIVE
       MULTIPLICATION:

      Accumulation Of Positives:
         Plus times Plus = Plus
             +M * +m  =  +R
              Interpretation:
                     Add +m  to the current value  M times.
                  OR GRAPHICALLY,
                     Relative to the current point,
                     go right M times in increments of |m|.  
               Example:
                   +3 * +2 =  +2 + 2 + 2 = 6

      Accumulation Of Negatives:
         Plus times Minus = Minus
            +M * -m  =  -R   
            Interpretation:
                   Add -m  to current value M times.
                OR GRAPHICALLY,
                   Relative to the current point,
                    go left M times in increments of |m|.  
             Example:
                +3  *  -2  =   0  + ( –  2  – 2  – 2)
                                =  0 + – ( 2 + 2   + 2)   =   -6

   2. THE  PRODUCTS OF DECUMULATIVE
       MULTIPLICATION:

      Decumulation Of Positives:
         Minus times Plus = Minus  
            -M * +m   =  -R
            Interpretation:
                Subtract +m  from the current value  M  times..
             OR GRAPHICALLY,
                Relative to the current point,
                go left M times in increments of |m|.  
             Example:
                 -3 * +2  =   0  +   -( 2 + 2 + 2)   =    – 6

      Decumulation Of Negatives:
          Minus times Minus = Plus
             -M * -m  =  +R
             Interpretation:
                  Subtract -m  from current value  M  times.
               OR GRAPHICALLY,
                  Relative to the current point,
                  go right M times in increments of |m|.
            Example:
               -3 * -2  = – ( -2) – (-2) – (-2 ) 
                           =  + 2 + 2 +2 = +6 

   Observe that I have identified two different types of multiplication,
   “accumulative” and “decumulative”. I make this distinction
   because accumulative  multiplication requires repetitive addition,
   where decumulative multiplication requires repetitive subtraction.

   Also, we note that the sign of the product resulting from the
   repetitive multiplication of a negative multiplicand  alternates
   between + on even repetitions & – on odd repetitions.  In other
   words, a successive number of subtractions of a negative number
   from itself ALTERNATES BETWEEN + & -.  This alternation
   does not appear anywhere else.  So this behavior is seen as
   unusual. 

E. RECONSTRUCTING THE PICTURE OF MULTIPLICATION:

   By insisting that the Multiplier always occurs in front of the
   multiplicand, we can clearly see that, among other things, a
   negative Multiplier means decumulation, whereas a positive
   Multiplier means accumulation. Aside from this fact, we might
   speculate that there could be other meanings in addition. What
   those could be, we are about to find out. 

   Moving on, we might assert that the Multiplier,M, reside on an
   M-axis different from the multiplicand,m, on a separate m-axis,
   with the two axes intersecting each other orthogonally at right
   angles. So the visual graphic of the Multiplier in relation to the
   multiplicand becomes a 2-dimensional planar picture with each
   axis having its own set of + & – directions, rather than just a
   simple 1-dimensional linear graphic.

             THIS                                  NOT JUST THIS

             m-axis
                  |  +                                                                              
       – ——0——- + M-axis        –  ———0———+ M & m
                  | –                              (We are not just talking candy here)

   Given this distinction, we can now begin to think in terms of:

      VECTOR CROSS-MULTIPLICATION,
          (aka, CROSS-MULTIPLICATION
               or
               CROSS-COMPUTATION 
               or
               X-MULTIPLICATION) ,

   versus

      VECTOR DOT-MULTIPLICATION,
          (aka, DOT-MULTIPLICATION
                or
                DOT-COMPUTATION   
                or
                *-MULTIPLICATION
                 or
                 SCALAR-MULTIPLICATION) .

   The difference is as follows.

   Vector dot multiplication results in a simple 1-dimensional product
   (called the dot-product) that resides on the same axis as the
   Multiplier & multiplicand. Up to now, current conventional
   multiplication has always been equivalent to vector dot
   multiplication for both accumulative and decumulative
   multiplication. But that is about to change, as we are about to
   change decumulative multiplication from vector dot to vector
   cross multiplication. The mathematical expression for computing
   the vector cross product is given as:

    R = M * m

   Vector cross multiplication results in a  product (called the
   cross-product) that is uniquely identified with a direction which
   is orthogonal to directions identified by the M-axis & the m-axis.,
   & whose numerical value is the simple product of the two
   numerical values further multiplied by the sine of the smallest
   angle, @, between the two vectors, M and m.  The mathematical
   expression for computing the vector cross product is given as:

   R = M X m = M * m*  sine(@) .

