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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 (http://www.etymonline.com/index.php?search=mathematics&searchmode=none)

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-
 myEtymology (http://www.myetymology.com/french/mathématique.html)


As 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"  (http://www.myetymology.com/english/arithmetic.html) .

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.pdf


What 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.PDF

About 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 (http://en.wikipedia.org/wiki/File:Euler's_Formula_c.png) R.  was used to refer to this method .

Euler (http://en.wikipedia.org/wiki/Euler's_formula) however was preceded by  Roger Cotes.

The use of the root sign (http://www.unisanet.unisa.edu.au/07305/symbols.htm#Radical) 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! (http://nsm1.nsm.iup.edu/gsstoudt/history/bombelli/bombelli.pdf)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

Piu via piu di meno, fa piu di meno.
Meno via piu di meno, fa meno di meno.
Piu via meno di meno, fa meno di meno.
Meno via meno di meno, fa piu di meno.
Piu di meno via piu di meno, fa meno.
Piu di meno via meno di meno, fa piu .
Meno di meno via piu di meno, fa piu .
Meno di meno via meno di meno fa meno.

Piu via piu di meno, fa piu di meno.
Meno via piu di meno, fa meno di meno.
Piu via meno di meno, fa meno di meno.
Meno via meno di meno, fa piu di meno.
Piu di meno via piu di meno, fa meno.
Piu di meno via meno di meno, fa piu .
Meno di meno via piu di meno, fa piu .
Meno di meno via meno di meno fa meno.

The issue has never been about finding the roots of negative numbers, but about what was the most understandable and consistent way to find roots: numerical or geometrical? Bombelli says , use the above rules of sign and you can use both!

The rules of sign is Bombellis operator, and encapsulate the neusis of the greek geometry. Bombelli could  see the geometrical application of his sign rules, but no one was looking for a geometrical representaion in algebra until its methods clashed with the Greek method, or rather the indian method clashed with the greek.

The operators are: sign   piu/meno  ±
            and sign sqrt sign    piu/meno sqrt piu/meno ±√±

These operators were immediately generalised to ±^p√± where p is any root but in particular prime roots.

So sign is a (mod 2) logarithm arithmetic for mutual action so
                                                   
So now if we make this sign (mod real2) and define sqrt as sign^1/2, pth root=sign^1/p then

Bombellis operator becomes
 
Now define the sign operator as rotating the measure anticlockwise through π then sign^1/2 is a rotation through π/2

sign^1/p is a rotation of π/p.

The measure is any system of measurement we place on a region to measure dynamic changes.It is used to give a measure to an object which we may call its magnitude,but it is just a scaled unit measure.

The measure is rotated relative to the object and  it also slips and slides, so some additional operators are implicit in Bombelli's operator, which are the affine transformations.

Numerically however we would use an iteration algorithm with a difference test and generate a continuous fraction/proportioning.

Thanks to Bombelli the rules of combining these approaches were made clear.

I have to admit being underwhelmed by Bombelli's operator.

I think this is because i have looked for it for a while and the idea is simplicity itself. A lot of myths have been laid to rest along the way, and i find hat it is a story as usual of apathy and disinterest not puzzlement and perplexity! What is there to be puzzled about? From the indians to the greeks to the early italians the introduction of negative numbers was a kind of nonsensical whimsy. Debt did not need to be recorded with negative numbers the chinese showed that, and it made sense not to call them numbers any way.Debts they were and as debts they had a utility for merchants and accountants. Even in india it was a fancy to go beyond debt calculations even if you could intellectually show an interesting result. NEgative results to solving quadratic or cubic equations were universally pooh poohed as meaningless. That was the opinion from Baghdad to Rome, Athens and beyond.

Because of this i suppose Bombelli toyed with the way forward for solving unsolvable equations and was willing to try anything. The problem was to find a way past what was everywhere being taken as a signal of "this far and no further"; " this way is fruitless and a waste of time!".

So i think Bombelli just pioneered and burst through what turned out to be not a wall but a tissue of conceits, lack of nerve, peer pressure and lack of confidence in exploring all ways forward.

Bombelli like all scientists had a mathesis, a praxis, but of course no one else had to follow it unless he became a great celebrity.
Bombelli had a passion to communicate Algebra to the engineers of his time, and that and his simple but effective solutions to polynomial equations of degree 5 and under did make him a celebrity in his local area of influence and beyond. But like my reaction his peers were underwhelmed by his operator rules, despite the advance in confidence they gave. Unfortunaely the distaste for negative numbers was too great to be overcome in a generation. It took time for scientists to find a use for negative numbers and that time takes one up to John Wallis via Descartes.

In the meantime Bombelli devoted the rest of his life to his operators, solving all sorts of equations through them.

The question of what these operators mean is therefore a later question, Descartes named them derisively imaginary,Cotes found their logarithmic nature intriguing and linked them to Napier's logs( of sines and cosines) and the logarithmic spiral, and Euler showed that the exponential function is linked to the trigonometric functions through them. He also renamed them i meaning imaginary. Gauss proved the fundamental theorem of polynomials using them, and Newton started to toy with the notion of vectors at around the time he invented the binomial power series that lead to Taylor.s theorem which Euler used in his identity.

A sea change therefore was necessary to reveal how fundamental Bombelli's operators really were.

The question what are they is a late question, not even properly considered in Bombellis time and requiring the great Algebraist Hamilton to give them a algebraic meaning as a group property of a fundamental nature.

Lets have a bit of fun,
Kasner (http://books.google.com/books?id=Ad8hAx-6m9oC&pg=PA91&lpg=PA91&dq=bombelli+geometry&source=bl&ots=vveKwBkpCZ&sig=ff9UwOsMcaVcdYBKTc0HI-1R3OE&hl=en&ei=Jbr8TJ6ZIsGGhQedoonVCg&sa=X&oi=book_result&ct=result&resnum=3&ved=0CB8Q6AEwAjgU#v=onepage&q=bombelli%20geometry&f=false) represents pretty much a popular view among lay mathematicians of the triumphal march of mathematics into "modernity". He is a good read actually but he still manages to comment in such a way as to make one feel that we are somehow better tan our ancestors!

In doing so he creates the impression that people were mathematicians by denotation, rather than what they were scientists and engineers and artisans and prodigies. The science of the day was mostly greek, The arithmetic mostly indian, the algebra mostly arabic , so to this mix was added numerology, astrology, and religious mysticism.

No one therefore had any love of or understanding of "less than nothing" or " cut from nothing", because every one expected "something"  from the gods even in the afterlife!

So scientists faced with this incredulity really had to tread warily about advancing an idea of "numbers" with an attribute of cut out of nothing. Even the indians with their veneration of the void had a hard time selling the idea of numbers cut off from zero, which return to the void as soon as they meet, that cannot be aggregated normally.

In india a rich imagination attributed colour, flavour,sensation and scent and even pressure or weight of fortune to numbers. The Chinese naturally attributed colour to their "number" rods, sticks, bones.

As you can see attributes are reflecting sensory meshes within the human animate. It makes no sense to attribute to nothing, never mind to something cut away from nothing!

The visual mesh gives magnitude and motion and equilibrium as attributes, and a system relativity and linkage. There are many other attributes,like system balance polarity,bi polarity etc. So how come we end up with something as dry as sign or mark?

Why can't we have different flavoured numbers?, or different coloured and scented numbers?

There is an "autistic" prodigy who has a different relationship with each number up to thousands of numbers in his imagination.
 
So can i smell a number? Of course! ijust take a unit volume of some odiferous substance and bingo i have a smelly unity that i can scale!

So why did we get rid of the smell of our "numbers"?  Because we wanted to abstract and emphasise one attribute that applied to all spatial objects that were in motion?   Rather we habituated ourselves to ritualistic motions and points of view that "fit" most of the experiences we got involved in in an activity. Builders noticed lengths of movement, chemists noticed smells and textures, with those , physicists noticed mass and density and equilibrium along with the builders,and relative motions,etc. Each individual abstracts his/her own sense of "number" which is related to a dynamic motion space of motile elements, substances and sensations.

We make sense of it, but how? And When did we start to do this as individuals and cultures?

The Bombelli operator is a version of the Brahmagupta shunaya operator. The hatred of negative numbers stems from there astrological significance of bad luck. In addition greek thought about unity meant that it was nonsensical to describe an element even a debt as cut of from "nothing". This misrepresentation of Brahmagupta's shunaya is as old as it is persistent. Shunaya means the infinite potential of the void.

So Bombelli's operator is an example of the powerful and mysterious workings of the void.

 So now if we make this sign (mod real2) and define sqrt as sqr^1/2, pth root=pwr ^1/p then

Bombellis operator becomes
 
Now define the sign operator as rotating the measure anticlockwise through π then sign^1/2 is a rotation through π/2

sign^1/p is a rotation of π/p and sqr^1/2 as {extending(translation)(a,b)bisection(a+b)circle(centre: bisection)translation(a) sign^1/2extend (to intersect circle)} where a*b is the magnitude i want the square root of.

Sqr¹ is defined as {extending(translation)(a,a)bisection(2a)circle(centre: bisection)translation(a) sign^1/2extend (to intersect circle)translate this length via this length to generate square area} where a is the magnitude i want the square  of.

The set of operations described is not tight notationally but it is clear that it is an algorithm which contains translations rotations and constructions like bisections which are probably formally equivalent to reflections. Of interest is the multiplication algorithm as a motion of a measure via (in) a certain direction for a certain length(not time!).

Bombellis operator rules can now be written

Piu via piu di meno, fa piu di meno.
Meno via piu di meno, fa meno di meno.
Piu via meno di meno, fa meno di meno
Meno via meno di meno, fa piu di meno.
  Piu di meno via piu di meno, fa meno
Piu di meno via meno di meno, fa piu
  Meno di meno via piu di meno, fa piu
Meno di meno via meno di meno fa meno.

From which i can define  {sqr^1/2sign¹}= sign^1/2 or sign ^3/2
 

And Brahmaguptas shunaya operator can be written in more general terms


Where equilibrium^1,0 refers to the balancing attributes in the void which facilitate dissolving and resolving into and out of the void,inverse attributes or condensing and sublimating attributes into and out of the void.

Whether i have to stick to a bipolar arrangement i do not at present know but for the time being the structure is that of 2 balancing opposites, requiring only a (mod2) arithmetic.

And where  transformation^1/p is an algorithm based on iterations of affine transformations with actual measurements being taken where necessary to construct or read off the measurement for construction. Therefore it relates to a constructive spaciometry and not an abstract one, although analogies should be applicable.

Since construction depends on measurement we see that unity is fundamental to this idea of transformation, and by contrast the void is fundamental to this idea of equilibrium.

The BG shunaya operator has some more rules which i will add later.

I am in the process of continuing with this thread but am in flux between some insights and their impact on the topic. Chiefly aggregation rhythms and the algorithms associated with them, their relation to logarithms and the structure of them, plus the dynamic magnitude paradigm and its relation to the iterator- convolution structure of algorithmic transformation.

The notion of recursive definition which has to be made explicit with an iterative counter the structure denoted by : convolution. Therefore i need to signify the three terms and their relationship, which is unavoidably "recursive", or some may think tautological.

The Logos Response to shunaya in all sensory meshes and particularly the proprioceptive mesh.

A clearer exploration and meditation on the "radial expansion/ spaciometric rotation"  relation and whether this is indeed a fundamental relation which provides a tie for all measurement  of dynamic unit magnitudes, through the relationship to roots of unity.

And whether a more general transformation group theory can be delineated which contains within it a theory of algorithmic measurement that will structure the units of measurement at all scales and across all situations.

