Kujonai also introduced the mod(n) categorisation which means that the logarithmic additions are mod(n) clock arithmetic. This means that the actions of the unary operators on themselves are added mod(n) and thus we can think of them as acting in a kind of multiplicative way. So a unary operator acting on another unary operator is a product mod(n).

For example sign acts on the real numeral 2: sign^{0}2 =+2 and sign^{1}2=-2

sign^{1}sign^{0}2=sign^{1+0}2=sign^{1}2=-2

sign^{1}sign^{1}2=sign^{1+1}2=sign^{0}2=+2.

The indices to sign i have called signals. They look and act like powers mod(2). In polynomials they may also get referred to as degree. These notational references speak of the history of notation more than anything else, but it is important to be clear from the outset that these indices refer to the action of an operator on an appropriate operator not to numeral manipulation,numeral products or numeral multiplication. Therefore i retain the term signal. This means that if sign is taken as the nomial polynomials in sign will all reduce to signal 1 polynomials mod(2)

sign^{0}2+sign^{1}2+ sign^{2}2+sign^{3}2

=2sign^{0}2+2sign^{1}2

signal 0 is defined as the identity signal and as we have seen the identity signal on sign means that a positive sign is symbolically attached.

Now i can use a consistent geometrical representation of a unary operator as long as i clearly define what that is. For this discussion i am going to define unary operators as acting in a plane only. I am implying that i can define them as acting in geometrical space but it is a bit of a tort to do so without establishing the definition in the plane first.

To have any geometry we need a specific orientation first. This orientation is essentially not definable by any geometrical reference frame i construct with it. So to be literal it is ground 0. Because of this every reference frame is relative to the observer, and it is only by agreement we construct a common reference frame .This orientation is the axiomatic orientation. However it helps me to see that a general property of space is orientation. i have a special capacity to fixate on a specific orientation and reference from that orientation.

Another fundamental property of geometrical space is extension . Orientation and extension are logically and practically inseperable and a sensory synaesthesia. The third fundamental of geometrical space for our discussion is rotation, and again this is inseperable from orientation and extension. However, customarily we ignore the sensory synaesthesia because we are not reference free, we live in a gravitational geometrical space and that determines our orientation fundamentally and frictional forces determine our rotation.

Nevertheless we need three fundamentals to establish a geometrical space : orientation, rotation, and extension. Of the three orientation is the ground 0 and cannot be defined,and extension can then be defined as being in a certain orientation (direction) and rotation defined as around a certain orientation (axis). I am going to use the notion of axis and axes to refer to an orientation different to and from the axiomatic orientation, and to give axes therefore there own extension and rotation. Because of this an axis will have a direction in the plane referenced from the axiomatic orientation.

I will define as a scale a division iteration such as: divide into 10 parts a unit length. This iterated will produce the real numeral scale along the axes. From the axiomatic orientation i define unary operator *cycle* mod(1) to be a rotation about any axis that returns to the axiomatic orientation . I define the unit length as sweeping out a circle radius 1 circumference 2π and the axis of rotation as being always right to its direction which in a mod(4) unary operator scheme will be identified as π/2, and the axes under cycle mod(1) i define as sweeping out a plane, which makes a normal to a plane (the axis of rotation under cycle mod(1)) the definition of that plane. AS the mod(n) n increases i can define a scale by division iteration: divide 2π into n parts. However we normally use: divide the unit (radius) into 10 parts divide the cicumference into parts that are equal to or fractions of the unit (radius). By this more complex iteration we construct a radian measure of rotation.

It is worth noting that geometrical space has all these attributes but we only distinguish them under some operation,and the vector notion ought really to include rotation as axiomatic and within its definition, thus a vector has magnitude direction and rotation.

As you can read i was struggling to get to grips with this topic of "imaginary numbers" based on inadequate information and limiting assumptions which i was keen to shake off but i did not know how.

today i can look again from a freer vantage point.

Greek mathematics was based on and derived from dynamic geometry.

Greek mathematics therefore represented a powerful mixture of pragmatism and analysis of empitical data. it was the mathematikos the learning and science of the greeks, and represented greek thinking about the cosmos. The greeks were lovers of philosophy of knowledge for its edifying nature,and studied and read widely in the Sophia of other cultures and languages, but their own wisdom was produced by mathematikos, scientific thinking and analysis and manipulation.

