Categories
TECH

LINUX WINDOWS COMPARISON

Linux and Windows have a great deal in common, except for the way they each address storage devices and their partitions. One must remember that a single storage device can have multiple partitions.

Windows uses a single drive letter to reference a device and a single specific partition within the device. The C: drive is really the main system partition, (not the hard drive itself), on the resident hard drive as most everyone knows. A perceived problem is what happens when you run out of letters. But then that is probably an unwarranted fear. It is highly unlikely to see a single Windows system with more than 26 partitions.

Linux, on the other hand, makes a distinction between device type, device letter, and partition number. So a specific device is identified by the combination of device type, such as “sd” for a removable flash drive, and a letter for the device.
Example: sda meaning flash drive 1 or mcd meaning media card 4.
But it does not end there because any one of these devices could have multiple partitions. Therefore, a partition number is appended to the device.
Example: sda2 or mcd1.
So we can see that Linux is not concerned with running out of drive letter like Windows. But who really cares?

It would not be fair to under-estimate Microsoft’s effort in providing program support for a wide variety of devices. No matter, it would seem Linux is all caught up in that regard.

In my view there are a couple of really big differences between the two. First, Linux can host multiple Windows systems very easily. I don’t know this to be true of Windows. Secondly, a full system backup is extremely important for the health and safety of any digital device.

For those who don’t know what a full system backup is, it is a recording on a separate storage area or device of everything on the digital device including operating system and user data. Google’s Android Cell Phone “FACTORY RESET” and Chromebook “POWER WASH” are examples of full system backups taken before any user system changes or data is introduced. The unfortunate thing about these factory backups is they do not include any changes a user may have made to the system after using the device or his user data. But Google does attempt to offer help in that regard by offering cloud services to dynamically backup any changes made by the user. No matter, this can all be done if the user has a pc which he can use to do his own backing up.

Getting back to our comparison, Linux has a very simple and efficient full system backup program called RESCUEZILLA that resides on a bootable flash drive. It can backup Windows as well as Linux in a very friendly user manner. It is the ease in which a full system backup is taken that makes Linux my favorite. Windows 10 relies upon Windows 7 System Backup which is integrated into the Windows system and far more cumbersome to use.

Aside from these differences, both Linux and Windows have a Line Command program, aka, a keyboard only textual user interface, and a Desktop Command program, aka, a mouse and keyboard graphical user interface. And both have a software libraries full of useful application programs, aka, “apps”.

The most confusing thing about Linux are the number of different “distributions” out there. Because Linux was developed freely by the open source community, a wide number of Linux “types” exist, all having different names like Ubuntu, Debian, Red Hat, Mint, etc . But regardless, they are all “FREE” and well maintained.

So obviously my preference is Linux over Windows.
Here are a couple of facts about Linux you might find interesting.
1. It was derived from a Univac mainframe operating system called UNIX and adapted to micro computers.
2. Google’s Android & Chromebook systems are Linux based.

Categories
CALL FOR ACTION SOLUTIONS TECH

WHAT OUR ELECTION SYSTEM SHOULD LOOK LIKE

Our current election system is a piece of crap. It is a hodgepodge of manual and batch systems ripe with fraud. That an online real time election system has not been made available by now is an absolute crime. So what would an online real time election system look like?

THE DATABASE:
We start with the actual database? What is a database, you ask?
It is a filing organization and access system.
Here is a diagram of what it would look like.

(Too small? On your phone, expand it. On your computer, right click and open image in new tab.)

So how does one read this diagram?

Each box represents a “record type” having a specific data content & format different from the other boxes that represent different “record types”. (Note: the word “record type” is used interchangeably with the word “record”). Boxes at the top are basic entity record types & accessible via their ids. Unconnected diagonal lines into some of these top boxes indicates they are directly accessible. As you can see, there are 4 basic entity record types: year, nation, state, & person.

Lines connecting top level boxes to lower level boxes are links from top level records to lower level record types, thus forming a chain of lower level records. The top level box is said to be the master record type of the lower level detail boxes. For each top level master record there can be any number of lower level detail records.

So for example, given a nation id of USA, one could directly access the USA record & then traverse the state chain to access all the states within the USA. Similarly, one could access all the counties within a state by traversing the county chain. And so on.

Although the default direction of traversal from master to detail records appears to be downward (or forward), upward (or backward) traversal from any detail to master can be possible before a person actually votes. However after vote occurs, traversal in either direction between a voter-ballot-link record and a voter-location-link record should be restricted to the voter for privacy reasons. Today’s average browser-to-website user should not find this method of navigation too difficult to understand.

PRE-ELECTION RECORD CREATION AND MAINTENANCE RESPONSIBILITIES:

Each state would have this database file system accessible to the public via their website on their own computer. The database would be initialized only one time with the geographic records of nation, state and counties, these being fixed entities. But before participating in an election, each state resident voter, including state & county employees, must already have or create their own reusable logon id record. Only state & county employee logons should be allowed to update the entire database for the purposes of setting up the database content.