In the process of which i hope to lay the ghost of "imaginary" firmly to rest!

Following up on the flip algorithm which is an algorithm we apply when measuring using a measure of choice: condition; flip: assign orientation marker; carry out aggregation rules.

I was thinking of a more general set of orientation markers(refer to polysigns for the notion) and the idea of a radial came to mind.

The marker is which is a bit long winded  but not as long as sign , and a bit more flexible than Tim Goldens  coordinate suggestion (http://www.fractalforums.com/theory/simple-algebra-to-the-3d-mandelbrot/15/)

Thus   +  = 0

  +  =  +

are  distinguishers for  roots of unity.

When Kujonai and Tim Golden introduced the polysign idea i had no knowledge of roots of unity. So at last i can say that the polysign notion is the exploration of roots of unity and beyond.
   is the distinguisher for the 4th root of unity

Thus ( )⁴ = ⁴ =

and ( )⁴ is the same as i⁴

This distinguisher separates the magnitude from the "sign" again and helps to give a clear link to the geometrical forms underlying all this.

It also i hope gives a clear link to De Moivre's formula for the roots of unity and might generalise to the sphere( horribly i think)

So any comments or further thoughts?

I really would welcome some input here.  ;D

I have moved it here because it is the same issue, and the other thread is more general.

The issue is that the root unit identities i define later have an action, but it is not multiplication. To define rules for this action i need to understand the De Moivre formula.

It is important not to make the same mistake i initially made in polynomial rotations which was not to realise that orientation and rotation though linked are not the same.

So the distinguisher is separate from the magnitude because the distinguisher is an orientation. Like sign the have no magnitude or action they just indicate/

I have chosen them to indicate the roots of unity.

Geometrically i have to construct the roots of unity and name them with these markers to distinguish them. So all the action takes place in the De Moivre formula and this is where i have to derive the action rules.

I have to do this circumspectly because everything else depends on it.

The firsr issue is that De Moivre already has in his equation, so i have to clarify if this is tautologically the same in which case why bother, or is it different and more fundamental?

So some help please would be appreciated. Tinab! :dink:

Here (http://ndp.jct.ac.il/tutorials/complex/node8.html#polar_roots) is a version of roots of unity to help us out.

Here (http://ndp.jct.ac.il/tutorials/complex/node3.html) is a set of propositions to guide the fundamamental understanding of the properties of the "complex" group of magnitudes.

Now notice how the presentation (http://ndp.jct.ac.il/tutorials/complex/node4.html) sidelines the polar version favouring the algebraic, placing algebra over geometry.

The polar form (http://ndp.jct.ac.il/tutorials/complex/node5.html) is the geometric representation chained and wrapped around with algebra so closely you struggle to see the simplicity of the geometry and the absolute trigonometry of it all!

De Moivre establishes the actions i am interested in here (http://ndp.jct.ac.il/tutorials/complex/node6.html).

So using this material i hope some one will help sort out the root s of unity distinguishers and then look at whether combining these orientations with unit magnitude creates any procedural problems.

√-1 and the Bombelli operators are necessary and sufficient to define all other  aggregations. So these are the rules of actions of the flip from one form of unity or unit to another. What is the conditiob for the flip?

Well in bombelli's day it was a stage in the calculation of a root which had a - under the √ bracket or rather the rhetoric stated R. -number.

There does not appear to have been any other flip algorithm for about a century when De Moivre uses it in reverse to flip from the new unity to the normal integer unity. Not a new thing,just an application of the Bombelli operators in a trigonometric setting and From the new units!

So the condition for the flip was the instances in the bombelli operator specifically dealing with the new units, and flip was being called constantly in this setting.

How come he chose the unit circle trig (http://library.thinkquest.org/22584/temh1600.htm)? It seems that his concerns with probability theory and with infinite series and roots  arithmetic and geometric series. He worked on Cotes paper (http://www.maths.tcd.ie/pub/HistMath/People/18thCentury/RouseBall/RB_Engl18C.html) on the nth roots of 1, and picked up from Cotes the notions involving trig functions and the √-1. Cotes (http://www.docstoc.com/docs/36055559/The-combinatorial-structure-of-trigonometry) in turn picked up his intense interest in the trig functions From Napier , through his Logarithms .

De Moivre was able through this education to work comfortably in a new "unity" along with Cotes using Bombelli's operator to direct flipping from the new to the old.

I am not able to say what De Moivre was thinking, but Cotes was definitely thinking of harmonising all measures, and did not tremble to include the so called "imaginary magnitudes" in Logarithms. Cotes was an astronomer and was keenly working to map the heavens. A reference frame and a a "calculator" was of great importance to him.

At this time sequences and series seemed to become prevalent. Where did they come from? Repeated fractions? Logarithms? Root formulae?

Also at this time Leibniz was exploring his monad  (http://en.wikipedia.org/wiki/Monadology) theory. The 16th century in the west was truly a renaissance.

There was a zeitgeist seeking a more fundamental harmony based on more fundamental unit revealed by differential calculus and conservation of energy, and momentum results in dynamic systems.

It seemed and seems inevitable that it would be found, and such rapid progress was being made towards its realisation that many felt justified in venturing the most crazy ideas, just like Bombelli. Soon science spawned an incredible technology for measuring and building mechanisms, and electronics. John Dalton (http://en.wikipedia.org/wiki/John_Dalton) updated the greek notion of the dynamic atom and the solution it appeared had been found.

But the initial explosion of invention and discovery was curtailed by calls and concerns for "mathematical" rigour, and the imaginary magnitudes were squeezed out onto a shaky limb and gradually replaced by vectors.The spherical trig of the 16th to 18th century went on the back burner until Quaternions which flared again briefly until vectors and tensors covered their potential glory.

So the way forward i think was laid out by De Moivre and Cotes, and in a sense buried in abstraction by Lambert.

I would start then from the work and method of De Moivre and Cotes who made √-1 the new fundamental unity and trigonometry and the circle the structure of this measure

When that is accepted then suddenly their are an infinite , dense collection of unities which we can accept from roots of the negative magnitudes which all are distinguishable by orientation.

Kujonai, Tim Gold, Fracmonk. Three members i now understand as exploring this area of roots of unity in the same vein as Cotes and De Moivre.

Because of this , i am now able to tackle Kujonai's system (http://www.fractalforums.com/theory/split-complex-ternary-algebra-xd/?topicseen). And i do it in the way Fracmonk outlines in his paper.

Kujonai could have done this i guess, just as Fracmonk has done, and maybe he should. All i can say was i was not able before to go in their direction, but their doggedness and persistence i am going to use as a guide.

My contribution is to try to free up the notion of sign, not the mark of sign or the notation of sign but the notion of sign.

The flip algorithm gives me an algorithmic basis to understand the difference we are distinguishing. Kujonai's use of mod() was so innovative it started me n this road of enquiry, Tim Golds explorations provided me with some pretty pictures that gave me a handle on the distinctions and rules. Fracmonk showed me how to go from different roots to programme them into Quasz and get a 3d picture. and Lazarus Plath has not got back to me yet, but his programme apps show me the tangible trochoidal effects of these roots of unity when used as the controls of relative spinning magnitudes.

Yes i am pursuing a strange new world, and i guess i want some help from any one who would care to join in, or comment . After all Tinab! :dink:

One of the things i have explored recntly with quasz is orientation assignment. Terry has  some controls he calls slice in the parameter section. Some of my recent gallery items have utilised this.

Doug  points out that Hamilton did not use quaternions the way at least i have in quasz and the way Terry has in his mandlebulb built in functions. Hamilton actually used a vector form with a unit vector for the reals and a unit i vector for the x, unit j for the y and a unit k for the z ordinates of the cartesian coordinate measure. However there is another level of confusion brought in by the Euler equation where x may refer to the x axis , geometrically it refers to an arc on the surface of a unit sphere, or in the planar case just an arc of the unit circle.

This is not really confusing, it is more that i have not been taught spherical trig or spherical geometry before!

Well yes i have done navigation on a spherical  earth in terms of eastings and northings but nothing of the rich significance of this area of math to our fundamental view and measurement of the universe!

Any way, that aside , Terry has made it possible to assign the cartesian ordinates(orientations!) to different roots of unity i, j, k!

He only programmed for the usual 3 so that's cool, but if i can get this orientation figured out this may be a possible extension he can make. These will then become generalised coordinate systems.

Laz Plath i think has figured out how to do this in his apps, so i would like to talk with him about it. 

I am going to work through this equation of Kujonai and illustrate the roots of unity, the flip algorithm, and the Kujonai operator!

Kujonai and Tim Golding have been working in the new fields of yoked magnitudes "discovered" by De Moivre and Cotes.

All these guys have started with the new units as the basis of their exploration, and they all take i as a unit magnitude with a set of aggregation rules that are what i call a flip algorithm.

Without the flip algorithm you can not relate your new units to each other , their yoked units or to the unit of natural scalars.

The first flip algorithm was i think the Brahmagupta- Bombelli operator for signed "numbers" or precisely : signed scalars.
Then the Bombelli operator for the square root of signed scalars which introduced the notion of imaginary magnitudes.

Utilising these flip algorithms Cotes began to explore trigonometrically these imaginary magnitudes as a more fundamental measure of magnitude. He did so with  Logarithms, and attracted the attention of De Moivre. Both were in  sense collaborators and friends and students of Newton, who shared time with them both.

De Moivre in collaboration with Cotes came up with the De Moivre Formula, in particular to solve a problem Cotes was tackling on infinite roots of unity. Cotes died before he could complete his work, and De Moivre finished for him. On the strength of his work with Cotes on infinite roots De Moivre was elected to the Royal Society.

De Moivre had added another rule of aggregation to the Bombelli operator. In effect he has generalised the Bombelli operator and so the flip algorithm. De moivre has revolutionised the notion of sign, but he starts from an established sign called men and piu, negative and positive eventually notated - and +. This last step is unfortunate, because + and - became symbols for aggregation, and the potential for confusion was realised right at that moment of adopting this convention.

Aggregation and sign are to different algorithms now, or we could define aggregation as an activity/operation and sign as an operation modifier. However my analysis places sign and aggregation in an algorithm i call flip, that is there is a process of parsing data and assigning attribute based on conditional or contingent sequences of actions. These attributes then are utilised by an aggregation algorithm to control or direct aggregation.

Flip tests a condition and assigns an attribute according to the condition tree. Once this assignment takes place measurements are flipped accordingly, and the aggregation structures start at that moment. The flip yokes/ relates the aggregation structures
to one another.

The initial attributes were men and piu and they came from aggregation calculations to find roots. Bombrlli cut through the confusion and pointed us back to the geometry of which the calculation was a measurement process. Men an piu had straight forward geometric reference: orientation relative to the measurement instrument.

Thus i venture that the fkip algorithm assigns orientation, and that is where we have to begin.

So lets play for a bit.
The chinese had red and black rods. Colour is entirely appropriate for a solid object distinguisher. So red was annihilated by black was the aggregation rule.
But measurement involves dynamic magnitudes! a tape measure, a moveable rod, etc and measurements are not static, so we utilise a static unit, static relative to us, and allow it to plethorate in any direction that needs to be measured.The aggregation rules are then set according to the direction and applied according to the project in hand.

So our everyday use of measurement is vectorial and orientation is assigned by a flip algorithm as we go about our measuring. We even write it down on sheets in orientation terms!( length, width, depth, height!)

These orientations are the attributes assigned by the flip algorithm and the aggregation is done accordingly. Bombelli's operator assigns the orientation height, the orientation width and the orientation length etc, but it assumes you and i know that bit! Bombelli focused on the relative opposite motion in measurement, which he knew was obtained by π radian rotation from his work involving neusis. What he did not know was his rhetoric would be changed to notation in such a way that much of what he said was "lost" or ignored!

