The discovery

The Gaussian-School of Higher-Arithmetic (1831, updated 2022); tutorial #3.

Introducing Advanced Natural (rotational) Number theory.

Every primitive logical human (infant) knew very well that we cannot take away one (real thing) from nothing. But we beat it into them and they gave in and accepted the number minus-one. Then every adult knew very well that there was no such number as the square-root of minus-one, but they gave in and just went along with the absurdity of it all. Well, the primitives or infants were right all along, there is no such number as minus-one where that is taken to mean the impossible "take away one from nothing". What really does exist however is the number-one times a rotation of two quadrants of polarity-rotation into inverse-one and the square-root of that is so obvious that even our colleagues in the neo-Pythagorean School of "flat-earth-arithmetic" might eventually be able to grasp it. 
Click on the above sketch to proceed with the arithmetic required for Quantum-Relativity.
Author's Notes:

It is worth pointing out here that the engineer Hero of Alexandria showed the existence of the square-root of inverse-one fully 200 years before the absurd mathematician Diophantus of Alexandria ever introduced the "clever" little Chinese dodge of ignoring the fact that zero minus one is strictly speaking a mathematically invalid operation. For the last 1,800 years, our so-called "mathematicians" have been indoctrinating young mathematicians and everybody else into their doctrine of nonsense. Hero must have been turning in his grave in sheer embarrassment that he ever even got involved in trying to help strange people who imagine that one may take a real number (of real things) away from nothing or that real negative lemons (for instance) actually exist. If I owe my neighbour a debt of a lemon, that is not my negative lemon, it is my liability and his asset, the orientation of the asset and the liability is RELATIVISTIC.

If you are not interested in RELATIVITY, then you are wasting your time studying this website and you can go back to your real negative lemons hypothesis and just go back to sleep.

As an aid to the AI interpretation of the above picture diagrams, here is the text that cannot be easily read by AI.


1) the main diagram


Title: The Flat Rotational Plane (FRP).

Subtitle: The FRP is merely a subordinated derivative of its logical source, the Exponential Plane.


Inside the grey boxes: (0 + i1), (0 + i0), (0 + i2)

Inside the ovals: (e^0.e^i1), (e^0.e^i0), (e^0.e^i2)

Inside the circles: 1.i1, 1.i0, 1.i2


In the feint purple box: only showing left rotation


Beside two of the circles in blue: -1, +1


Inside the three main numerical labels: lateral (with arrow pointing upwards), direct-1 (with arrow pointing right), inverse-1 (with arrow pointing left).

Inside the grey box with black outline: taking the square root of -1


Inside the circle to the left of the red box: 1.i1

Inside the red box: In the neo-Pythagorean school, they omit referring to the unit-magnitude and to the unit-rotation, and so they end up calling this just “i”, which is so comical that it is really tough to remain polite. 


The remark labelling the large blue circle: the flat locus of unity, 1.iω that is (0 and iω) on the exponential plane.


2) the colour key


Title: The key to the Gaussian School algebraic colour coding hints.


label of blue patch: flat numerical magnitude (i.e a finger-counting value).

label of green patch: relativistic ratio-metric magnitude, an exponential value.

label of the purple patch: numerical polarity rotation in the left or anti-clockwise direction.

label of the brown patch: numerical polarity rotation in the right or clockwise direction.

label of the red patch: numerical polarity rotation in invalid angular units, i.e. π radians


the remark placed under the colour keys: i is the universal-polarity-rotator; in flat (finger counting) arithmetic it can only be used in the grammatical forms: a.ir meaning; magnitude times a-rotation-of-left-rotation (in quadrant-units) and; a.ir meaning; magnitude times a-rotation-of-right-rotation (in quadrant-units).


Inside the sub key to the box shapes:


above the grey oblong shape: a grey box encloses the complete master exponential value.

above the circle and oval shapes: a circle or oval encloses the subordinated equivalent flat (finger counting) value.

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