   So we now have two methods of multiplication, with
   cross-multiplication giving us a clearer 3-dimensional/directional
   picture shown as follows.

                   + m-axis              + R-axis  = CROSS PRODUCT AXIS 
                            ^                        /\
                             |                      ‘ 
                             |                  ‘             
                             |            R1 = (M1 X m1) * sine(90) /
                           m1       ‘     
                              |    ‘
      -M————–0———— M1 ——–> + M-axis
                        ‘     |    @ = -90   
                    ‘         |
                ‘             |
            ‘                 |
      -R                  -m   

   We now proceed to examine the deeper meanings of the
   cross-multiplication method.  

F. ABOUT THE ANGLE, @,  BETWEEN M & m.

   We’ve started out saying that M-axis was orthogonal to
   m-axis for the sake of simplicity. But the cross-product
   approach says that such is not always the case when it comes
   to vectors, because @ can take on any value between +90
   degrees and -90 degrees as the shortest path between the
   sides of the angle. And this has consequences for both the
   numerical value of the resultant, R, its dimension & its
   positive versus negative directions.        

   Before we go any further, we need to have a clear understanding 
   of how we view angles from a fixed observation point. Then we
   need to know what the sine of an angle is. And finally, we can
   discuss what role the of the angle between the Multiplier &
   multiplicand might be.     

   1. ABOUT PLUS & MINUS ANGLES:

       Envision the face of your clock where the M-axis is a straight
       line running from 12 to 6 in a negative direction & the m-axis
       is a straight line running from 9 to 3 in a positive direction.
       Progressing clockwise, we consider 12 o’clock to be +0
       degrees, & relative to it we recon 3 o’clock to be +90 degrees,
       6 o’clock to be +180 degrees, & 9 o’clock to be  +270 degrees.
       But progressing counter-clockwise from 12 o’clock, we
       consider +270 degrees to be -90 degrees & +180 degrees
       to be -0 degrees. So in this scenario, 12 o’clock is the reference
       side of any angle from it. And because we have aligned the 
       M-axis with 12 o’clock, the M-axis is also the reference side
       of any angle at which the m-axis intersects it.  Furthermore,
       should the M-axis be in a direction other than 12  o’clock,
       then the M-axis should remain the reference side of the 
       angle, @.  

       Therefore, the plus or minus direction of the angle,@,  between
       the M-axis and the m-axis depends upon whether or not we go
       clockwise or counterclockwise from the M-axis to the m-axis.
       And the shortest path from M to m will dictate whether we 
       proceed clockwise or counterclockwise from M.

   2. ABOUT THE SINE OF AN ANGLE:
       Now what about the sine of @? Without going into too much
      detail about what is meant by the sine of an angle, it is enough
       to say that the sine( +0  degrees) is +0, the sine(+90 degrees) is
      +1, the  sine(-0 degrees) is -0, & the sine(-90 degrees) is -1.
      So the sine of an angle acquires the same sign as the sign of the
      angle. If the angle is negative,  its sine is negative. If the angle is
      positive, its sine is positive.

   3. WHICH WAY JOSE, PLUS OR MINUS?:
       We now have to determine in what direction the product
       points, plus or minus, along  the resulting orthogonal axis. 

       Traditional vector math calls for the application of the RIGHT
       HAND THUMB RULE.  Finding this to be a little too
       nebulous to explain, I will only mention that the index
       finger should be the multiplicand. I leave it there.

       As an option, I would suggest discounting the sign of the
       Multiplier and applying the sign arising from the sine(@) 
       to the sign of the multiplicand to determine the sign of the
       resultant.  