Thus Bombellis operator was assigning orientation to measurement, and he assumed students would know that. owever i o not think that Bombelli knew the full significance of his orientation assignment, but he was getting there. He it was who used the terms adjugate and conjugate etc.

Think our "numbers" could have been called north and south numbers or east and west numbers!  :embarrass:

This links me directly back to trigonometry, where distinguishing  the orientation of measurements was taken as a basic requirement. A requirement without which surveying, land measurement, wall construction, astronomy etc could not have been done. Tis lead to the codification of oriented measurements, by the application of ratios in a right angled triangle, the sine ratio.

The sine ratio in particular linked three oriented measurements to the orientation of the hypotenuse. This orientation was called an angle, and the angle measure and the sine ratio developed together, indivisibly. Thus he angle measurement is the orientation of the hypotenuse, and our measurements flipping from one orientation to another are distinguished by the angle measure between them.

The flip algorithm therefore could and should replace + and- by radian measure related to the sin ratio primarily but the trig ratios generally. The sin is the primary ratio historically best developed.

Thus we could have 0^c numbers and π^c numbers!

Omg it is so wonderful

De Moivre and Cotes realised that the trigonometric ratios as used in spherical trig and circular trig and Logarithms behaved in the same way as Bombelli's operator. They knew that the Greeks and the Arabic mathematicians had been using this operator for centuries, but not in the form Bombeeli and Brahma gupta put it in. They were the first to see the connection and to use it , but even they did not realise what a big deal it was, because nobody had invented the number line concept we use today to define numbers! so they were not excluded from a measure concept, and never have been!

The Number concept is what caused all the difficulty with the so called imaginary magnitudes! For cotes and DE moivre they were clearly trigonometrc magnitudes in the everyday sense and were a way of riting cown compactly trig calculations!

They never got a chance to fully explore what they had discovered!

I am now fascinated by the interplay and relationship of Roger Cotes,  Abraham DeMoivre and Newton and John Napier. These four are the Britsh Scientists that shaped the Face and heart of Mathematical Science as we know it today. Had Cotes lived we would have learned a thing or two!

I believe Cotes was working on a foundational revolutionary Theory of measurement. In every way he reminds me of Albrecht Duerer who combined the Measurement theory of his day into his major work, and spotted the importance of the cardioud curve for natural description. Cotes spotted the importance of the logarithmic spiral for natural description, and his only major work was on Logarithms.

But the four of them laid the foundation for vector theory probability theory, Mensuration theory,Complex field theory, computation theory, Tensor theory, Analytical theory. the list goes on and on...

The death of Cotes took away a driving force for De Moivre , who had survival concerns of his own, and Newton at this time in his life could not be bothered to explore new realms, So thheir groundbreaking work was shelved, and only parts of it remembered until Caspar Wessel (http://en.wikipedia.org/wiki/Caspar_Wessel) "rediscovered" the obvious and natural and centuries old geometric link which inspired Argand and Gauss and Euler.

Wessel deserves his late accolade because he knew from his work with trig and surveying that the Bombelli operator was all about direction  "On the Analytical Representation of Direction"

  (http://www-history.mcs.st-andrews.ac.uk/Biographies/Wessel.html)

And guess what? He also inspired Sophus Lie (http://www-history.mcs.st-andrews.ac.uk/Biographies/Lie.html=).

So in short, the work is done. Replacing + and - by orientation markers which directly reflect Cotes innovative radian measure is not only possible it is historically demanded!
Cotes Harnonium Mensuram reflects a theory of measurement that goes backto Albrecht Duerer an the 16th century Italian Mathematicians, who are heirs of the Greek Arabic Indian Chinese Axis or network of influence.The simple devising of the radian measure harmonised Spherical and circular trigonometry through the arc.

Some crazy relationships that Cotes     an de Moivre noticed i am sure ,as this is par for the course for spherical trigomometry.

Sin 0 * sin = sin 0

But sin * sin =sin
 and  sin *sin * sin = = sin) - sin()
 
in other words multiplying sin by itself tracks the observer around the unit clrcle circumference proportionally like a clock ticking! But this is exactly the process Napier used to calculate his logarithms!

What do we have? Crazy! clock arithmetic is related directly to logarithmic arithmetic!
 And measurements in the 0 orientation have no effect on each other when multiplied through  or with sin but measurements in any other directions or orientations rotate each other when multiplied with or through sine!


Therefore sine is going to be one of my fundamental actions for orientations as a group, and through spherical trig i hope to explore the group properties of 3d orientations. Of course if i use sine i use or imply all the the trig functions, and all the logarithmic functions and all the arithmetic and geometric series that can be linked to sine

What i love about trigonometry is the crazy whorls of relations it gives!

From trig i can see Bombelli operators, Kujonai operators, Symmetry relations , flip algorithms , log like relations, cross product and dot product relations and so vector relations and mod arithmetic relations. This is especially neatly organized by the unit circle/ sphere. In every sense the use of + and - as direction signs is confusing.

In orde to remove them i must make clear the use of aggregate gate signs +(mod()) and disaggregate gate sign -(mod()). But before i do look at this

Imagine that and are orientations. as are

Then sin =sin =0

and sin =sin =1

This is because i do not use + or - for sign  i use orientation. Each case is clearly distinguished by the orientation! Isn't it?

What can i say about sine and unity?  I say sine is a unit for scalar plethoration. Any sine scalar i can denote  with a prefix and any opposite sin scalar with a prefix thus i naturally write
 

for trigonometric sine scalars where is an angle magnitude not an orientation.

Now a similar examination of the cosine behaviour gives

Then cos =cos =1

and cos =cos =0

Each case is again clearly distinguished by the orientation isn't it?

I can now  say about  cosine and unity that it  is a unit for scalar plethoration. Any cosine scalar i can denote  with a prefix and any opposite cosine scalar with a prefix thus i naturally write
 

for trigonometric sine scalars where is an angle magnitude not an orientation.

The aggregation structure for these scalars will be as expected


Where the +(mod()) gate has not yet been determined but involvrs π/2 in some way, maybe as an angle magnitude.

This seems strange , but it is more familiar as cos + i sin!

It also highlights the special nature of this aggregation because now it seems simple to inquire after
cos + i sin or cos + i sin

etc.

I like this draft result because it suggests the way forward.

Any one ot any suggestions?

It is good to have imagination, but even better to have premises on which to imagine.

Gerardus Mercator (http://books.google.com/books?id=husGAAAAYAAJ&pg=PA378&lpg=PA378&dq=Gerardus+mercator+john+napier&source=bl&ots=jrcgZ3mxoB&sig=0g8fdqlXT4NLw-W5jEOs0l8ofFM&hl=en&ei=Q2c6TdPaJaKM4gbmjryxCg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBUQ6AEwAA#v=onepage&q&f=false) has now to be includedin the background story and Newton has to take a pole position!

Bombelli and Mercator were contemporaries with Napier. Napier influenced them both including Tycho Brae and Kepler . There are two Mercators so I will clarify later. Napier was inspired by the earlier Mercator and the plight of seafarers in their desire for safe navigation. His method of constructing logarithms applied particularly to the sphere using spherical trig he could formulate calculations for finding position.

However the later Mercator was in need of a way to draw flattened maps from a sphere to a plane and needed some way to do this. He eventually discovered the logarithmic series, but not easily. Newton among his early works discovered the logarithmic series, looked at roots of polynomials and welcomed imaginary numbers which means he utilised Bombelli's operator.

He extended Descartes rule of signs for finding / calculating the number of root to a polynomial to include imaginary numbers.

The ground was set for de Moivre by Newton . De Moivre was able to take Newton's ideas, sketches, geometrical synthetic notions and place them in a new form which would become named analytical. De Moivre did come up with the De Moivre equation before cotes, but that hardly mattered because it had been suggested by Newton before either of them! . Newton's work on the logarithmic series brought  crashing together the:- sine, and roots of equations, and series; in the forefront of any student mind  influenced by him. De moivre showed how to extract an infinite number of roots, based on Newton's ideas, and continued to apply Newton's ideas in wider fields than Newton had time to plough.

Why did De Moivre link cos and sin as he did?

Because Newton suggested it by his use of Bombelli's operator.

Bombelli's tool or model vector was a set square, a right angled carpenters measure. Napier's tool was a huge Bombelli vector. Newton grasped the utility of the trig ratios in all sorts of measurement requirements, and conveyed that in his work. When Cotes began to converse with him, Newton came to realise he had someone capable of understanding his work. Newton opened up to him and De Moivre as he rarely did to any other. He told and showed them things we are only now becoming aware of in his scrapbook!

In any case Cotes once again tackles the general solution for roots of polynomials and the problem of navigating the globe, in both cases he utilises the logarithmic series based on the sine as a starting point. De Moivre was there helping him and explaining how Bombellis vector worked as an analytical tool.

was De Moivres construction of a set square! He did not need to carry one physically, this odd aggregation structure actually did the trick.

Meanwhile Cotes was able to demonstrate a neat solution to the latitude problem based on the logarithmic spiral, which he developed from Newton's influence and his own table of logarithms and differential identities based on the methods of fluxions, which as I have pointed out are applied trigonometry of moving objects.

So it is hardly surprising that he comes up with in his Logometrica with De Moivre around, because what this means is: take napiers method of making logarithms and apply it to the sphere and you will find the lines of longitude intersect a loxodromic spiral according to this rule. An equiangular spiral is what results.

At the time these "numbers" were thought of as perfectly acceptable magnitudes, if properly distinguished and aggregated correctly. Descartes never understood them, nor did he want to but Bombelli made them "real" magnitudes to be learned and used. Newton did , and he taught De Moivre and Cotes to do the same. The three of them therefore had unusual success because they embraced these magnitudes readily. If De Moivre had not done so I doubt if Cotes would have come up with his logarithmic, loxodromic spiral equation a generation before Euler.

Many are told Napier used base e for his logarithms, but he did not. However Cotes calculated his own logs using the logarithmic series Newton showed him, and he did use and calculate e a generation before Euler.

Despite sounding complicated and looking long winded logs can be calculated for any number of geometric curved ratios, and it is indeed a marvel that the sine ratios can be put into such a sequence . It is even more amazing that they can be measured by  a series! Geometrically though it is very easy to see how and what to do. Consequently Newtons discovery of the binomial series is a first result i think from his growing understanding of sequences and the applicability of logarithmic construction to solving curve compounding from tangents.

Newton did not invent curve compounding, he simply saw further than his contemporaries the geometric construction for the compound interest curve and went from there.

Many forget that calculus always has been a geometrical tool, and i can safely say that i was surprised to see it used in geometry when i first came upon it, because i was mis-taught its relevance to trigonometry.

Anybody know when "imaginary" numbers became "complex" numbers and why?

I get the feeling that they are still a "dirty little secret" as far as maths goes.

So orientation does nothing!(Apart from distinguish that is, and that is very important and fundamental).

 To multiply different orientations is not possible. But to rotate to different orientations, and to multiply " via" different orientations does make sense

a*b=ab

a*b=ab

a*b =?. we could define it as a*b*sin( -) where is perpendicular to and and < . The result is an area magnitude.

 *a=()a is a rotation by radians anti clockwise along the circumference.

/a=()a is a rotation by radians clockwise along the circumference. The circumference for all these actions is along a great circle in a  unit sphere.

now () = ()

so in general /a=()a is a rotation by radians clockwise along the circumference, but we have to take into account mod(2π) when writing the final orientation.

