Given the most recent blatant congressional spending bill, it appears that it is the desire of congress people to deflate the value of the dollar, thereby causing inflation. Why? One must ask if congress people are not increasing their own coffers via their ownership in assets (like gold) at the expense of the American people. Such a concept explains the SOROS MOTIVE.
This concept is introduced based upon the belief that there can be no confidence in an electronic election count system until each an every state provide what would be called the virtual precinct that provides an after-the-fact ability for the voter to validate his own counted ballot, in addition to providing a method for auditing the vote count within the voter’s precinct. So how would such a system work? First, we need to identify the two big concerns to the average voter.
Concern #1 is identifying exactly who is eligible to vote and what is the total count of eligible voters at the time of an election. Certainly a county government is able to identify its eligible voters by referring to a county’s property tax roles. But this only covers property owners. What about renters? It makes sense for a state government to obtain this information by referring to its DMV driver’s licenses, assuming there are no non-citizens allowed a drivers license. But what about those who do not own property or own a car, ie, the homeless, who may be citizens?
So we see there remains a problem in identifying all those who should be eligible to vote in any given election. One solution has been offered called VOTER ID. However, it does not solve the problem of ballots being issued to dead voters. To solve this problem, I suggest that to be eligible to vote in a general election, an eligible voter must have voted in the primary election within the same year of the general election. In this manner, one can be reasonably assured of an accurate total electorate count within a state. Of course, there may be deaths and resident transitions within the few months between the primary and general elections. But why should a state allow any new residents to vote immediately when they can vote in any subsequent elections. In implementing such requirements, one can expect that the count from the primary will not be any greater in the general election.
Concern #2 is what can only be called the opaque electronic count system. It is not sufficient for a vote count to not be publicly audited. Transparency is an absolute necessity to provide trust in the results. This can only be done by the state providing to the voter a virtual precinct.
Democrats are deliberately destroying our democracy by exacerbating election difficulties via open borders.
Probably one of the greatest difficulties in conducting an election is knowing who is eligible to vote and who is not. Counties are the first resource to determine this, because they are the central hub for identifying property tax payers. But this only identifies property owners, not renters and homeless people.
So, to identify citizen non property owners who actually reside in a state, vehicle operator licenses are a second resource for identifying eligible voters.
However, there is still a hole wherein there are homeless citizens who may be eligible, but do not own property or have a drivers license. Such citizens should be required to register as homeless citizens and prove their citizenship.
It should now be clear to every US citizen that the Democratic Party has become a traitor to our democracy by obfuscating the election process and employing the inflow of illegal entrants in the hopes of maintaining power. They are not without help from some very stupid Republicans, aka, RINOS. Those now working in government need to ask themselves how far they are willing to go for a worthless paycheck.
It is clear that Democrats have not inspired trust in our election system by maintaining open borders. That fact, coupled with the use of electronic election systems leaves the US citizen with very little confidence that elections are not rigged. It can be said that no matter which party is in control of the election process, without the electorate being able to validate their own counted ballot and see the accumulation of actual ballots counted for a particular candidate after the fact, there can be no confidence in the electronic tabulated election results.
To conclude, until the borders are closed and until the voting public can audit/validate their own votes after an election, this country will be in a chaotic mess.
Due to my age, what is historical to me is probably ancient to you,
which makes me “conservative” and you probably a “progressive”. Many “progressives” and cultural bigots of today want to rewrite history & discard ancient things, such as the Bible, statutes of our founding fathers and maybe even the Constitution, in a manner similar to the ISIS idiots who destroyed the ancient Syrian city of Palmyra. Claims of advancing forward frequently involve the destruction of the past, either deliberately or unintentionally. But how can you claim to have advanced forward if you don’t know how you got there in the first place or what promoted another culture to excel? And so the truth gets lost.
Of course, the collection of too many things also tends to bury the really important points. This is where breakpoints to compartmentalize information become essential in maintaining the whole enchilada. And hierarchical filing structure helps to collect related breakpoints.
These are not new ideas as evidenced by the National Archives (NAR). It is merely common sense. BUT SUCH RECORDS SHOULD NOT BE ACCESSIBLE TO ONLY THE PRIVILEGED FEW, as has occurred with the recent findings of Joe Biden’s emails proving his criminal activity. Secrecy, censoring and non transparency can be just as destructive of the truth.
Consider this. Just think of how much better off we would have been had the historical library in Alexandria not been deliberately destroyed.
https://www.mymcpl.org/blogs/historical-libraries-library-alexandria
Maybe there might have been proof or disproof of the existence of ufo’s and et’s. I use this as an example , not as a sarcasm, because there are today real questions regarding ufo’s, which might have been answered by documents in the Alexandria Library. But for the lack of that library, we can only assume the answer to be “NO”.
By the same token, we are destroying our own libraries by proliferating them with the nonsense of WOKEISM. Is not the proliferation of self evident garbage to bury important findings the same as burning down the library?
My electric bill for January is $180 for 655 KWH.
It has been a cold cold month. I have one computer on at all times with two electric utility heaters that kick in and out depending on thermostat settings. Plus, I have two air filtration devices of which I run only one for no more than 8 hours a day. My water heater is all electric. I never use stove or oven. I only use a microwave very minimally. No dishwasher or wash machine. One refrig.
Without changing habits, summer usage is like half. So I have to figure the water heater and two heaters account for the difference, ie, 300 KWH.
I’m sorry Biden, BUT I AIN’T GOING TO FREEZE MY ASS OFF FOR YOU AND YOUR DUMB POLICIES. NOR DO I EXPECT ANYONE TO FREEZE THEIR ASS OFF FOR ME.
