Unions

 

Look For The Union Label

Look for the union label
When you are buying that coat, dress or blouse.
Remember somewhere our union's sewing,
Our wages going to feed the kids, and run the house.
We work hard, but who's complaining?
Thanks to the I.L.G. we're paying our way!
So always look for the union label,
it says we're able to make it in the U.S.A.!

U.S.A.! U.S.A.! U.S.A.!

The lyric above is of course the old commercial jingle for the International Ladies Garment workers union. Union organizing is having a bit of surge and pushback at the present time. The workers at the Volkswagen plant in Chattanooga, Tennessee voted to unionize which prompted much hand wringing by anti-union governors in the South. Trader Joe's, Amazon, Starbucks, and many companies beloved by liberals are fighting union organizing efforts by their own employees, prompting a call to boycott their products by liberals.

I am addicted to Trader Joe's products. I am an Amazon Prime member. And while I am a New Englander and prefer Dunkin' Donuts coffee, I will admit to enjoying a hot caffeinated beverage from Starbucks, including their mother store in Seattle. I will not be joining any boycott, but I also do not support their efforts in opposing unions. It seems to be centered around the fact that they are treating their workers well so how dare those workers unionize! Silly corporations! It is not about your present management. It is about how management might act in the future. They are not making fun of you. They are making fun of her. https://www.youtube.com/watch?v=VkcdU4gxU38

Walt Disney famously fought against union organizing efforts at his studio in the 1930’s, for many of the same reasons. How dare his employees! We are a family! And besides aren’t unions full of communists and criminals! 

Any healthy system should be as large as possible.  If a corporation depends on workers to produce its products, then why shouldn't those workers act together as a system?  I believe the phrase is we all hang together, or we will all hang separately. There may be bad unions, but that does not mean that all unions are bad. Walt may have been wrong, but that does not mean I won’t be watching Disney movies or going to Disneyland.  And that will be my position unless the behavior is so egregious that by buying those products, the buyer becomes complicit in the actions of the seller. Until then, IMHO, unions are as American as Apple Pie.

Billionaire Tax?

 

Billionaire

Oh every time I close my eyes
I see my name in shining lights
A different city every night oh
I swear the world better prepare
For when I'm a billionaire
Oh ooh oh ooh for when I'm a billionaire
Oh ooh oh ooh for when I'm a billionaire

But should there even be billionaires?

When I was growing up, the aspiration was to be a millionaire. Inflation has contributed to a change, but Reaganomics also created more billionaires when those in the highest tax bracket unintentionally could retain more of their income. Just as not understanding the difference between a mean and a median can lead to economic disasters, https://dbeagan.blogspot.com/2018/08/wonderful-world-dont-know-much-about.html, not understanding the difference between an effective tax and a marginal tax has led to those in the highest tax bracket retaining more of their income. This error over time has led to millionaires becoming billionaires.  An effective tax rate is a first derivative. A marginal tax rate is a second derivative. Confusing the two, is like confusing speed and acceleration. They are NOT the same. Acting as if they are the same is the basis for the Reagan’s income tax codes. The tax rate in any bracket is the marginal tax rate, not the effective tax rate. When the effective rate varies from 0 to 40%,  the  marginal tax rate should vary between 0 and 100%. This is not being punitive at high brackets. It is merely how math works. A 100% bracket does NOT mean that the tax rate is 100%. It merely means that in that tax bracket the full highest effective rate applies.

The income tax brackets are supposed to be merely a convenient means to compute the effective tax. Otherwise a formula would be required. The tax brackets allow that formula to be approximated by straight lines and the use of simple multiplication. Beause of the error, those in the highest brackets are retaining more of their income than was intended by the effective rates. If that income has been retained each year, and it has been over 40 years, and this retained income has been also been compounded, is it any wonder that millionaires have become billionaires.

Should Congress create a tax on billionaires? Congress created those billionaires in the first place by enacting those erroneous tax codes, so Congress should certainly tax them. But at the same time, Congress should fix the income tax code so that millionaires don’t unintentionally become billionaires in the first place.

Solutions

 

Ain’t Necessarily So

It ain't necessarily so
The things that you're liable
To read in the Bible
It ain't necessarily so

And the things that you read in Math textbooks ain't necessarily so.