Now for yoked pairs
()a + ()b+()c+()d
=()(a-c)+()(b-d) or

()(c-a)+()(d-b)

And for yoked triples  

()a + ()b+()c+()d + ()e+()f

=()(a-d)+ ()(b-e)+)(c-f) or

()(d-a)+ ()(e-b)+)(f-c)

And yoked 4's for comparison
()a+()b + ()c+()d+()e +()f + ()g+()h
=()(a-e)+()(b-f) + ()(c-g)+()(d-h)  or

()(e-a) +()(f-b) + ()(g-c)+()(h-d)

I have to admit, the notation is off putting in my view. Any suggestions?

The ^c is starting to look like a power instead of a distinguisher.

Now i would agree to calling these complex magnitudes! ;D

The pattern is now being established for aggregation and the +(mod(rotate π)) or +(mod(2π)) aggregation gate is fundamental and clear.

The fractal structure of the different yoked roots is also emerging, as i can further simplify in the roots which are 2ⁿth roots of unity, thus leading to compactness of notation.

The flip algorithm is clearly in action  and is fundamentally necessary.
The relation to mod() arithmetics is there but not clear yet.

And the operation of rotation is only partially explored.

The logarithmic connection has not been addressed yet, and the relation to De Moivre's magnitudes only hinted at, and it is not clear yet if these behave like De Moivre magnitudes.

So any one care to help? Bib maybe?

All jump in whenever you like! ;D

Just thought: relativity demands that I have an orientation measured relative to any surface as well as radially to a sphere.

What do yo think to ^r for radial ^c for circumference measure placed before the greek symbol thusly

, ?

The radial is the orientation radial, and the circumference measure is an arc of a great circle on the surface of the unit sphere?

what would you like to know about roots of unity?
the nth roots of unity can expressed as
exp(iπj/n) = cos(πj/n) + i sin(πj/n)
where 0 <= j < n
  from bethchen (http://www.fractalforums.com/let's-collaborate-on-something!/and2350and2339and2381and2337and2354and2348and2375and2341-(maand7751and-t4935/msg25553/#msg25553)



Nice! :-*

a*b =?. we could define it as a*b*sin( -) where is perpendicular to and and < . The result is an area magnitude.

*a=()a is a rotation by radians anti clockwise along the circumference.

/a=^r()a is a rotation by radians clockwise along the circumference. The circumference for all these actions is along a great circle in a  unit sphere.

now ^r() = ^r()

so in general /a=^r()a is a rotation by radians clockwise along the circumference, but we have to take into account mod(2π) when writing the final orientation.


Now this is getting somewhere.

But i have to be very careful because i am using 2 magnitudes now not directions "suggesting" appropriate magnitudes.
is a unit magnitude direction distinguisher assigned or attributed to only one radial on a unit great circle.  

is a magnitude of a part of the unit great circle circumference . Now it is assigned a direction relative to rotating around a perpendicular or orthogonal axis, orthogonal to every one of the radii in the circle.

So i  also have another constraint on the , and that is a great circle rotation direction. I could use for anticlockwise and ^c for clockwise rotation.
The relations above then become:

*a=^r()a is a rotation by radians anti clockwise along the circumference of a great circle.

now ^r() = ^r()=^r()

  and i see here the familiar Bombelli operator --=+, in this case written as disaggregating a clockwise rotation is the same as aggregating an anticlockwise rotation.

This is written so formally to convince you that the distinctions i am trying to make have a real sense that is symbolically the same as the traditional Bombelli operator.

so in general /a=^r()a is a rotation by radians clockwise along the circumference, but we have to take into account mod(2π) when writing the final orientation. This is strange, but for now i will just say it is a definition.

I know i am close because i woke with this :



Which is complicated to write, especially in latex, but is geometrically straightforward, and it is based on these discussions (so far with myself!- any body out there want to help?) and the method Napier used to construct his logarithms.

How can i get to Cotes formula from here?
What constraints or definitions do i need to make this work consistently?

Here  (http://www.fractalforums.com/mathematics/foundations-of-mathematics-axioms-notions-and-the-universal-set-fs-as-a-model/msg26361/#msg26361) is some blue skying i did earlier which might help.

Help, anyone? :sad1:

I made a mistake of sorts, but it is very revealing.



means that the result of finding the sin value is oriented to the ^rπ/2 radial of the great unit circle.
However the calculation depends on a right angled triangle, and every line in the triangle has an orientation. The orientation radial of the hypotenuse is required to calculate the magnitude of the opposite side of a right angled triangle, That is what i forgot

o=rsinø

so this part of the procedure should read

=
 These are expressions to be read not rushed. When read steadily they enable one to construct a geometrical figure which represents the action of the procedure. it is becoming apparent that these are algorithms controlling or describing the dynamic transformations of these magnitudes

These dynamic magnitudes have 3 descriptors of dynamism: orientation radial,rotation radian magnitude, and spatial magnitude.

These are consequently more complex to deal with, but also more rewarding, as you feel the transformations are occurring before your eyes and at your request.

The corrected algorithm is :


This makes perfect sense as a logarithm table, so why is included in DeMoivre and Cotes?

The answer to my question is going to be found here (http://docs.google.com/viewer?a=v&q=cache:NVRtElo41fUJ:www.maa.org/editorial/euler/How%2520Euler%2520Did%2520It%252046%2520e%2520pi%2520and%2520i.pdf+De+moivre+Cotes+wallis+circle+roots&hl=en&pid=bl&srcid=ADGEEShPZtvKnv8oPfwMNKC-zfUUve7ZETWearwKFXRW27f0VuxrGX_RsM4h9u-q9ylQPF3xtGSlypdcr7ozmmAaPhOxb8K4GmU1MikU1d7XOIZwbcmpPjPIQcPYCmTx4obwqBoRAQPN&sig=AHIEtbSyUECO_0dU2JYWMYSMwOxEUF6Tfw).

Anybody seen this  (http://docs.google.com/viewer?a=v&q=cache:Ju5NmS6MUcIJ:home.sandiego.edu/~langton/elog.pdf+Bernoulli+and+imaginary+numbers&hl=en&pid=bl&srcid=ADGEESjXXQh08NtnZsxXvdSwlMZjKr5V8zuSuDmswH3VcokwVMG1SDdVt84MNQtdJ_HTJiYkeq2t5OiHhxa6UMZ8DssOKcq9a_DZWSK_0eso9KhCXcKWnfkKNwCcx6eEAn4KHi4BbNAE&sig=AHIEtbSrunHwomYebSxE0HGD5SlV0Ywg1g) before?

In fact the answer is with Newton and Wallis (http://books.google.com/books?id=VO82AAAAMAAJ&pg=PA15&lpg=PA15&dq=Newton+wallis+logarithm+roots&source=bl&ots=2uAq3HdZlH&sig=MxLzCj6LEWcohC9hfNvU__x1P7c&hl=en&ei=a7dATfyZKoqv4Ab4sujLAg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CBkQ6AEwATgK#v=onepage&q&f=false) as i once surmised,

Wallis (http://www-history.mcs.st-andrews.ac.uk/Biographies/Wallis.html) himself being the source of so many of Newtons ideas

And here it is Newtons spherical trigonometry (http://books.google.com/books?id=eQfZV7a7cUsC&pg=PA168&lpg=PA168&dq=Newton+wallis+spherical+trigonometry&source=bl&ots=W3hhm5Wm0Q&sig=TsreF7bC_6HuA1VJ8TIqT1yojJE&hl=en&ei=lcRATabLB8KJhQf_rIWSCA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBUQ6AEwAA#v=onepage&q=Newton%20wallis%20spherical%20trigonometry&f=false) the same which or a progenitor of which inspired De Moivre and Cotes


De Moivre was following Newton in using trig functions to solve quadratic and higher equations.

It is clear from my investigation that both Cotes and De Moivre were privy to papers and idead from Newton that others never heard or saw. Newton himself was inspired by Wallisa and schoeten in the acceptance of complex numbers, but i think it was De Moivre who actually combined the trig and complex in the style of Newton to solve equations Newton did not have time to. Similarly Cotes applied Newton's logarithm algebra to solving harmonic means and ventured into spherical trig logarithms, using his own differential identities and logarithmic tables he constructed himself in the style of Newton,

It was De moivre who in 1707 came across a version of and informed Halley and cotes of solutions he had obtained relevant to their search for a longitude convention to augment the navigational spherical trig,

So i need to look more carefully at the properties of these notations to see if i have missed something else in relation to solving polynomial equations of arbitrary degree 

The thing i missed is Viete's Law of Cosines. (http://en.wikipedia.org/wiki/Law_of_cosines).
There is no doubt that Wallis and Newton knew it and employed it in calculating the compounding of the tangents to an orbit.
De Moivre Took it to its Ultimate Conclusion.

Wallis had already revolutinised Conics, by using a Cartesian approach, and through his formula for the circle, guessed that certain polynomials/multinomials whose equations could be classified as describing points on a circle in the x,y plane would have solutions involving imaginary magnitudes. It lead him to conjecture that imaginary magnitudes had a representation on the plane!
He saw that it would apply to all closed conic curves and venttured to explore parabolic and hyperbolic in these terms.

He trumpeted the algebraic Geometry over the use of cones and planes, that is over the use of carpenters models created by rotation and slicing. Therefore he trumpeted his algebraic tools over the carpenters tools. He investigated calculus before Newton, but was only able to lay the groundwork for Newton to take it further, as he hoped.

Newton of course did so, and immediately applied it to solving equations of the nth degree in cartesian form, and thus discovered polar elliptical and other coordinate systems, a generalised coordinate measure, inspired by Wallis's work, in 1686 onwards while at his family farm. Newton put these notions to one side to concentrate on his fluxions!

De Moivre then had a rich background to draw on, including Walis's groundbreaking work in trigonometry both spherical and planar, and a loyal if not tentative tutor in Newton. He also came into tje orbit of the brilliant Cotes who shining like a star next to Newton was in his ascendancy before his sudden death. The two collaborated very closely it would seem, sharing results and notions to each others mutual benefit.

Meanwhile, Wallis kept a wary eye on the continental mathematicians in case they stole a march on Newton and his Ideas. It was Newton's tardiness to publish that lead to the controversy with Leibniz, and the man who knows the truth of the collaboration between english and continental mathematicians was Wallis, and he knew how easily continental mathematicians plagiarised others work without acknowledgement!

The political times were those of the puritans for whom uprightness of moral character was fundamental. Thus to claim a lesser's work because one was of nobler birth was morally reprehensible to Wallis, and politically fraught with implication. Cromwell had deposed the king for moral Corruption.

Actually the thing i missed is polynomial factorization (http://www.google.com/search?q=complex+factors+of+unity+De+Moivre&hl=en&client=opera&rls=en&prmd=ivns&ei=l1JDTZWoJZLH4gbF983nDw&start=20&sa=N).

It is not surprising really with everyone shouting"roots of unity"

I actually have to start with acceptance of i as a "number" which it is not, it is an oriented magnitude with a bombelli operator controlling the orientation and thus the aggregation structure and summation. The orientation is assigned by a flip condition and the intention and construction of the measurer, and only dynamic systems flip or are motile, thus i is part of a dynamic measuring system and concept.

So as a "number" i would expect factorsor ratios, and this is where D Moivre started: he wanted to discover the unit of these complex, or imaginary numbers and familiarise himself with the way they added subtracted, multiplied and sivided and formed factors: in short how they behaved as numbers or magnitudes.