G. REDEFINING ACCUMULATIVE  VS DECUMULATIVE
   CROSS-MULTIPLICATION:

   Having identified two different, but similar forms of
   multiplication, we now ask,”Are we using the correct form
   of multiplication for each?”. After all, we see some unexplainable
   differences between decumulative & accumulative operations.
   So let’s try applying vector cross-computation to multiplication
   instead of dot-computation. 

   We can now see that cross-multiplication not only results
   in a product pointing in an orthogonal direction away from 
   the directions of the Multiplier & multiplicand, but can
   yield an absolute value entirely different from today’s
   conventional multiplication, especially if the sine(@) is
    other than +1 or -1. Therefore, we ask “Which value(s)
   +1 or -1  would yield the same results as todays
   multiplication”.

   The answer(s) are clear. For accumulative
   multiplication, we need a sine(@) = +1, ie, @ = +90.
   For decumulative multiplication we need sine(@) = -1,
   ie. @ = -90. With this understanding,  we now modify the
   current conventions by simply replacing  the * operator with
   the X operator and adding the (sine @),   making @ = +90
   for  accumulative  & @ = -90 for decunulative
   multiplication. 

   Let:
            M = multiplier/operator

            m = multiplicand/operand

            “*” means times,
            (not to be confused with “**” which means exponent of)

            “X” means vector cross multiplication,
            (not to be confused with  variable “x” )

            “@” is the smallest angle between the M-axis & m-axis.
                    It is plus (+) if the shortest distance
                    from the M-axis to the m-axis is clockwise.
                    It is minus (-) if counterclockwise.   

             R = resulting product

   1. ACCUMULATIVE CROSS-MULTIPLICATION:
       For accumulative multiplication, +90 degrees is appropriate.
       In order for the resultant product, R, to become the same
       value as determined by vector dot multiplication, the value
       of sine(@) must equal +1, which means the angle, @, between
       the +M-axis and +m-axis must be  +90  degrees.
        @ = +90,   sine(+90) = +1 

      Accumulation Of Positives: 
         Plus times Plus = Plus 
         R =  +M X (+m)
              = |+M| * (+m) * sine(@) 
              = |+M| * (+m) * sine (+90) 
              = |+M| * (+m) * (+1)
              =  |+M| * (+m)
              = + (M * m)

      Accumulation Of Negatives:
         Plus times Minus = Minus 
         R =  +M X (-m)
              = |+M| * (-m) * sine(@) 
              = |+M| * (-m) * sine (+90) 
              = |+M| * (-m) * (+1)
              = |+M| * (-m)
              = – (M * m) 

   2. DECUMULATIVE CROSS-MULTIPLICATION:
      For decumulative multiplication, -90 degrees works.
      In order for the resultant product, R, to become the same
      value as determined by vector dot multiplication, the value
      of sine(@) must equal -1, which means the angle, @, between
      the +M-axis and +m-axis must be  -90. 
      @ = -90,   sine(-90) = -1

      Decumulation Of Positives: 
         Minus times Plus = Minus 
         R =  -M X (+m)
              = |-M| * (+m) * sine(@) 
              = |-M| * (+m) * sine (-90) 
              = |-M| * (+m) * (-1)
              =  |-M| * (-m)
              = – (M * m)

      Decumulation Of Negatives:
         Minus times Minlus = Plus 
         R =  -M X (-m)
              = |-M| * (-m) * sine(@) 
              = |-M| * (-m) * sine (-90) 
              = |-M| * (-m) * (-1)
              = |-M| * (+m)
              = + (M * m)

   Note that I did not recognize or apply the sign of the Multiplier. It
   was unnecessary when the sine(@) was included. Of course, I
   could have made @ = +90 for the decumulative operation. But
   then there still needed to be some explanation for the differences
   from accumulative cross-multiplication.

H. THE MYSTERY OF THE FLIP-FLOPPING RESULTANT: 

   1. ABOUT  SUCCESSIVE MULTIPLICATIONS.: 
       The fact that the resultant product of M1 X m1, R1, always
       resides in the direction orthogonal to the plane of the M-axis/
       m-axis, only one possible direction is left in which  R may
       reside, that direction being identified as the R-axis. And if that
       resultant  product, R1, now becomes the multiplicand, m2, of
       a 2nd such computation involving a new M2, then the direction
       of the new resultant  product, R2, must be on the same  axis as
       the previous multiplicand, m1. And  if that product, R2,
       becomes the next multiplicand  m3, on a 3rd such computation,
       then the direction of the new resultant product, R3, must be in
       the same direction as R1.  In other words, given a succession
       of repetitive vector cross-multiplications & where the
       resulting product becomes the next multiplicand, the R-axis
       switches positions with the m-axis & reverses its negative &
       positive directions.          

