Category: TECH
With the influx of illegal migrants and in view of the wide spread fraud in the 2020 election, a great concern has arisen regarding the integrity of our national election system. What is herein is to suggest a solution to that dilemma.
Our current election system is a piece of crap. It is a hodgepodge of manual and batch systems ripe with fraud. That a secure 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 database look like?
What is a database, you ask? It is an electronic filing organization that provides entree point access to “External-Master Records” which in turn provide navigatable link chains to more “Internal-Detail Records”. A database “schema diagram” is a map of the accessibility & connectivity of “record-types” contained within the database. Think of a “record-type” as a single template representing any number of records consisting of the same record format, a “record” being defined as a fixed length string of data broken into contiguous fixed length data fields, each data field consisting of a contiguous fixed length string of textual characters. Record-type accessibility is accomplished in one of two ways:
1) externally via a unique key field, such records being called “external master” records, or
2) internally via link chains which are data fields within each record pointing from one record to the next, such records being called “internal detail” records. There can be any number of different link chain data fields within any record-type.
3) A record-type that is accessible either way is a “hybrid master” record-type.
SECURING THE VOTE VIA AN ADMINISTRATOR MINI DATABASE:
Here are two versions, A & B, of the same Voter-ID section of the overall election system database shown in C. They represent two possible configurations of what could be called the basic administrators mini-database that appears within the far right side of the overall database, C.
This mini database provides a cross-reference system entirely necessary for use by administrators at those levels of government concerned with citizenship identification, specifically election administrators, social security administrators, medicare administrators, & US State Dept administrators. It provides the ability of these various government agencies to uniquely identify and maintain the identity of citizens as distinct from non-citizens.
A.
In version A, you can see two different record-types represented by the two big boxes with rounded corners, ie, an upper external master header record-type connected to a lower detail voter-id record-type. The header record-type is a single master record & externally accessible given its voter-id = 0. It is the “head” or “master” or beginning of a sorted link chain. Via the line connection from this single master record to the lower level detail voter-id records, this master record 0 allows access to the entire list of other voter-id records in voter-id number sorted order. You should note that additionally each individual voter-id record is externally accessible given its unique voter-id number. This makes the lower level record a “hybrid master”. So every detail voter-id record is a naked hybrid master record as well, because there is no indicated master to detail record-type indicated below it. Regardless, it should be clear that any given detail voter-id record is also directly accessible given its unique voter-id number and indirectly accessible by going through the sorted list of voter-id records.
B.
Every voter id record should be validated at least every 4 years by either the US State Dept, the Medicare Admin, or the Social Security Admin. As such, the voter-id record must also specify reference to a passport id, a medicare id or a social security id. Any one of these 3 federal departments should have the ability to flag the voter id record with a status flag, ie, active, dead, expired, etc. In addition to these data fields being included in the voter id record, the record should include an expiration/ renewal date.
Version B shows two connection lines that replace the single line in version A. This is two allow administrators at different levels to communicate, specifically between a more local election administrator & a more distant federal administrator who has access to citizenship records. The need for two link chains (as indicated by the two connecting lines) is to separate new voter-id requests initiated by a local administrator from those voter-ids that have already been validated by one of the 3 federal agencies.
“But”, you ask, “Why not use just one link chain between the two record-types & simply flag the voter-id record as new or approved?” Answer- Do you really want to go through Africa to get from LA to New York? I dont think so. Having two link chains expedites the feds job of identifying new voter-ids requiring their approval, where just one link chain would not queue them to look at specific records. In approving a voter-id record, all they need do is zero out the voter-id record’s new link field and plug in a new link in the approved link field. Furthermore, when an approved voter-id record becomes expired, it automatically goes back to the new voter-id chain from the approved voter-id chain.
THE OVERALL ELECTION SYSTEM DATABASE, C:
Here is a diagram of what an overall National Election Database should look like. Can you find the administrator mini database/ You will see a couple of other record types I’ve included below the basic structure.
C.
(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? (If you feel comfortable with what you have read so far, you can skip to the next section called THE ORGANIZATIONAL HIERARCHY.)
Let’s begin with an analogy in the form of a picture on paper representing an invoice containing a list of purchased items. The top line appears only once & contains information common to all items listed as purchased . The top line is the header, or “master” record. The items listed are “detail” records. Now consider a stack of invoices that all have the same format. They all have the same “master record-type” with the same “detail record-type”.