In De Moivres day number , magnitude, quantity arithmoi all meant the same thing essentially, just in the different languages that were used for scholarly communication. It was virgin territory and his investigations would overturn the concept of number down to this day. However human animates are a sentimental breed and they have clung to the concept number, even defining it so rigorously that they excluded imaginary magnitudes. They were only accepted because there was nothing else they could do, and they enriched the idea of number in odd and curious ways,by deining abstract objects like rings groups and fields,and sets in order to try to cope with the burgeoning and incredible outfall from these" complex "number" calculations and observations.

People worked with complex numbers with trepidation, because their natural tendency was toward the infinite sum of infinite series, and infinite sequences of factors!

This is what De moivre Noticed from the unit circle work Newton did


Therefore De Moivre took as the first factor of unity and as the second factor

It was immediately clear that there were infinitely many factors of unity,but they were yoked in conjugate pairs. These conjugate pairs were always oriented radians to each other .

The question then becomes is the seperation significant, or just a fluke? are all complex factors of unity apart indeed factors of unity?

Do they need to be apart?

These investigations naturally lead to the roots of unity theorem, and the dynamic flip rotation around the circle.

Number theory. (http://books.google.com/books?id=9hG-qF-L0vsC&pg=PA138&lpg=PA138&dq=complex+factors+of+unity&source=bl&ots=zBkZA9UMHv&sig=txD0tN0DVNGmYRMcjZxDpxgyoNM&hl=en&ei=k0dDTbvFNsT14QbmuOAC&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBUQ6AEwAA#v=onepage&q=complex%20factors%20of%20unity&f=false).

I have found some papers of the late (very late! :embarrass:) Mr. Henry John Stephen smith, Some presented to the london mathematical society in around 1880+ which deal very fully with this topic of complex factors of unity and much else besides, in fact is life's work, with biographical sketch.

I dare say that had  De Moivre, Euler and Gauss not had the preeminency that these would have obtained a wider fame.

Nevertheless they show in great detail the extent of number theory and algebra at the time of William Hamilton, and the use of modular arithmetic notation, as in Kujonai, and the continuing doubt of mathematicians in the usefulness of these considerations even at that time.

I think Mr  Smith,Krummer, et al have demonstrated that " numbers" are not based on natural counting numbers at all, nor on a Number line concept, but rather firmly and incomprehensibly if you believe in numbers on imaginary or complex numbers of a polynomial form, which exhibit modular behaviour.

Now, unless Russel And A.N, Whitehead have stated and proved in symbolic logic the same thing, their work is in need of a slight revision.
AS i do not believe in "numbers" so called, that is following Dedekind et al, i have no problem in accepting the trigonometric relations that underlie all or conception of magnitude, to which Wallis, Newton, Cotes, Napier,Bombelli, Brahmagupta have so ably contributed their insights and which Wessel, Euler, Gauss, Bernoulli Leibniz have Given some groundbreaking exposition.

Therefore Mr.Smith's, Krummer's,et al Labours, and Sir William Hamilton's have such eminent predecessors on which to draw, that it is hardly surprising that they made such advances in the complex magnitude basis of our measurement system.

This just makes a meal out of some dry bones! Vectors (http://www.math.mcgill.ca/labute/courses/133f03/VectorHistory.html) developed under the whip hand of Hamilton, although  Moebius and Grassman, and Rodrigues had similar pursuits and concerns.

The reason behind this push is that we measure in 3d, and we compare and contrast in 3d, and we distinguish in 3d. A "one or 2 dimensional number system" is not enough!

We actually require an n dimensional magnitude measuring system!

We have one now, called a tensor system. More obfuscation, and mystifying of what Artists and artisans, Musiscians and Dancers Do everyday!

We all measure as an act of existence! "I sense therefore i am", kind of thing. What we do is undoubtedly convoluted and iterative, and therefore to write it down is like unraveling a ball of wool! Better to apprentice a student and show and guide him/her in play mode than to task them to read overmuch!

Dirac, when he started Quantum mechanics Made certain he had a good grasp of what we do when we measure, developing what looks like a vector math, but was in fact his measuring tool!

It comes down to being real simple: i am a measuring tool: 'man is the measure is the measure of all things". I devise measuring tools and the algorithms that are needed to operate them, thus i choose and decide ad observe and promulgate certain activities. These i iterate, or observe to be iterating by establishing certain flip algorithms, which control my subsequent aggregation of my iteratively produced data.

My tool enables me to create data, calculate data and display data in various formats, and verify the accuracy of my calculated outputs by test measurements.

What? This souns like a computer? Too right! The universal machine as Turing imagined it was always going to be able to mimic human ability. As such it becomes mans ultimate measurement system.

It hard to not see the inevitability of computer systems arising as tools of measurement, but it is nevertheless a startling and stunning achievement which highlights for me a fundamental fractal convolution(truth?): the computation will iterate and produce fractals!

"Its all greek to me", " Its all based on 0 and 1"," It is only addition, subtraction and counting" all are aspects of this dynamic magnitude measurement system we are experiencing and playing with, artistically.

The sine and cosine and tangent have a suitably well established flip algorithm supported by the Bombelli operator. This observation actually allows me to rewrite the Bombelli operator in terms of conditions based on π/2 and its multiples.

However there is one new relation i would like to explore

"Imagine that and are orientations. as are

Then sin =sin =0

and sin =sin =1

This is because i do not use + or - for sign  i use orientation. Each case is clearly distinguished by the orientation! Isn't it?

What can i say about sine and unity?  I say sine is a unit for scalar plethoration. Any sine scalar i can denote  with a prefix and any opposite sin scalar with a prefix thus i naturally write
 

for trigonometric sine scalars where is an angle magnitude not an orientation."

To clarify among the jumble: i am imagining that  sin is the magnitude of the unit scalar in that direction. Thus for any direction there is a natural unit magnitude using the sine ratio.

Now a similar examination of the cosine behaviour gives

Then cos =cos =1

and cos =cos =0

Each case is again clearly distinguished by the orientation isn't it?

I can now  say about  cosine and unity that it  is a unit for scalar plethoration. Any cosine scalar i can denote  with a prefix and any opposite cosine scalar with a prefix thus i naturally write
 
"

Again to clarify among the jumble: i am imagining that  cos is the magnitude of the unit scalar in that direction. Thus for any direction there is a natural unit magnitude using the cos ratio.

Thus combining the two gives a combined unit cos + sin which may or may not have its uses for orientations between 0 and

I know this is unfamiliar ground because it is unfamiliar to me, but the flip algorithm helps to specify the process.

The flip algorithm governs a procedure of assigning orientations of a measuring instrument and the conditions of the flip to the different orientations of measurement. Allied to the flip algorithm is an aggregation operator, like the Bombelli operator, like the Kujonai operator, Like the Fracmonk operator that controls the aggregation structure ans the aggregation mod() gates,determining when orientations can be flipped to other orientations to enable summation, or the orientation a product or a reciprocal type product flips to .

So the flip algorithm is involved in measurment and calculation and it is an if-then procedural algorithm, a conditional statement of operation.

Thus i have suggested an orientation notation, now we need to determine a Bombelli operator in teerms of that notation.

the addition operator flips are straight forward

+mod() becomes -mod()

  -mod() becomes +mod()

as they say "vice versa".

The multiplication ones are also straight forward for integer magnitudes

   * gives =*
* gives =*

It is immediately clearer how the Bombelli operator is  in fact orientating and one may now ask how come?

The answer is Brahmagupta when he wrote 1-1=0, and the indian and the greek use of the unit circle in trigonometry. Otherwise using the chinese colour distinction would have meant any distinction could be used to represent annihilating magnitudes, and the +mod() gates indicate a structured arrangement of unit standards, thus mod (1) cannot sum with mod(2) type magnitudes.

Multiplication is purely about logical assignment of orientation, but it turns out that the unit circle and the trig ratios provide a geometric construction of the logic which applies it consistently to orientation. This application of the circle to measurement has a very long history starting with the sumero Akkadians and the Dravidians, and for a time was a universal mystical, magical symbol competing with the mystifcal magical spiral. Myth has always been associated with the two forms.

However the circle and its divisions into 360 sectors was an early Akkadian Achievement for astronomical measurement. Whatever else the circle as the ultimate foundation for measurement has proven itself throughout the millenia, but it is the spiral that has proved itself as the foundational technological form. Now we know that the circle and spiral are indivisibly linked at a fundamental group structure level.

so sin*sin=sin² =1-cos /2=1-cos/2

So under the trig multiplication identities we get a similarity in behaviour of the associated angle magnitudes, and this subtlety is enough to signal the fundamental symmetries involved in this type of attribution.

This is a case of prosthphaeresis (http://en.wikipedia.org/wiki/Prosthaphaeresis) and links directly to logarithmic and clock arithmetic behaviour of the trig functions. Modular arithmetics and congruence relations,symmetry and relativity, self similarity and logarithms all arise naturally out of the trigonometry of the unit spherical surface, and hyper geometries lie just beneath its surface.

To my mind Cotes saw this and made it his aim to study it fully.

I have explored this by way of exercise, as i much prefer the radial^r and circumference ^c notation.

I hope that the structure of the flip algorithm in it 2 parts is clear and the ease in which it may be used is clear. I hope also that it is easily seen how the Bombelli operator is adapted to this notation, and why the trig relations are a model of the rules, and indeed how this may be generalsed to both Kujonai's operator and to Fracmonks.

If i get time i will do the exercise for Kujonai's operator to show how i got the images posted in the gallery under his name.

http://upload.wikimedia.org/wikipedia/commons/4/4c/Unit_circle_angles_color.svg (http://upload.wikimedia.org/wikipedia/commons/4/4c/Unit_circle_angles_color.svg)

(http://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Unit_circle_angles_color.svg/600px-Unit_circle_angles_color.svg.png)
"   This is space ,Jim...but not as we know it!..."
Thanks jim

If you remember CAST you will know the old fashioned flip algorithm related to the positive sign

All in the first quadrant
Sine in the second quadrant
Tangent in the third quadrant and
Cosine in the fourth quadrant.

The inherent and powerful symmetry lies before your eyes! Imagine now a dynamic version of this  using a colouring algorithm.
Realise that De Moivre hit upon the unique power of this space, the unit circle wraps all the reciprocals of its oriented magnitudes within itself. They are not within the disk or outside the disk, but precisely in the circumference of the circle. Thus this impossible space is an infinite infinitesimal dynamic system which is unstable and unpredictable and able to implode or explode, through the slightest variance.

So in the sphere's surface we have that infinite possibility surface which dynamically generates all fractals from the Cosmos to the tiniest wormhole, and beneath its surface is Hyperspace!

"This is space ,Jim, but not as we know it" is based on a quote from Startrekkin  across the universe.

http://www.youtube.com/watch?v=OrYbkbWRA2I

The basis of the Bombelli operator is found here (http://books.google.com/books?id=Op9pnGlsBGgC&printsec=frontcover&dq=complex+spherical+trigonometry&source=bl&ots=23kgPIuHPU&sig=mIzcVnZFOyrwaPciC9YdSBVtFw4&hl=en&ei=o-ZITY2PMsOLhQfT6oioDg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CDgQ6AEwCDgK#v=onepage&q=complex%20spherical%20trigonometry&f=false) on page 18.

It is rendered amazingly simple by this treatment to give a trig meaning to √-1 and to found the complex numbers securely on these trig relations.

I might also add that of course i have discovered nothing new, just cleared away the mountain of confusion covering these simple observations!

One fancy i have is that Newton absolutely based his 3 laws on these trigonometric relations particularly those of spherical trig, and that the first law states conditions of dynamic and static equilibrium, not in terms of straight lines but in terms of tangents to a curved motion! In other words he had in mind the trigonometric relations of the tangent to the unit circle underpinned by the sine and cosine ratios.