As far is Biden is concerned, we can all eat bugs, ride electric bicycles and huddle together to prevent hypothermia. Meanwhile, he will enrich our lives with drag queen story hour. And if you object, he will have you thrown in jail for a lifetime.
Why Is It Important To Restrict Access To This Country?
Three Words:
Portable Nuclear Bombs
Over the years we have witness an increase in a mentality that is willing to blow itself up along with its preceived adversaries in an act of alleged justice. This makes it imperative that access to dangerous technology be highly guarded from falling into the wrong hands. This is the real problem facing humanity today. And we must not forget that fact.
Joe Biden appears to have forgotten that by allowing anybody and everybody into this country without proper vetting. In so doing, he has increased the probability of some nutjob acquiring access to dangerous technology.
Then there are those who say “Well, nuclear bomb technology has become so proliferated that it makes no difference what we do. So we might as well give up and let the nutjobs have their way.” WRONG. As long as there is one ounce of rationality left in the world, that glimmer of hope needs to be maintained.
One thing is for certain. To make “global warming” the central theme in global politics rather than protecting the world from nuclear proliferation is completely irresponsible. And what does DEI have to do with protecting the world? NOT ONE DAMN THING. DEI = Divisive Efforts and Injustice. It is interesting to note that DEI spelled backwards is IED, Improvised Explosive Device.
The RIGHT STRATEGY to defend against nuclear attacks from the air is Reagan’s STRATEGIC DEFENSE INITIATIVE. But to protect against nulear IED attacks, the border must be highly restricted with RBA, not open borders. And you cant have either protections if you persist in pursuing green energy, electric cars, and DEI to the exclusion of all else.
Why Is It Important To Restrict Access To This Country?
Three Words:
Portable Nuclear Bombs
Over the years we have witness an increase in a mentality that is willing to blow itself up along with its preceived adversaries in an act of alleged justice. This makes it imperative that access to dangerous technology be highly guarded from falling into the wrong hands. This is the real problem facing humanity today. And we must not forget that fact.
Joe Biden appears to have forgotten that by allowing anybody and everybody into this country without proper vetting. In so doing, he has increased the probability of some nutjob acquiring access to dangerous technology.
Then there are those who say “Well, nuclear bomb technology has become so proliferated that it makes no difference what we do. So we might as well give up and let the nutjobs have their way.” WRONG. As long as there is one ounce of rationality left in the world, that glimmer of hope needs to be maintained.
One thing is for certain. To make “global warming” the central theme in global politics rather than protecting the world from nuclear proliferation is completely irresponsible. And what does DEI have to do with protecting the world? NOT ONE DAMN THING. DEI = Divisive Efforts and Injustice. It is interesting to note that DEI spelled backwards is IED, Improvised Explosive Device.
The RIGHT STRATEGY to defend against nuclear attacks from the air is Reagan’s STRATEGIC DEFENSE INITIATIVE. But to protect against nulear IED attacks, the border must be highly restricted with RBA, not open borders. And you cant have either protections if you persist in pursuing green energy, electric cars, and DEI to the exclusion of all else.
In political jargon, people like to use the linear terminology of “far left” or “far right” to condemn the opposite side. They go so far as to label the opposite side as “radical extremists” or “dangerous cult”. They tend to see “communism” as “far left” verses “nazism” as “far right” How convenient for them…. and HOW STUPID.
Such thinking is myopic, unintelligent and nepharious. It is flat earth thinking that fails to see that in the end both directions eventually meet in an irrational mode of behavior and accusation, as if completing a full circle where one direction is indistiguishable from the other. In mathematics it is well known that, given a rational radius, there is at least one irrational point around the circumference of a circle. So it is with political directions.
The Biden Admin is doing everything it can to prevent Israel from accomplishing its task to wipe out Hamas. All in the name of protecting “innocent” Palestinian citizens. Well are those citizens really so innocent, since they freely elected Hamas to be their government?
One might argue that not all Palestinians voted for Hamas.
QUESTION: DID ANY OF THEM DO ANYTHING TO PREVENT HAMAS FROM MURDERING JEWS? It seems not. So it can be said they are complicit in the murder of jews, even though they had no part in it. Even Biden has said, “Silence is complicity”.
There are those who claim that by retaliating, Israel is a war criminal. That will not be determined until it is over. And if any such judgment of that is to occur, it must be within the same context as WWII in which Winston ChurchHill destroyed the French Navy after it had been acquired by Nazi Germany, killing hundreds of “innocent” French sailors. Is ChurchHill to be rendered a war criminal? I think not.
Certainly, those protesting and demonstrating on behalf of Palestinians DO BEAR RESPONSIBILTY FOR THE MURDER OF JEWS AND SHOULD BE HELD ACCOUNTABLE AS SUCH. But what about the silent Palestinian? Are they to be considered the same?
Unfortunately in war, it becomes a numbers game and the non-involved do become the victims of war. How many lives of non-combatants do you spare vs how many of your own do you spare. This becomes the over riding answer in war. Its called “colateral damage”, ie, unintended consequences.
Perhaps we need to ask ourselves
“WHAT IS THE PRICE OF OUR NON INVOLVEMENT?” Do we want to become the unspoken victims of war, as in the movie SHENANDOAH, or a combatant?
What ever the answer, we should be sure there is in fact a war going on before making a choice. And to answer that question, all we need do is ask ourselves, “Am I safe walking down the street alone at night in my country?”
BOTTOM LINE IS
TO CLAIM NON INVOLVEMENT IS TO ACCEPT ANY CONSEQUENCES.
For those who do become involved, the overriding objective should be winning the peace, not just the war itself. There is no glory in war. But winning the peace does not mean you surrender your principles. Nor does it mean surrendering or betraying your country as Biden has done.
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”.