VOTER REGISTRATION & JOINING AN ELECTION:

Of particular importance before each election is the validation and creation of the voter-ballot-link records for each logon record. This process establishes the voter rolls for each specific election. It begins with a person submitting a voter-id-ballot-request record to an authorized state employee who initiates a search of other states by checking each states registered-voter-year-check record. If no such record is found & everything else checks out on the person, the state employee creates a single year-voter check record, followed by setting up the voter-ballot-link records leaving blank the candidate-selection button in each voter-ballot-link record.

VOTING:

The database is now ready for the voter to make his candidate selections by clicking the appropriate candidate-selection button in each appropriate voter-ballot-link record, at which time a write-permission lock is placed on the voter-ballot-link record. This would prevent anyone from changing candidate-selection button and the corresponding link to the selected candidate record.

POST VOTING RECORD ACCESS:

For secrecy/privacy purposes all cast voter-ballot-link records should be locked from updating by anyone other than the voter and as soon as they are counted. With this one exception, all should be able to view any record at any time, thus providing complete transparency and veracity of an election. In other words, everyone can look anywhere but not touch.

THE NETWORK:
Each state would have the same file system within a website on their own computer. In addition to state residents being able to create and logons to the state website, each state computer would have access to other state computers to eliminate double dipping voters who might try to vote in more than one state. Currency of voter rolls is essential.


In addition, currency of vote tabulation is essential. As soon as a voter has locked his voter-ballot record, his vote should be counted and tallies by candidate should be made available. Washington DC should be connected to each state & have access to each state database for the purpose of rolling up ballot totals for each national candidate by state.

PROS:
What would be the advantages of an online real time election system?
First, the voter would cast his ballot directly into the database, thereby eliminating anybody other than the voter from touching his ballot.
Second, the election should be completed within 2 days.
Third. since there is only one machine per state involved, auditing software should be comparatively easy as well as identifying any other anomalies.

CONS:

There will always be some ding dong who insists on having a paper ballot which necessitates having a batch input system.

Categories
TECH

INTERNET BASICS

Rules For Securing Your Own Email:

1. You can’t secure your email if you use a cloud email service except via the tools provided by the email service

2. Your router and your firewall are all you need to secure your own email server. This requires a simple understanding of how internet addressing works.

First, understand that a domain name has no real significance in addressing. All addressing boils down to an IP NUMBER and a PORT NUMBER. The ip number, v.w.x.y, is the equivalent of a building address. The port number, z, is the equivalent of an apartment number. The two are expressed together as v.w.x.y :z.

With respect to email, the port numbers 25 and 587 are used to send email. The port numbers 143 and 993 are used to retrieve email in an IMAP fashion as opposed to 110 and 995 in a POP fashion.

So how do I block out unwanted hackers or spammers. First, my ROUTER must block out all unused ports, ie, empty apartment numbers. But it has to let through all traffic to the ports in use, ie, ports 25 and 587. This is accomplished via PORT FORWARDING, which is a function in your router to direct traffic addressing a port to its ultimate computer program, ie, destination.

Secondly, I have to use the firewall to block out the ip numbers (addresses) of unwanted visitors trying to hack my computer. These are found by inspecting the email server log. The hackers become quite obvious in the log.

Oh, wondering about my ip address? It is the ip assigned to me by my internet service provider. The isp forwards all internet traffic calling out this ip address to my router which begins sorting out what ports go to what computers I have on my local network.

What is my local network? That’s everything I have wifi-ed or physically attached to my router. And how do ports get connected to programs? Once the router directs a specific port to a specific computer, the programs (apps) have to be told what ports to “LISTEN” to.

So what has domain naming got to do with anything? First, we have to know that a domain is a fancy word to identify the digital hardware and software resources belonging to a specific owner. So I have two main domains that are identified as trcooper.com and aaarrrg.com. These names get resolved (translated) into the ip addresses where they reside(hosted) by an INTERNET DOMAIN NAME SERVER (DNS) which is nothing more than a lookup service, like a phone book or 411.

So, getting back to the actual residency of my domains, the trcooper.com main domain is on the GoDaddy computers, whereas aaarrrg.com is on my home computer. These are where my websites reside.

But what about my email server? Where does it reside? This is where SUB-DOMAIN naming comes in. My email server resides on my home computer which is identified as a sub-domain of trcooper.com and identified as mail.trcooper.com. The dns server will direct all traffic calling out mail.trcooper.com to my home computer and not the GoDaddy computer. So it’s all like a company that has multiple addresses.

The only thing I have not covered is encryption which provides the ultimate security in digital communication. Another word that goes flying around is the word “protocol”. I find this word being so widely abused that it has almost lost its meaning. Basically, it is a predefined way in which two communicants can verify they are speaking to the right person.  One of the first protocols was the “ack”- “nak” , acknowledge- no acknowledge. Different strokes for different folks. Ground control to major Tom. At the current time, it appears as though a protocol called TLS (transport layer security) is replacing SSL (secured socket layer) in two-way digital communications.

Categories
COMMON SENSE SCIENCE SOLUTIONS TECH

The DIRECTION OF MULTIPLICATION

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”.