His third law details the nature of static Equilibrium, and his second law details the nature of Dynamic Equilibrium, in which force or pressure arises only as a means of maintaning equilibrium in  a dynamic state. Having achieved Equilibrium forces disappear but motion continues forever!

Thus Newton describes the laws of a Dynamic Motion field in which constant motion everywhere is immediately assumed.

Action at a distance therefore is not a mysterious force, but a relative  point of view in a dynamic motion field. Newton understood that, but wanted to find out the true state from God's point of view. This he found impossible to conceive and so declared all a mystery. We accept the conundrum as a fundamental outfall of relativity, which was only possible because Einstein had no qualms in placing God inside the reference framework. Thus God had the same undecidability conundrum as a human observer, and the problem goes away!
Of course that has consequences for a theory of God, but that is for theologians to sort, not scientists!

Product-to-sum

http://upload.wikimedia.org/math/e/1/c/e1c9a58f3d0a4a4372642cfae4acc7c4.png

http://upload.wikimedia.org/math/9/1/9/91970e07f76159bf710f39265995622e.png

http://upload.wikimedia.org/math/d/2/1/d21208f87b9c55b68e4cb36e4ec1cc8f.png

http://upload.wikimedia.org/math/1/b/9/1b9930390f8fc72cd312648e41ce3e71.png

Sum-to-product

http://upload.wikimedia.org/math/2/7/b/27baa5fb481ec2dcaf6cbb38b56f2570.png

http://upload.wikimedia.org/math/7/6/4/7649e6e5a976127426eaff7b31f54b6f.png

http://upload.wikimedia.org/math/d/d/a/dda26c1ebede67da7e54dd4890d665ff.png

Using these products we can define√-1*√-1 as √(cos π*sin3π/2)

I am actually a bit annoyed at mathematicians over last the 5 centuries!

The simple and obvious conclusion on investigation of complex numbers is that they represent the algebraisation of Spherical and Plane trigonometry, the calculation part of Geometry.

Any mathematician, and particularly geometer who has not used or referred to the use of complex numbers in the proof of certain geometrical theorems or facts, has not advanced very far in his subject. But what is particularly galling is that having noticed this use i have not heard anyone explain that the formal equivalence of the two needs to be expounded from the very outset of education in trigonometry!

There are of course historical reasons, the main being the rivalry and secrecy between Cardan et al,

".....It is unclear exactly how Bombelli learnt of the leading mathematical works of the day, but of course he lived in the right part of Italy to be involved in the major events surrounding the solution of cubic and quartic equations. Scipione del Ferro, the first to solve the cubic equation was the professor at Bologna, Bombelli's home town, but del Ferro died the year that Bombelli was born. The contest between Fior and Tartaglia (see Tartaglia's biography) took place in 1535 when Bombelli was nine years old, and Cardan's major work on the topic Ars Magna was published in 1545. Clearly Bombelli had studied Cardan's work and he also followed closely the very public arguments between Cardan, Ferrari and Tartaglia which culminated in the contest between Ferrari and Tartaglia in Milan in 1548 (see Ferrari's biography for details)....." http://www-history.mcs.st-andrews.ac.uk/Biographies/Bombelli.html

Bombelli himself was not open unil he had published, but then again he was not a mathematician but an engineer and probably felt intimidated.

Bombelli is so crucial to this whole subject because he publicly announced that he used the Greek methods to solve the problems.

I was unaware of the history of Spherical trig until i explored this topic, but now i am aware i am annoyed!

I did introduction to trigonometry when at primary school! Admittedly analytical trig was too advanced for a 10 or 11 year old in general, but ratios are not, neither are scalars or navigation etc.

I believe the mystical and the mythical plays its part in all sciences and that includes mathematics! The mythical √-1 has generated an alternative mathesis - methodology- of dealing with trigonometry namely the complex numbers, and in addition: vectors, a derivation from complex numbers right at the start. This may not have happened as it did if this "mystery" √-1 was not promoted to mythical
proportions!

However there is no longer any need to be ignorant of the simple truth that trigonometry and trigonometric concerns underly, underpin, and found the notions of complex magnitudes and vectors. This leads to a direct relevance of Pythogorean concerns about unity; an inestimable disproportionate impact of Eudoxus' proportional theory; the importance of distinguishing arithmoi from unity; the plethorate nature of measurement as aggregations of unity; and the Shaping effect on all of mathematics of Euclid and Plato!

Though one is eager to admit all sorts of sources to mathematics, Indian and Chinese in particular, it cannot be avoided that by conquering "all the world", Alexander the great  lead  a dominant and impressive cultural campaign called Hellenization, that influenced scholars in India and China and reflects back echoes in mathematics to us today.

Chief among these influences and collaborations was and is spherical trigonometry and its application to astronomy, astrology, and  the building of great monuments and temples and all artistic endeavours.

Somehow the introduction of Brahmagupta's "negative" numbers confused the plot, and got lost in translation by Arabic intermediaries.

We end up with Bombelli who rhetorically introduces notation ±√- and mathematicians lose the plot! Descartes grumpily calls them imaginary, while others scratch their heads in bewilderment. Nobody connects geometry with geometry, nor later, trigonometry with trigonometry!

Geometrically "meno" was associated with the rotation through π anticlockwise, thus with the circle theorems of Euclid and the Trig of Euclid. No one thought to give π/2 rotation a sign other than the perpendicular sign of geometry, and that is where the connection fails!

If Bombelli had gone that one step further in his madness and put | or even we would have had no difficulty in drawing on the insights of trigonometry much sooner.
√- √1, or √-√a for the general case is what we have been looking at all this time. meno di meno and piu di meno are the whole sign. We do not do square roots of "negative" numbers we do square roots of Directed numbers! The signs give us the directions and +√- is the perpendicular direction anticlockwise, but also the clockwise is -√- .

Everything else is trigonometry as well!

I have been able to demonstrate the trigonometric basis  of the complex "numbers".

First we need a spherical trig grapher (http://www.flashandmath.com/mathlets/multicalc/paramsphere/surf_graph_sphere.html)

Secondly we need to make clear some conventions. Angles are measured anticlockwise. The minus sign will be used in the normal sense, but will not be used to alter aggregation, nor in a Bombelli type vector or operator, it will be rarely used and means opposite direction of "motion" rather than orientation.

The The radials move away from the origin of the sphere in a positive direction . Moving in a negative direction is moving toward the centre and out nto the radial in the π radian "positive" orientation on the other side of the centre.

The square root of a magnituse includes a dynamic reorientation of the radial of the magnitude by π/2 radians.
The complex "numbers" are therefore representative of aggregations of the form of spherical polar coordinates, namely , where and have the condition that they must always be measured orthogonal  to each other.

Now, of course , the app uses the negative sign through and through, however the calls the formula makes do not involve calls to complex arithmetic , and the formula at least can be written to remove reference to even negative signs, so that only addition and subtraction aggregation are referenced. In addition i have to set phi to run from -1*π to 0 radians so that the app is drawing anticlockwise angles through and through.

The iteration(4 stages)
(0.4*(0.5*(0.12*(1.07*(cos(s)+sin(t))^2+cos(1.05*pi*s/s)+sin(0.5*pi*t/t))^2+cos(1.05*pi*s/s)+sin(0.0*pi*t/t))^2+cos(1.05*pi*s/s)+sin(0.5*pi*t/t))^2+cos(1.05*pi*s/s)+sin(0.5*pi*t/t))^2+cos(1.05*pi*s/s)+sin(0.5*pi*t/t)

is based on

1.07*(cos(s)+sin(t))^2+cos(0.0*pi*s/s)+sin(0*pi*t/t)  = (1.07*(,)^2+c)

Which is a spherical surface(1.07*(cos(s)+sin(t))^2) + some variation with a constant translation(cos(0.0*pi*s/s)+sin(0*pi*t/t)) along a specific radial. The surface is just outside the unit sphere surface.

Non of these scalars make much difference to the surface drawn, but they do affect closure of the surface. It is recursion/ convolution that makes the difference. This literally means the relational properties of the surface are altered not by scalar but by convolution.
We naturally count the end product of each convolution as an iteration. I suppose the most accessible analogy is that of the specific marque of a specific form of a car take the Ford fiesta, or the Volkswagen Beetle. Each vehicle is convoluted, much recursion goes into producing each iteration. And of course, recursion is about bringing together things into a relatonship that is detailed and complex.

We take for granted the convolutional power of a modern graphics system, able to compute the position of each of millions of points/pixels directly or relationally.

You will notice that at each iteration i have reduced the radial of the surface that is being convoluted. This has been done by trial and error, which is how i discovered the required ratios.

You should play with them and with increasing the number of iterations.

Directed dynamic magnitudes therefore have dynamic trigonometric ratios of the most complex convolutional relations, and we can now draw these relational convoluted surfaces, that is tensors, with utmost ease nowadays.

The plane trig basis for the complex numbers is equally demonstrable

First the plane grapher (http://www.flashandmath.com/mathlets/calc/param2d/param_advanced.html) is set to polar coordinates.

Then the following relation is iterated:

(1*cos(pi/2+4*t)+1*sin(1*t))

(1*cos(pi/2+4*t)+1*sin(1*t))^2+cos(pi/3)+sin(pi/2+pi/3) is

(0.2*(2.9*(0.081*(1*cos(pi/2+4*t)+1*sin(1*t))^2+cos(pi/3)+sin(pi/2+pi/3))^2+cos(pi/3)+sin(pi/2+pi/3))^2+cos(pi/3)+sin(pi/2+pi/3))^2+cos(pi/3)+sin(pi/2+pi/3).

There are so many things to vary with different outcomes that these truely deserve the name complex.

I have a issue with the term "number", and much prefer the term scalar.

The indians were the first to develop the sine ratio! before them the ancients and the Greeks used the Chord. For me the chord is more intuitive and suggestive of navigation measurement and orientation. However logarithms and prosthaphaeresis may not have been so obvious  from the chord.

The other thought that occurred to me is that Brahmagupta may not just have been stating the obvious when he wrote 1-1=0
He may very well have been thinking co-sin(π/2+π/2)- sin π/2= 0, before the cosine was seperately figured. The cosine was undoubtedly used as the sine(chord) at right angles to the direction of initial measurement of the sine(chord).

It seems a moot point, but since these things are so clear in Euclid and Ptolemy it is hard not to think that Brahmagupta, and Bombelli, both saw the behaviour of the trig ratios and derived the dehaviour of directed numbers from them. The unique insight of shunaya allied with the greek insight on unity seems to be evident in Brahamagupta's collected work. and Bomelli certainly accurately described the rules for multiplying by √-1.

There is a further directional element to addition and subtraction the aggregation/ disaggregation proceses: Aggregation tends to be a gathering to a central location. That is aggregated objects move in the opposite direction to the positive direction of the ray of orientation.

The Bombeeli operator covers over to directed motions: translation and rotation, both accessed by the "multiply" command.So these notions of directed motions in the actual operator definition are to be explored.

Bombelli's use of the word "via" for multiplication needs to be referenced against the gnomon.

So very much in keeping with the practical nature of greek geometry the gnomon would be used to "establish, hold, support, represent, define" a rectangular "area". The 2 sides of the gnomon would be designated "via" which is of course " road, way, direction, route"and as a preposition "by way of, or by; but not as in 'near' ". Thus an area was "calculated" by a gnomon. Literally it was measured by the "via", ordered and arrayed by the "via", and the "calculii" (latin) the stones of "reckoning" counted.

Thus geometrically multiplication was an ordered rhythmical iteration described by the 2 via of a gnomon.