The equation y²=x², seems like it has a simple single solution, y=x. but actually there are two solutions y=x AND y=-x. The second solution is due to the fact that i²=-1. This  suggests that an imaginary axis could be important in resolving this paradox.

Minkowski proposed a method to transform the three dimensions of space into a single dimension. In the two dimensional,  2‑D, version, the other dimension is typically time. This gives rise to inverted triangles fomed by light travelling on a surface in space‑time described by Minkowski space. This is often expressed that reality must be within two light triangles that intersect at the origin. If that space-time surface is rotated about another dimension, then it is suggested based on the conclusion of the first paragraph that this dimension might be the imaginary axis. Thus in 3‑D Minkowski space, there would be dimensions of space, time, AND imagination, while in conventional space, the dimension of space is transformed into length, width, and height, such that there are five dimensions.

Rotating a flat 2‑D flat, Euclidean, surface about an imaginary axis still produces only one solution, while paradoxically there are two solutions. This suggests that the rotation of a flat, Euclidean, surface may not be correct. To resolve the paradox, it  is proposed that the rotation of a hyperbolic surface is required. This still gives rise to a cylindrical 3‑D space, but then there are two solutions,  y=ln(0 ± 2*cosh(x)), if the surface must pass through  the origin.

The rotation of two triangles on a flat surface which intersect at the origin produces two inverted cones whose peaks intersect at the origin. The rotation of two triangles on a hyperbolic surface produces a two-sheet hyperboloid whose sheets also intersect at the origin. However one of the sheets will have the opposite sign of the other sheet and any solution in that sheet will also have a solution that is the opposite sign in the other sheet. It is noted that a change of sign is equivalent to a rotation on the imaginary axis of π. This because of Euler’s Formula and the fact that sin(0)=0 with a rotation of π, is equivalent to sin(π), but cos(0)=1 while cos(π)=-1.

Since cosh(x) is a odd function, that is cosh(x)=cosh(-x), and logarithms are undefined for x<0, then unless there is a rotaion of π when passing though the origin of 0 between the two sheets of the hrperboloid, it will apear to an observer on one sheet of the hyperboloid that there is only one solution, y=ln(cosh(x²)).  The derviative of this solution is tanh(x²).  This repeats with a period of πi.  If x<2/3*π, then the solution is virtually identical to y=x when x>0.  A Nash Equilibrium discontinuity occurs at 5/6*π. The approximation can be used until this discontinuity, at which point the uncertainty  becomes significant and the approximation is no longer valid. Since the absolute can also be stated as a multiple of π, this can be restated that the universe is flat locally, but is hyperbolic universally.

Reality

 

I Can See for Miles

Well, here's a poke at you
You're gonna choke on it too
You're gonna lose that smile
Because all the while

I can see for miles and miles
I can see for miles and miles
I can see for miles and miles
And miles and miles and miles

But is there a limit to what you can see?

A moving object in the reality of space-time can be described by its vector. A vector has a magnitude and a direction. That magnitude can be expressed as a scalar, but by definition that magnitude can NEVER be negative. Thus a vector can be opposed by the same magnitude in the opposite direction, but it can NOT be opposed by the opposite magnitude in the same direction.

This has a bearing on space as a dimension. A particle can be described as a vector. But its position in space can NEVER be negative. An exponential distribution, which also does not allow negative values, can  be viewed as a magnitude. The three dimensions of space: length, width, and height; should NOT be expressed as negative numbers. They can be expressed as relative, positional, negatives with respect to a fixed location, but this is merely a translation, not an allowance of absolute negative numbers. Time as a dimension by contrast can be expressed as negative numbers with now defined as zero, the past as a  negative number, and the future as a positive number. If the three dimensions of space are collapsed into a single dimension such as in Minkowski space, where time is retained as separate dimension, the limitation with space, x, no negative numbers,  but with time as both positive and negative values still applies. If this surface, constrained in space and unconstrained in time, is rotated about a third axis, say an axis of imaginary numbers,  that surface becomes a volume. Rotation is infinite but repeating, and rotation can be either positive or negative depending on the direction of rotation. Reality can be considered to be an imaginary number of zero. A clockwise rotation might be considered to be positive, a daydream,  while a negative, counterclockwise,  rotation might be considered to be a nightmare. The rotation of the surface of space-time about an imaginary axis defines a volume that is a cylinder. A vector in cylindrical coordinates can be expressed as re^iθ, where r is the real radius in space-time and θ is the angle with respect to the imaginary axis. This is a special case of Euler’s Formula,  e^ix=cos(x)+i*sin(x), where r =1 and x=θ, and thus re^iθ=r*cos(θ) + r*sin(θ)*i. It is true that θ=arctan( (r*sin(θ))/( r*cos(θ))) regardless of the surface being rotated. It is also true that r²=( r*cos(θ))²+(r*sin(θ))². But this has a different solution for r depending on the type of surface which is being rotated.