Thus mathematic

late 14c. as singular, replaced by early 17c. by mathematics, from L. mathematica (pl.), from Gk. mathematike tekhne "mathematical science," fem. sing. of mathematikos (adj.) "relating to mathematics, scientific," from mathema (gen. mathematos) "science, knowledge, mathematical knowledge," related to manthanein "to learn," from PIE base *mn-/*men-/*mon- "to think, have one's mind aroused" (cf. Gk. menthere "to care," Lith. mandras "wide-awake," O.C.S. madru "wise, sage," Goth. mundonsis "to look at," Ger. munter "awake, lively").

math.

online etymology dictionary Etymology of the French word mathématique

the French word mathématique

derived from the Latin word mathematicus (mathematical; astrological; mathematician; astrologer)

derived from the Greek word mathematikos, μαθηματικός

derived from the Greek word mathema, μάθημα

derived from the Greek word manthanein, μανθάνω (to learn (in any way))

derived from the Proto-Indo-European root *mendh-

myEtymologyAs is established mathematiics derived from greek science based on geometry: a Geometrical science.

We are more familiar with Academic geometry than Greek science because Learning required travel and empire. Euclid was made famous by the arab empire, and his geometrical methods filtered slowly to the west by islamic influence and an education in the classics which required travel. But al kwarzim's work Al Jibr promoted indian numberr solutions and this defined algebra in the early days as Arithmetic, that is to do with

"a number as reckoned up" .

What this means is only just becoming apparent to me. We would call it a scalar number, the result of a division of a proportion,a quadratic number! It is not a "number" it is a ratio a proportioning of different portions of space. Arithmoi represented proportions of geometric figures in particular scalar values proportioned against a unity.

This fine distinction was not easy to grasp and is why confusion arose over negative "numbers" from india and arithmoi from greece. The square root of negative numbers were always going to cause a problem because of the clash of these two systems or "mathesis", doctrine of how to calculate,or as it is now called algebra.

People learned in both systems had to have a classical education and had to travel to read the sources. Therefore information was spread with a great deal of invention and interpretation, and a rigid protocol of academia.

Bombelli was fortunate to be able to travel in his work and to get sight of original papers. Bombelli also wanted to popularise algebra for italian engineers. thus over many years e wrote "L'Algebra the major part of the art of arithmetic". In this book Bombelli set out his mathesis

www.math.uoc.gr/~ictm2/Proceedings/pap285.pdfWhat Bombelli did was have a mad idea and this was to lead to the birth of modern Algebra and to predate Descartes.

www.math.unipa.it/~grim/Pre-mod_algebraQuad11.PDFAbout the time of Bombelli Napier developed his logarithm of sines, again by dynamic geometry of proportions, movement of points and lengths. Both men dealt with what the greeks called proto arithmoi, ratios that had no reduction and were pure scalars of one. These arithmoi lead to numbers that could not be represented by ratios of integers through the application of methods of mean proportioning. These methods of mean proportioning Napier accomplished by his "bones" and skill at calculations and Bombelli by his use of neusis in geoemtrical constructions, but until

Euler R. was used to refer to this method .

Euler however was preceded by Roger Cotes.

The

use of the root sign was promoted by arabic and german influence, so it is clear that in the west the word itself clearly conveyed the fact tha an algorithmic operation was being called .

Where Bombelli made a real contribution is by linking the R.Algorithm to the Elements of Euclid,rather than just a numerical estimation of a repeated fraction.

Bombelli therefore linked square rooting to geometrical manipulations called neusis thus showing that the Greeks had solutions to these questions by geometrical means. He therefore popularised Euclidean geometry.

What Bombelli showed was that you could find square root by not doing the operation of square rooting negative numbers,

but rather treating them as an operation to be canceled out when they were being squared or compared, otherwise they were allowed to just hang about!They were not part of the normal numbers they were adjugate to them yoked until needed.

So from the greeks Bombelli had a geometrical solution that gave him radicles of a certain magnitude but multiplied by these adjugate numbers, while from Cardano's input he had an algebraic analysis to guide his neusis. All he had to do was keep his nerve and notice how the sign rules worked under multiplication

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