Physically the iteration would result in a tiling or stacking, and the tiling would proceed in a "via", each "row" beinc thought of as a single unit. Thus one of the via becomes a row  the other may be termed a via. Similarly, one of the via may be termed a column and the other a via.

Bombelli's operator shows that intuitively he thought of these via as having a direction, which he called "less" or "more". He intuitively as did the greeks associated dynamic magnitude with a direction  or if static , an orientation.

So how do we describe the second notion of motion in via?

Somehow, both Brahmagupta and Bombelli could intuit a curved via, a circular one.

I do not mean to egg the pudding, as for thousands of years astronomers and scientists had been relating the curved via of the heavens to geometrical observations, but what Brahmagupta did in the first instance, under i think greek influence as in the case of Hipparchus or Ptolemy, along with some chinese notions, was to relate direction to the circular via in a multiplicative way.

The only way this makes sense is with a notion of 0 (shunaya) and a notion of Ptolemy's chord relations.

It becomes even clearer (as mud gets clearer!  ;D) when you drop to the 1/2 chord ratio we call sine, but the indians who first did this called "limb"(?).

From spherical geometry and trig it became clear that "multiplying" these ratios, that is swinging around the centre via the circumference usisng the radius as a "row" gave a calculable result.

Thus rotation of the radius was a via as well but was subtly different. Only Brahmagupta went on to explore this difference philosophically as Shunaya.

sin(x−y)=sinxcosy−cosxsiny

sin(x+y)=sinxcosy+cosxsiny

2sin2x=1−cos2x

Were all derived by Ptolemy in chord form.

Thus π/2 forms a natural and ubiquitous distinction in the theory of circles and the multiplication via a curve.

The notion of "multiplying" via a gnomon leads to the measurement of "area". By this i mean that we are measuring area not as it seems "multiplying lengths". This "operation" of multiplying is an abstraction from measuring a rectangular area.

It seems odd to point this out as multiplication is usually taught as "giving" an area or a volume etc.

However as an immediate geometer, i have no qualms with dispensing with the abstraction in favour of what is actually happening in front of my very eyes and self.

In this way geometry is very empirical and natural and does not need to be thought of as complicated at all.

Now how i explain what is happening or what i do to get a result may be very complicated and complex, but that does not make the geometry essentially complicated or complex, only my procedural basis for measurement, and my"method of discourse" .

These if anything are what turn childs play into a horrible agonia called mathematics!

So a gnomon is used to measure an area, so what does a circular gnomon measure? tha area of a circle which by inspection out to be the circumference via the radius or the radius via the circumference. in boh cases this will be too big when transferred to a gnomon. and as it turns out it is precisely half the measurement of the circumference via the radius.

Thus Circle area =

Dynamically the geometry suggests this as correct as whichever "via "you use, triangular variation occurs.

Triangles are therefore obviously fundamental in a dynamic geometry, and we get the full set of Euclidean arithmoi as being triangles rectilinear forms reducible to quadrilaterals and circles and of course there solid counterparts the tetrahedra, pyramids, parallelepipeds, polyhedra and the sphere .

We have later revisionists and redactors Like Plato et al  to blame for any confusion in the basic simplicity of Euclids conception.

Negative  (http://www.pballew.net/arithme1.html)

Negative numbers, and the equivalent word for negative were introduce by Brahmagupta, a Hindu mathematician around 600 AD. The Latin root of today's word is negare, to deny. The negative numbers, in this sense, denying or invalidating an equivalent positive quantity.

The negative numbers were themselves denied for a long part of mathematical history, and only slowly came to be accepted. The first record of the operational rules for what we today call positive and negative numbers came from the pen of Diophantus (around 250 AD) who referred to them as "forthcomings" and "wantings". His work may have been drawn from proposition five in Euclid's Book II of the Elements in which Euclid demonstrates with geometric figures what we would write in modern algebra as (a+b)(a-b)+b2 = a2. This, of course, is easily recast as the more common identity (a+b)(a-b)= a2 - b2. Diophantus would accept negatives only as a way of diminishing a greater quantity, but did not accept them as independent quantities and would not accept a solution that was negative. Al-Khwarizmi (850 AD), whose writings brought Arabic numerals to the west, used a similar approach with negatives allowed in-process but not as a final result.

Descartes, around 1636, used the French fausse, false, for negative solutions. Thomas Harriot had described negative roots as the solution to an alternate form of the equation with the signs of the odd powers changed. Today his idea would be expressed by saying that the appearance of -c as a root of f(x) was only to be understood to mean that c is a root of f(-x).

In Mathematics: The Loss of Certainty, by Morris Kline includes the following argument against negative numbers by Antoine Arnauld (1612-1694), mathematician, theologian, and friend of Blaise Pascal; "Arnauld questioned that -1:1 = 1:-1 because, he said, -1 is less than +1; hence, how could a smaller be to a greater as a greater is to a smaller?"

Franz Lemmermeyer wrote in a posting to the Historia-Matematica newsgroup that Gleanings from the History of the Negative Number by PGJ Vrendenduin suggests that a number line with both positive and negative numbers could be found in the work of Wallis (1657)[This is certainly true as seen here]. Another posting to the same list quoted Kline's "Mathematical Thought from Ancient to Modern Times":
"Though Wallis was advanced for his times and accepted negative numbers, he thought they were larger than infinity but not less than zero. In his 'Arithmetica Infinitorum' (1665), he argued that since the ratio a/0, when a is positive, is infinite, then, when the denominator is changed to a negative number, as in a/b with b negative, the ratio must be greater than infinity."
Even as late as 1831, De Morgan would still write that one "must recollect that the signs + and - are not quantities, but directions to add and subtract." [ Albrecht Heeffer refutes this position, held by Kline and many others, in a post to the math-history list. ] In a recent book by Gert Schubring see clips here he also supports a view that Wallis' understanding of negatives was much broader than generally credited.


According to a post from Laura Laurencich, the Incas had a method of indicating both positive and negative numbers on their quipus as documented by the Jesuit Priest Blas Valeria in 1618."

And Bombelli was of course very inspired by Diophantus as was Fermat.

But it is clear that Wallis was inspired by Brahmagupta.

It is a surprisingly simple misdirection, but one which bears revealing (http://www.pballew.net/arithme4.html#root). There seems to be little evidence that the greeks ever sought the "square root" of much at all. Their concerns were different to those of the Arab Al Khwarzim the patron of algebra and the Algorithm.

In fact the word square is a late gloss on Exquadrataeia which explains why so "quadrature" is so heavily used in early writingIf anything the greek had a love affair with the right triangle and developed all rectilinear forms, (arithmoi) in respect of that fundamental. Greeks were no stranger to bits of string stretched tight or knotted in certain ways, Gnomon from which they referenced gonia- a kind of gnomon based on various crossed lines not "Kathetus" that is perpendicular.

The circle and the inscribed right triangle or cyclic quadrilateral seems to have been their ultimate analytical device from which they derived their form of complex magnitudes namely trigonometric ratios, especially on the theory of Eudoxus and Ptolemy. They of course conquered all before them both physically and geometrically and had theorie of the conics and spirals and decimal aggregation structures long before others even thought of them.

Unfortunately Al Kwharzim, appears to have been more impressed by indian than greek presentation, and it is indeed beautiful, but his presentation obscured the everydayness of greek exposition with the mystical subtlety of the subhasutras.

Even though i have come this far with the notion that somehow √-1 means a rotation by π/2, i am not afraid to ask if the whole idea of rotation is spurious with regard to the complex numbers C.

I can justify rotation around the unit circle as arc lengths on the great circles of a unit circle and the consequent trig relations reflect that but the √-1 seems a bit arch.

From the point of view of gnomon values i can justify the square root as being a geometrical mean of the rectilinear form expressed as a rectangular form. Apart from direction of the row of the gnomon there is essentially no difference in the construction to find the geometric mean of a negative area. In gnomon terms the roots are a pair+1 and +1 or +1 and -1 and the question of the (√-1)² does not arise as the geometric mean is the same. But there seems to be a unique arrangement of the geometric means so that multiplying them out gives me Bombellis operator, and this requires  rotation by π/2 of the geometric mean!

Now if i use curve or trig ratio values values the chord of positive length one occurs at arc π/3 and the chord of negative length one occurs at 5π/3. can i find a root of the chords that necessitates rotation?

At last I have come to the place (http://nrich.maths.org/6908&part=) where the shaping of trigonometry was done. It is an amalgam of all knowledge skilfully blended but not faithfully transmitted to the west due to religious intolerance and book burning. Therefore westerners have developed many erroneous ideas including the notion of false or imaginary magnitudes. In addition Arabic predilection dictated the mix of information style and method in trigonometry for a long time, but despite looking Indian trigonometry is essentially Ptolemaic.

This is something so simple that it at first seems silly.

The geometric mean of the line is the line.

Thus  =  or or

Now that makes the quadrature of the directed lines [-1,0), and (0,] have a curious link with trigonometry and in particular the area -1*+1= -1 have some strange "roots".

This type of development shows how historical changes can have a confusing effect on later generations of mathematicians/scientists/artisans.

We can write = which would be correct except for signage. This thus represents a comparison and an "Aequalis"of magnitude. Similarly for

Thus the correct relation is
=

Howeve this does not notationally allow us to "square root" but is the correct geometric mean of the line[-1,+1]

Geometrically, by Euclid we have to choose or

While this appears a difficulty over signage to us it represents the exact meaning of the calculation/construction.

Thus i can conclude that the problem is our modern angst over notation rather than meaning.

It is quite clear and fundamental that the signs are not there for their own purposes, but to represent direction. The symbolic choice of + and - for both direction and aggregation was destined to confuse and does confuse. Apparently even Euler was confused by it, And Descartes contextualised the operations and constructions so geometrically that he never got confused by it, thus his term False numbers for negatives and imaginary for roots of negative values. He avoided the confusion not the utillty. Wallis, Newton, De Moivre and Cotes and possibly De Fermat all followed Descartes in this. They were strong and gifted geometers!

Al Kwharzimi was the one who introduced the term root or radix into the algebra he created, and he applied it to binomial equations. Thus he maintained a solid link with the geometrical solutions for finding the geometric mean. However later scientists pushe on to cubic and quartic polynomials, but retained the use of the word root and its geometrical association. It was Bombeelii, and then Descartes who publicly showed the use of the geometrical constructions as relevant to solving the roots of many equations, But nobody liked or understood the signage especially the "evil" negative numbers!

What we have been left with is the revulsion of past mathematicians, scientists and artisans over the changes to their beloved and familiar  "numbers", and their total disbelief in the negative numbers, despite their utility. This was indeed like Alchemy, Magycke, and Divination, the work of the devil and the Moor, and fit for what? To be Burned!

No wonder the arabic empire saw Europe as a dark and Backward place, full of superstitious nonsense, and unenlightened by the wisdom of Allah!

Fortunately for us a few brave souls struggled on to disseminate the knowledge of the Arabic empire to those who would and could listen, Watch and learn.

We are our Father's children, but someday we must grow up!

The basis of our dynamic magnitude system is Arabic Indian Greek trigonometry.

The rules of the geometric mean are a*b=c*d where a line A is split into a+c by another line B  which is mutually split into b+d, if the ends of the lines are on the circumference of a circle.

Thus the value of a*b is given by c*d.

In the case of finding the "square root", that is properly the "exquadrature radix", the magnitude that enables a square to be constructed c=d=x

Thus a*b= x*x.

This same rule applies to the quadrature of the line[-1,+1] in trig form

=

But we have to pick

or

as the "square root"!