If the surface being rotated is flat, Euclidean, it has the conventional solution of

r=√(r*cos(θ))²+(r*sin(θ))²),

but this is only because on such a surface

cos(r)=cos(r*cos(θ))*cos(r*sin(θ))

and the elliptical/circular identity that 1=cos²+sin².  If the surface being rotated was spherical, then the solution would be

cos(r/R)=cos(r*cos(θ)/R))*cos(r*sin(θ/R))

where R is the radius of the spherical surface. As R approaches infinity this also becomes, r=√(r*cos(θ))²+r*sin(θ))²). However if the surface being rotated is hyperbolic, the solutions must satisfy,

cosh(r)=cosh(r*cos(θ))*cosh(r*sin(θ)).

If the surface being rotated must also pass through the origin, then this has two solutions,

  r=ln(0 ± 2*cosh(r*cos(θ))). 

This is because while sin(0) and sin(π) both pass though the origin, are zero, but  cos(0)=1 and cos(π)=-1; the hyperbolic identity that 1=cosh²-sinh²; and cosh(x)=cosh(-x), i.e. cosh is an odd function.

What is conventionally described as two intersecting inverted light cones in Minkowski space assumes that light travels on a flat, Euclidean, surface. If it instead light travels on a hyperbolic surface, the two inverted cones, become a two-sheeted hyperboloid where the sheets intersect at the origin. There is one solution in each sheet of the hyperboloid. However within each sheet of the hyperboloid, the solutions may appear, as 0 and ln(2*cosh(r*cos(θ))) because a particle in one sheet can not pass though the origin unless it is rotated by π, i.e. changes signs, and the other solution is in the other sheet. The average of 0 and ln(2*cosh(r*cos(θ))) is locally r=√(r*cos(θ))²+(r*sin(θ))²).  This approximation holds true until r is more than 5/6 of the absolute, is hyperbolic globally. Thus Pythagoras’ Theorem is true locally, but is not true globally, if the surface of the universe is hyperbolic. And you can see for miles, but if you are in one sheet of the hyperboloid, then you can not see past the origin into the other sheet and you might thus think the solution in that sheet is zero.

Currency

 

One

One is the loneliest number that you'll ever do
Two can be as bad as one
It's the loneliest number since the number one
No is the saddest experience you'll ever know
Yes, it's the saddest experience you'll ever know
Because one is the loneliest number that you'll ever do
One is the loneliest number that you'll ever know

Got it. How about Three? And how about neither “Yes” nor “No,” but “It depends”?

In game theory, a game of solitaire without rules is not a fair game. With rules, it becomes a game of two, eenif thee is only one player. But when there are two players, but one player decides not to play by the rules, then it also not a fair game. Only in a game of three players, where there might appear to be only two  contestants, but they are both playing by the rules, is there a fair game. One, no fair game. Two, not necessarily a fair game. Three (or more), always a fair game.

An economic transaction also requires three elements: a buyer; a seller, and goods being exchanged. The goods can be specific, in which case the transaction will only take place if the buyer and seller have different goods that they wish to exchange, i.e., the barter system. If both the buyer and the seller have the same good, or one party does not wish the specific goods that the other party has, no exchange will take place. But if the buyer has a general good,  e.g. currency which has a value equal to the good being offered by the seller, then an exchange can take place.

The buyer has to trust that the goods being offered are as represented, and the seller has to trust that the currency being offered has the value that is being represented. To satisfy the trust of the buyer is why sellers are subject to regulation of their goods. To satisfy the trust of the sellers is why currency is subject to regulation. Currency can be either commodity or fiat.

Historically in most cultures, precious metals, commodities, have been used as the currency. To guarantee the weight of the precious metal, and that this is indeed the precious metal represented, the sovereign of a nation may coin/mint currency. To make it possible that various exchanges are possible, it is also likely that the sovereign will coin many denominations which can be totaled for the value of the exchange, and enforce laws against counterfeiting that currency.