   2. REPETITIVE
       DECUMULATIVE CROSS-MULTIPLICATION 
       OF NEGATIVE  REAL NUMBERS:

       The placement of the product appears as a positive on the 
       R-axis & as a negative on the m-axis  in alternating order
       due to the right-hand thumb rule flip-flopping with each
       iteration of computing the cross-product.

       This explains why a repetitious negative times a negative 
       equals a positive R1 on the R-axis, followed by a negative
       R2  on the m-axis,  followed by a positive R3  back on the 
       R-axis. It gives the appearance of a pulsating R-axis
       acting as a binary switch between + & -. 

   3. RAISING IMAGINARY  -i TO THE Pth POWER.

       If we conduct a succession of decumulative-cross 
       multiplications of -i , assuming @ = 90 degrees, we get:

        (Cycle begins)

       (-i)**2 = -i  X -i   =  +i**2  =  -1   ( R1 to the R-axis)
                              ______________|                    
                              V
       (-i)**3 = -i X -1  =  -1 X -i  = +i     (R2 to the m-axis)
                               ______________|                    
                               V
        (-i)**4 = -i X +i  =                   +1   (R3 to the R-axis?)
                               ______________|                    
                               V
        (-i)**5 = -i X +1  =                    -i   ( R4 to the m-axis?)
                                                             |
        (Cycle starts over)                       |               
                               _______________|                    
                               V
        (-i)**6 = -i X  -i   =                    -1  (R5 to the R-axis?)                                                             

        Powers of (-i) confirmed by internet.

        Of great interest here is the observation that the successive 
        multiplications oscillate between real rational numbers and
        imaginary irrational + & – i.  We must ask, ” is i the basic 
        unit of measure in the world of irrational numbers?”. 

        NOTE:  e**i*pi = -1     where e is Eulers irrational constant.

I. CONCLUSIONS:

   1. We have identified two distinct forms of multiplication, ie,
       accumulative vs decumulative multiplication, the difference
       being the accumulative form is a series of additions whereas
       the decumulative form is a series of subtractions. 
       The sign of the Multiplier, M, identifies which form it is.

   2. We have identified two methods of multiplication, dot-product
        multiplication and cross-product multiplication, We have
        adopted  cross-product as the proper method to be used in both
        accumulative and decumulative multiplication. In doing so,
        we recognize the angle between  between the Multiplier &
        multiplicand to be +90 degrees for accumulative multiplication
        as opposed to -90 degrees for decumulative multiplication. 
        As a result, the sign of the Multiplier does not enter into the 
        computation of the product. 

    3. The angle, @, from the Multiplier to the multiplicand is
           normally +90 degrees in order to make the sine(@) = +1,
           thereby confirming that the M-axis is normally orthogonal
           to the m–axis, albeit not eliminating other possibilities for
           values of angle @, resulting in a wide variety of product
           values and plus or minus direction.

   4. The fact that both operands, M & m,  reside on a different axis
       as vectors means that the communitive law no longer applies,
       disproving the idea that a minus times a plus is the same as
       a plus times a ninus. It becomes like saying
       6 cats are the same as 6 dogs.

   Nothing has been done to change anything outside the realm of
   conventional arithmetic & mathematics.  Rather we have found
   old precepts to be applied in new ways to open the door to
   understanding some areas that left us perplexed. As a result, we
   have uncovered a new way of perceiving multiplication, resulting
   in the identification of decumulative multiplication as distinct
   from traditional accumulative multiplication. We have  uncovered
   some interesting details about how we can graphically interpret
   multiplication that involves what we call “direction” Finally, we
   have shed important new light on an entity that has kept its
   meaning hidden from us for so long,
   ie, the imaginary number,  “i”.