A database is an electronic file cabinet for all these invoice “record-types”. To be more specific in electronic terms, a “record” is a fixed length string of data broken into “data-fields”, all representing something or having some particular meaning, and a “record-type” is the representation of the collection of all records having a common fixed data content, format & key sort field different from all other records. To look a record-type is the same as looking at the format of a specific record. A record-type is a template or example of what a record looks like. (Note: the word “record-type” is used interchangeably with the word “record”).
Returning to the database diagram, each box represents a “record-type”. You will see boxes around the periphery that have a little short arrow pointing at the box headed by an ellispse, a square or an “x” . These are “External Master Record-Types” which are the basic entree points into the database via their unique id data-fields. The ellispe heading on the arrow means read accessible to all. The square means update accessible to administrators only. The “X” is exclusive to the citizen logon.
You will see that the nation record-type (of which there there is only one for the USA) is directly read accessible by anyone based upon its nation ID. You will also see that each state record-type is directly read accessible by all based upon its unique state ID. The same holds true for the election year and zip code record-types. Another external access point is via each individual citizen’s exclusive login point. Finally, there are three other administrative access points necessary for validating citizen requests.
Outside of the External Master Record-Types, all other record-types are Internal-Detail-Record-Types accessible only through “Link Chaining” which is shown by a link chain line connecting a higher level master box to a lower level detail box. You might notice that each state record is not only accessible as an external master record, but is also accessible via the “Link Chain” from the nation record. By “Link Chaining” I mean there is a link field in each master record-type that points to the first record within a detail record-type, which in turn has a link field that points to the next record within the same record-type, and so on until the last record of that record-type which may or may not point back to the original master records. Note that an internal detail record-type in one link chain can be a master record-type in another chain. Hence, a completely accessible hierarchy is achieved.
Referring to the database diagram, with respect to link chain traversal, the default direction of link chain traversal is always forward from master to the first detail record and from current detail record to the next. if there is no arrow pointing to a detail record-type, it means that the master record is retrievable from any detail record by continuing to traverse the chain to the last detail record which contains the link back to the original master record. However, if there is an arrow pointing to the detail record type, it means the last record in the chain has its next record link null, thereby making it impossible to retrieve the master record of the chain.
To complete the picture, 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 first accessing the state record & then traversing the county chain. And so on.
Today’s average browser-to-website user should not find this method of navigation too difficult to understand.
THE ORGANIZATIONAL HIERARCHY:
In the governance of our national elections, the smallest entity is a precinct within a zip code, which is within the governance of a city, which is within the governance of a county, which is within the governance of a state, which also governs congressional districts & is within the governance of the feds.
Clearly, the record-types in the left hand side of the database diagram illustrate this organizational hierarchy of our election systems. The hierarchy is structured as relational levels with level 6 being the national level, level 5 the state level, level 4 the county & district level, level 3 the city level, level 2 the precinct level. This leaves levels 1 & 0 representing the individual citizen level.
TYING PEOPLE TO THE ORGANIZATION &
RECORD ACCESS PERMISSIONS:
We note here that anybody can query any level in the organizational hierarchy of left hand side with read only permissions once that side of the database is created by the administrators. However, we remain left with the task of identifying those administrators in the first place.
The right hand side of the database diagram deals with the people record-types. Any individual, regardless of their level, (federal, state, county, city, precinct or voter) should be able to create a logon in order to make an online candidate/voter ballot requests to an administrator in the current or next level up.
At first, each citizen logon will be tagged with an accessibility level = 0 to indicate the citizen logon has no valid Voter-ID and will not have access to any record in the database other than the logon & candidate/ballot request records.
With respect to the citizen logon level indicator, level-1 will indicate the citizen voter or candidate has a valid & current Voter-ID. Level-2 will indicate the citizen is a precinct only administrator. Level-3 will indicate a city only administrator. Level-4 will indicate a county & district administrator. Level-5 will indicate a state administrator. And level-6 will be a national or federal administrator. Administrators will have update accessibility to their own level and the next lower level, but only read accessibility to any other level above or below.
We now ask who must have what to have or request update access to the appropriate portions of our organizational hierarchy in our election system database. Every federal, state, county, city, & precinct election administrator must be a citizen with a current and valid Voter-ID. Also, every candidate & voter must be a citizen with a Voter-ID. In other words, this system is by & for US citizens only, and they must prove their citizenship to an election administrator in order to get their Voter-ID recorded in the database.
This can be done by submitting their proof of citizenship in person or by submitting proof of citizenship to the online system via a candidate/ballot request. Proof of citizenship is a passport or medicare card, naturalization papers, and/or a birth certificate, along with a picture. Drivers licenses do not count.