But precious metal as a currency has some issues. It has a physical weight that it is difficult to carry in large amounts. Instead notes may be issued, that represent the value of the precious metal backed by that note/bill. But the note/bill is still the value repeated by the issuer even if it no longer has the physical characteristics of the commodity.

The sovereign must then decide if its currency is to be backed by a commodity, e.g., precious metals, or backed by only its word, fiat. Both have their problems. A currency backed by a commodity has the danger of economic panics/bank runs when there is hoarding of the commodity that prevents the exchange. But the sovereign must also be trusted not to print/create currency only for their own needs, which is the danger of hyperinflation, e.g., the Weimar republic. It is the responsibility of the  sovereign/central banks to set the domestic currency to the value of economic transactions. E.g. the US Federal Reserve Bank sets the US Dollars in circulation/money supply to the amount of money in the domestic economy at a point in time in time which will  support the exchanges in that domestic economy.

The British Empire dominated the Global Economy for much of the period immediately prior to the 20^th century. Issac Newton, as Warden of the Mint, set the currency value of silver with respect to gold in 1717, despite the fact that silver was also used as a currency up until this point. This is arguably the origin of the gold standard for currency. This is despite the fact the British unit of currency is the Pound Sterling, which at one time was a pound of sterling silver. The US Dollar was based on a Spanish coin of silver, nicknamed pieces of eight. Two bits, two eighths of that coin, was a quarter of a US Dollar. Much of US paper currency was at one time silver certificates. But despite the historical context, the gold standard prevailed and the United States, for example, did not allow the free coinage of silver.

The United States domestic currency was backed by gold until 1933, at which point domestically it became a Fiat currency when domestic individual ownership of gold was prohibited. At this point, the international currency remained as gold. During WWII, international conditions resulted where most of the gold was being held by the United States. In order to prevent the stifling of international trade, the United States joined with other nations in the Bretton Woods agreement. There the international fiat currency was established as the US dollar fixed at an agreed upon value of gold. However there was no provision for raising this rate based on the growth in international trade. Consequently the accumulation of US dollars by foreign nations resulted. This was the basis for the “Nixon Shock of 1971”, when the Bretton Woods agreement was abandoned, and the US Dollar floated with respect to gold. This effectively returned international trade to  a commodity, gold, even though most international transactions primarily remained in US Dollars. This remained the international situation until the advent of the multinational Euro was introduced.

This resulted in some clearly identifiable periods of US domestic, multinational European, and international currency.

Time Period	US Domestic Currency	European Multinational Currency	International Currency	Notes
1793 -1933	Commodity/gold	None	Commodity/gold	Minimal International Trade
1933-1935	Fiat	None	Commodity/Gold: WWI disruptions	Much of the world’s gold in the US
1945-1971	Fiat	None	Bretton Woods: US $ as Fiat	Growing International Trade
1971-2002	Fiat	None	Nixon Shock: return to Commodity/gold	Accelerating International Trade, mostly in US $
2002-Now	Fiat	Euro	Commodity/gold	Robust int’l trade. The top 4 currencies, US $ (45%), Euro (15%); Japanese Yen (8%); and British Pound (7%), represent almost 75% of international trade. All other national currencies are individually each less than 4%, with most individually less than 1%.

 Perdiods: 

1.      US domestic currency Commodity: International currency Commodity

Domestic currency hoarding and bank runs; no domestic inflation: Minimal int’l trade

2.      US domestic currency Fiat: International currency Commodity

No domestic bank runs: no deity inflation: WWII distortion of int’l trade

3.      US domestic currency Fiat: International currency Fiat as US $

No domestic bank runs: modest domestic inflation; growing int’l trade

4.      US domestic currency Fiat: International currency Commodity, Transactions in US $

No domestic bank runs: significant domestic inflation; robust int’l trade

5.      US domestic currency Fiat: International currency Commodity, Transactions in mixed currency

No domestic bank runs: modest domestic inflation; robust int’l trade

Periods 1-2 provide background for US monetary policy and might serve as useful for additional analysis. Period 3 provides data for when the US Domestic currency and was a fiat currency and the International currency was also a Fiat currency but its value was fixed in 1945 and was not adjusted to reflect increases in international trading. In period 4, the US Domestic currency remained a fat currency, but the International currency was returned to a commodity/gold currency with little competition for the US $ as the international trading unit for this commodity currency. In Period 5 the US Domestic fiat and International commodity currency situation remained the same but the introduction of the Euro was a stronger competitor to the US Dollar for an international currency-based economy.

The Triffin dilemma, where the domestic fiat currency is also the international fiat currency was arguably in effect only during period 3. The return to international commodity/gold currency in period 4 is arguably why US domestic inflation soared during the early years of period 4. During this period the domestic fiat currency was a function of the domestic economy, but the international trading currency was unconstrained, but international dollars competed with domestic dollars for domestic goods. This dichotomy where the amount of US Dollars in circulation was constrained by the domestic economy but was unconstrained in the international economy is arguably the source of high infastion in Period 4 and the persistent inflation during Period 5.  The introduction of the multi-national Euro in Period 5, where some Euro members sovereign states grew at less than the total of all member states but the Euro could not adjust for those domestic economies is possibly the reason for the monetary crises in Greece, Italy and other domestic economies in the European Union.

Truth IV

 

Make Believe.

Others find peace of mind in pretending,
Couldn't you,
Couldn't I?
Couldn't we?
Make believe our lips are blending
In a phantom kiss - or two - or three.
Might as well make believe I love you,
For to tell the truth, I do...

I don’t want Make Believe. I want the Truth.

A random distribution, function, has two parameters: the location, µ, and the range, σ. If that random function was considered to be a wave, then the location could be considered to be the phase, and the range is a function of the amplitude. If the Truth is absolute, the sum of all random functions, then each of those waves has a phase and an amplitude, except that the phase of a single wave may inter with the phase, cancel out, another wave. Thus to an observer that is not the absolute it might appear that the absolute has no phase. But the Truth still has a phase and an amplitude, even if that non-absolute observer can only percieve one parameter.

Thus saying that something is True, has an amplitude, or is False, has no amplitude, is misleading. Truth has both an amplitude, which  is a function of σ, AND a phase, which is a function of µ.

If you have played Clue, Mastermind, Wordle,  or many other games, you might be familiar with the concept of a Truth Table. Saying that a Truth Table has only one dimension is misleading. Something that has a phase, but no amplitude might still be the truth if you can use: fact checking, regulation, etc., to establish what the amplitude should be. Something that has an amplitude, but no phase specified, might also be the truth if you can use: voting rights, suffrage, unionization, or etc., to establish and increase the size of the sample. Thus an incomplete Truth, by having no amplitude or no phase, might still be found to be True, if you can establish the missing parameter.

To use Clue as an example, you might know Colonel Mustard is the killer, but you might still need to establish the weapon that he used, assuming that the dimension of the room is not needed. In Wordle, knowing that you have the number of letters correct AND the the positions of those letters correct is what defines a winning word. Then you know will know the Truth, the Whole Truth and Nothing but the Truth, while knowing only one dimension, when there are two, might only be Make Believe.

Zero

 

Zorro

Out of the night, when the full moon is bright,
Comes a horseman known as Zorro
This bold renegade
Carves a 'Z' with his blade,
A 'Z' that stands for "Zorro"

What about a 'Z' that stand for “Zero”

Is zero

1) a mid-point between two absolutes, -∞ and ∞;

2) a coordinate transformation, a relative, positional notation, or

3) the absence of an absolute.

If there are negative numbers, then zero can be the mid-point between two absolutes. If there is only one absolute, then there is an absolute zero (which is an absence of that absolute), and negative numbers are not allowed. Relative, positional, zeros exist when there is an absolute zero  (as in Temperature measured in degrees Kelvin) and the zero is merely a coordinate transformation to a different scale (e. g. 273.15˚ Kelvin is 0˚ Celsius). Zero degrees Celsius is merely a convenient reference point, i.e., the freezing point of water. -40˚ Celsius does not mean that there is NEGATIVE temperature, just that with respect to the temperature in Celsius at which water freezes, the temperature is 40 degrees below that temperature.

Thus there are really only two kinds of zero:

·        mid-point zero, Case (1)  and

·        absolute zero, Case (3).

If negative numbers are allowed, then the zero is a mid-point and there are two absolutes. If there is only one absolute, then there are no negative numbers. Case (2), a postitional zero, is thus merely a subset of Case (3), where negative numbers exist with respect to a reference point, but there is no -∞.