HOW TO GET A VALID VOTER-ID FOR YOUR LOGON:
Depending on the level of the request, the appropriate administrator would be able to confirm whether or not the requester has a current Voter-ID. If the requester does, then all the administrator has to do is create candidate or ballot record linkable to the request. But If the requester does not have a valid current Voter-ID, then he must submit proof of citizenship in his request. With this information, the administrator must check to verify it is okay to assign a new Voter-ID to the requester and then create candidate or ballot records linkable to the request. It should be that simple.
Of course, not every body may have a cell phone or computer. In this case they must go into the county registrars office to apply. And if they can’t do that, then a ballot notary is required.
FEDERAL DATABASE RESIDENCY & ADMINISTRATIVE RESPONSIBILITIES:
The database would be maintained on a federal computer, where the feds might initialize Voter-ID records from a cleaned version of the State Department’s passport system allowing only citizen records to be used, non-citizen records being excluded. In addition, the feds would initialize the Nation and State ID records for each state , these being fixed entities. And for each State ID, the feds would create backdoor level-5 sign-ons with passwords to be used by the state for its administrative duties. This is reiterated in the following.
A CASCADING INITIALIZATION OF THE ORGANIZATIONAL HIERARCHY:
Original access will begin at the national level-6. For database initialization purposes, a backdoor administrator level-6 logon will be provided to create the National-ID record. And from this backdoor level-6 logon it will be possible to change any level-0 logon to another level-6 or level-5 administrator logon once the level 0 logon owner submits proof of citizenship either in person or via the logon online candidate/ballot request process. Any level-6 administrator will create the State-ID records, along with creating a backdoor level-5 administrator logon for each state.
STATE RESPONSIBILITIES:
In a similar fashion, each state’s backdoor level-5 logon will be able to change any level-0 logon to another level-5 or level-4 administrator logon once the level 0 logon owner submits proof of citizenship either in person or via the logon online candidate/ballot request process. Any level-5 administrator will create the County-ID records & District-ID records, along with creating a backdoor level-4 administrator logon for each county.
COUNTY RESPONSIBILITIES:
In a fashion similar to the state, each county would setup City-ID records, along with creating backdoor level-3 logons to be used by city administrators in conducting its elections.
CITY RESPONSIBILITIES:
In a fashion similar to the county, each city may elect to setup Precinct-ID records, along with creating backdoor level-2 logons to be used by the precinct administrators.
OFFICE/PROPOSITION RECORDS:
Obviously for each level of governance from city on up there exists departments & offices with positions to be filled, some of which are filled via elections. And so each level of government is responsible for defining those positions to be filled via election. These are the Office-ID records, and each year some or all of them may have vacancies to be filled by candidates.
This brings us to the creation of a candidate-link record by the appropriate level administrator. And how do we determine that?
The office record should have a field that identifies its level.
CANDIDATE/BALLOT REQUESTS:
I have previous mentioned these record types before without being specific.
A CANDIDATE REQUEST will call out the specific Nation-ID, State-ID, County/ District-ID, & City-ID where applicable & the specific Office-ID. Assuming the would-be candidate is qualified & has a valid Voter-ID, the impacted administrator level will create a candidate link record for the year in question, thus establishing his candidacy.
It should be noted that there are times when there are non-candidate measures at the city, county, or state levels to be voted upon. These are identified as Propositions. They are identified in the same candidate link record-type which is accessible via the voter to the same candidate link record-type as used for candidate-to-office links.
A BALLOT REQUEST is simply a request to his lowest level of election administration to create links between his ballot request master & the candidate link records applicable to his geographic area.
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.
TALLYING UP THE TOTALS:
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.
KEEPING THE VOTER-ID ROLLS CLEAN:
Currency of the voter rolls is essential and a responsibility of the feds. Voters who have died would have their Voter-ID record flagged as such. But aside from that, there are bad players who would try to vote more than once. This should be averted by looking out for duplicate Voter-ID and ballot requests in & across all states.
PROS:
What would be the advantages of an online real time election system?
First, the election should be completed within 2 days, if not sooner.
Second, the voter would cast his ballot directly into the database, thereby eliminating anybody other than the voter from touching his ballot.
Third, a voters ballot choices would remain unknown to anyone but the voter.
Fourth, since there is only one machine involved, auditing software should be comparatively, making it easy in 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.
EARLIER POSTS:
The Real Threat: https://www.aaarrrg.com/wordpress//?p=3161
The Virtual Precinct: https://www.aaarrrg.com/wordpress/?p=2632
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.
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”.