Vectors have both an amplitude, radius, and an angle. The radius expressed as a scalar can NEVER be negative. If you turn a vector by 180˚,  then you still have a radius of 1, not a radius of -1.

Are there mathematical functions which also do not allow negative numbers? Absolutely. An exponential distribution has a Probability Density Function, PDF, of λe^-λx and a Cumulative Distribution Function, CDF, of 1- e^-λx, both with x≥0. Logarithms are undefined for x<0.

Let x be the position in space. The dimension perpendicular to the x-axis might be the value of the PDF and CDF, but for convenience let’s refer to this axis as time. While the PDF and the CDF of an exponential distribution will not ordinarily be zero, in converting to Minkowski space, the time axis is with respect to a fixed point, i.e. now; negative numbers are defined as the past; and positive numbers are defined as the future. For example, a time of 2024 CE only means that -2024 is 2024 years before the Common Era, or 2024 BCE. Alternatively that measurement can begin at the Big Bang. Negative numbers could then be the time before the Big Bang, and positive numbers could be the time since the Big Bang. Reality is the set of choices that have been made. Thus negative numbers could be the set of choices before that reality and positive numbers could be the set of choices that will be made after that reality.

Thus there are three dimensions in Minkowski space:

·        space, which does NOT allow negative numbers;

·        time, which allows negative numbers; and

·        unreality, imaginary/imagination, which allows negative numbers. 

A complex number in three polar dimensions can be expressed as re^iθ .  Transfomed to cylindrical coordinates where the cylindrical volume is formed by rotating the real surface (r, space-time) about the imaginary, i, axis, this is r*cos(θ)+r*sin(θ)*i. Thus Euler’s Formula, e^ix=cos(x)+sin(x)*i, is the special case where r =1 and is on the real surface which is rotated about the imaginary axis. If that surface is flat, Euclidean, then x=θ and r=√((r*cos(x))²+(r*sin(x))²). Since cos²+sin²=1, this becomes r=r. If the surface that is being rotated is spherical, then then as the radius of the spherical surface,  R, approaches infinity, its  limit is also r=√((r*cos(x))²+(r*sin(x))^2. But if the surface being rotated is hyperbolic, then it is     

  r=ln(cosh(r*cos(x))*cosh(r*sin(x))±√((cosh(r*cos(x))*cosh(r*sin(x)))²-1)).

The origin (0,0,0) has two solutions for the coefficient of the imaginary axis. Sin(0) and sin(π) are both 0. But cos(0)=1, while cos(π)=-1. And cosh(0)=1. This mean that a rotation of a hyperbolic surface that passes through the imaginary plane at the origin also has two solutions

r=ln(cosh(r*cos(0))*cosh(r*sin(0)) ± √((cosh(r*cos(0))*cosh(r*sin(0)))²-1)) 

and

r=ln(cosh(r*cos(π))*cosh(r*sin(π)) ± √((cosh(r*cos(π))*cosh(r*sin(π)))²-1)).

Using the hyperbolic identity that cosh²-sinh²=1, and thus sinh=√(cosh²-1), and the values for 0 and π , these can be simplified to r=ln(cosh(r) ± sinh(r)) and r=ln(cosh(-r) ± sinh(-r)). Because cosh is an odd function, that is cosh(x)=cosh(-x), these both can be expressed as r=ln(0 ± 2*cosh(r)). In Minkowski space it is conventional to describe a light cone and an inverted light cone whose peaks intersect at the origin. But this assumes that the surfaces being rotated are flat, Euclidean. If the surface being rotated is hyperbolic, then the light cones become two sheets of a hyperboloid where the two sheets intersect at the origin. There is one solution in each sheet of the two sheeted hyperboloid. The radius appears negative in one sheet and positive in the other sheet, and there is a rotation of π when passing through the origin between those two sheets. But from  the perspective of each sheet, its radius is positive, and radius of the other sheet is negative, and its rotation of the imaginary axis is 0 and the rotation of the imaginary axis in the other sheet is sheet is π  from its imaginary axis. Thus while there is one solution in each sheet, it may be percieved differently, relatively, in each sheet.

This is all because zero as the coefficient of the imaginary axis is a positional, not an absolute zero. This zero does not signify nothing and can thus can be ignored.  It signifies a relative postion that can NOT be ignored.