The question is does + *

or  - *

work the same as +i * and -i *?

Of course we have to remember the rule of the geometric mean to be fair, not our modern signage rules, that is the Bombelli operator for √-1.

So - *

must be written as

+ *.

Any body care to explore?.

An observation: It does not matter how we found the Complex numbers we cannot get away from " interfernce". Which is to say we have to interrupt all our other rules o allow them to be.

So for example -1= √-1*√-1= √(-1*-1)= √1= 1

We have to interrupt the flow of the notation and manipulation and say
"Wait a second. What construction or calculation am i using here, and what are its rules, and which takes precedence?!"
Only by doing that can we maintain a link with real world geometric constructions not symbolic ones. Or rather more interestingly we can give a geometric interpretation to every symbolic manipulation!

Now there's a thought!

In line with that, as i have mentioned elsewhere, addition or aggregation is a directed motion toward a centre and subtraction or disaggregation is  a directed motion away from a centre. Thus in our current notation we have a conflict of signs and directions.

In this particular instance the difference of 2 squares requires we subtract the second term in the second bracket, regardless of its direction!

This leads to interesting insights. The difference of two squares usually gives the square area on a Gnomon, which has  sides related to the "via" or "jya" or "legs" or "half chords" of a right angled triangle in  its escribed circle. But using the geometric mean of the "-1" directed area, leads to the difference of 2 squares becoming the sum of 2 squares: relating to the hypotenuse and the factors of unity, and both legs of the right angled triangle; emphasising their orientation.

This is a kind of Heisenberg uncertainty principle where strange relationships can be teased out but not easily unified except in the geometry.

This is also the ordered face of Chaos and confusion, something mathematicians/scientists have up until the 20th century strenuously avoided.

There are two issues remaining, seeking an answer in my mind.

The first is why cosx+isinx? Or rather how and by whom did it come about?

I have had an initial stab at it earlier in the thread, and my lines of enquiry take me along 2 main routes: Wallis and Newton, using the information From the latest sources, real cutting edge stuff by Regiomontanus, and Vieta, De Fermatt and Descartes, combined with the interminable calculations of the trigonometric and logarithmic tables. The discipline of calculating combined with the theory of what the calculation was all about provides a potent mix and hands on unedrstanding of Sequence, and sequences of data and series as in series of calculations.

This discipline provides an inestimable appreciation of iteration as fundamental to the measurement and construction of the universe, as well as a profound understanding of the scalability of unity the similarity at all scales of the calculation and the minute infinitesimal steps one might make to define precisely the smallest measure!
This great labour gave rise to such a familiarity with calculus that ordered formulae could be written to guide the calculation, and these became based on continued fractions, infinite and finite series , multinomials and interpolation and extrapolation schema of finite differences.

Every one of these burdensome calculations owed there existence to Trigonometric ratios, a fact that many scientists and mathematicians of Newton's time had lost sight of in the detail. Fortunately Newton and Wallis in particular had not, and Neither had De Moivre and Cotes. Also Bernoulli was heavily into the quadrature of the circle as was Wallis and he knew why the long interminale calculations were taking place, and he taught Euler what he knew. But only Wallis and Newton et al it appraers had made the connection at the theoretical level as well as the calculus level, due to their gift of being able to easily do long calculations in their heads!

Thus i think perhaps Wallis in deriving his conical equations (circle, Ellipse, parabola, hyperbola) converted rcosø to x on the x axis and rsinø to y on the y axis deriving x² + y²=r²,

This i think somewhere Newton used in a geometrical Diagram and De Moivre factored into (x+iy)*(x-iy) to solve some equation and to generalise it to higher powered  multinomials.

Thus Newto and Wallis probably without much thought regularly worked from a diagram similar to the unit circle with the sine an cosine marked on it , but it was De Moivre who factored it in the course of Solving Equations, and he similrly moved between the trigonometric form and the cartesian like form.

Without seeing Regimontanus i cannot say what influence he may have had nor Vieta. However it was not common, because the Royal society expressed great surprise at the trig functions appearing in this way in the solution of multinomial Equations, and promptly welcomed De Moivre into the club!

It is also attested to that De moivre helped Cotes to solve a problem in navigation concerning the latitude, introducing him to this factor of unity. I believe then that Cotes genius took him at a pace in a direction De Moivre had not considered and working together they found the trigonometric relationship with the Logarithms of Napier, namely

iø= ln(cosø+i*sinø)

And what is now called De Moivre's Theorem, but properly the De Moivre- Cotes theorem.

The other route is simply that De moivre discovered the factorisation of cos²ø+sin²ø=1 while looking at the difference of 2 squares.

Now the other issue is rotation. The gradual tendency from the greek geometry was to reduce the explanation to a static fixed on the page block, probably a trend enhanced by the printing press and the cost in time of doing over many diagrams.

Thus Although motion and rotation was what was being explored and measured, there does not appear to be a connection between the ratios and immediate motion of rotation. The aim seems to have been to be able to measure a fixed position in the sky and on the earth, and then through sequence of data to calculate a fixed position at a given time frame.

Many astronomers made mobile models of the universe based on these measurements and their estimates of radial distance to the stars, but very scant mention seems to be made of the notion of describing rotation  per se by using trigonometric ratios until Rodrigues and Hamilton's time. So when did √-1 become entangled with rotation of the plane, or rotation around an axis?

Ok so this is my conclusion, this is how i roll :surf:

Wallis worked ou how to represent the conics algebraically. He clearly started with the circle and formed x² +y² = r²

He did a polynomial factorisation to find the roots and got (x+iy)*(x+iy)=r²

From this he deduced 4 roots 2 on the x axis and 2 on the y axis. The ones on the y axis he realised had to be the ones that Descartes called imaginary, and he said as much. Of course nobody listened or understood but Newton, Who told Demoivre and Cotes.

De moivre took it and explored it in solving polynomials and also explored it in terms of the relaively new sine and cosine tabulations, because the tabulation process generated infinite polynomials/multinomials. These were among his favourite research projects and he discussed them with Newton and with Cotes. Newtonexplained to De moivre the geometrical interpretation of them and the cyclical nature and how they appeared in the binomialseries and in the theory of Fluxions. The theory of floxions was Newton's personal version of Greek dynamic Geometry. nothing stood still! Rest and motion were therefore aspects of relativity to Newton et al. De Moivre was able to see their applicability to the theory of probability, and was of course uniquely placed via his tutoring in Newton's Binomial series etc.

Cotes joined the discussion and picked up on some of De Moivre's work and in collaboration with De Moivre and Wallis took it into the realms of Napierian Logarithms. The De Moivre Cotes theorem was generated. Of course Bernoulli at the time had got close but was in a fog, and a dispute with Leibniz about how to go forward with the  negative numbers with regards to logarithms. Newton was also in dispute with Leibniz so only Cotes and De Moivre really were at liberty to concentrate and push forward in exploration.

Historically cosx+isinx took its status from solving for the roots of polynomial/multinomial equations Especially the conic section equations discovered by Wallis, which Newton absolutely knew inside out and used to describe Gravity. De Moivre made the link to probability and Cotes the link to napierian logarithms. As Newton said, had he lived we would have learned something particularly in relation to Newtons Laws.

Cotes had Found something which he discussed with Newton, who genuinely became excited by his young friends line of inquiry. Newton had always been embarassed by his appeal to action at  a distance and could not answer the observation that he had only described orbital motion not shown a cause. In addition Kepler had given a more geometrical description of the action of the "force" or "vis" an undefined notion, which Newton could derive from his laws but could not explain Why. Cotes it appeared could and it was through the Cotes-Euler equation

ix= ln(cosx+isinx)

Both Newton and Cotes had got as far as calculating e= 2.718... in correspondence, and cotes had derived a set of tables of logarithms based on this and some tables of differentials which i think solved for elliptical curves and others. Thus i think Cotes was about to show Newton, how Kepler's law derives from logarithms to the base e  of the circle curve/ orbits In other words how all the conic curves including the hyperbolic derived from napierian logs of the ordinary geometrical circle!

It is a moot point, by Cotes improvement to the logarithmic spiral intimates that he had an exact way for calculating it, and this suggests to me that Cotes was able to calculate a radius to an angle on a form of polar coordinate graph, namely the surface of the earth, which he then projected to a plane using a Mercator projection. Thus by logarithms circles were shown to be a special form of spiral and all classical greek forms were now capable of a single form of measure. This was Cotes Harmonium Mensuraram.

Logarithms are not strictly equations, they are procedures for calculation, and frameworks for calculating the exponents of bases.

Thus they are useless without a prior art, that is an immense calculation of exponents. But it is the experience of calculating the exponents that reveals to the orderly mind the orderly arrangement of "numbers", and the logarithmic interconnection of all magnitudes through a rhythm, a rhythm that is harmonious and musical and which now had a name "Logarithmos". By finding and choosing the base e Cotes had found something that explained action at a distance, and Newton was excited waiting for the proof!

Unfortunately Cotes Died, and De Moivre was not privy to all the directions Cotes wanted to take this discovery in, so beyond completing Cotes' work in hand De Moivre did not know how to take it further.

There has always been an unfortunate divide between scientists of geometry who generally see what they are doing and scientists of symbolic algebra who are generally blind to what they are doing! The Arabic scientists used and lived by trigonometry over abstract symbolic algebra, a later Western development brought about in the main through starvation and frustration of receiving the pure source of trigonometric knowledge via the arabic empire and its european centres of influence.

Thus much of western invention was based on misguided and mixed up notions of Arabic empirical science. This lead to the curios developments in mathematics in the west which made the topic an agonia not a flowing, lyrical joy! However despite the strangulated hernia that the west had with regard to trigonometric knowledge, nevertheless they employed it to the study of the world and received corrective medicine over time from a freer dissemination of arabic knowledge and eventually created a powerful hybrid, even if it was flawed.

Mathematicians in the west have sensed the flaws in their mathematic for centuries, but few have attributed it to the failure to base their mathematics on a solid and sound trigonometric basis, Observing the Eudoxian principles of the theory of proportion.

AS to the matter of rotation: simply read this (http://nrich.maths.org/6908&part=) in its 3 parts to realise that trig has always been about rotation and from the time human animates began to record sequences of data and pictures of stars in the sky, geometry has always been dynamic and about motion.

For my money i think Hipparchus and Ptolemy are the first ones to derive  a systematic measuring tool called the table of Chords specifically designed to measure rotation. They inspired the Indian Astronomers who inspired the Chinese. Millennia of Data could now be utilised to formulate a measuring tool for, and a system of displaying, motion on a sphere. It was the arab empire that melded the system that was most efficient and elegant theoretically and set in train a vast centuries long calculation that iterated better and better "exactitudes" for the measuring tool . The table was called the table of sines after a mistranslation (http://www.pballew.net/arithme4.html#root) of the indian jiya.

This vast system of sequences and series and calculations was the absolute fundamental cutting edge science of measurement and magnitude. Thus to withdraw from it through "ignorance" lead to misguided western approaches.

The fact that the west took its day in the empirical sun, does not alter the fundamental basis for a theory of magnitude and measure, which is trigonometry, in alliance with geometry.

From this fundamental basis Napier derived his scheme for logarithms, the most successful scheme in history and the basis for the theory of logarithms of all sorts and bases!

From a deep appreciation of the geometric calculation of the sines Newton devised his calculus of fluxions, Leibniz his differential geometric calculus. Descartes devised his geometry after the Arabic formulation synthesized by Harriot and vieta.

One cannot overlook the fundamental nature of trigonometry and Eudoxian theory on the development of all calculation and tools and machines of calculation, and all sciences.

Without the sine curve and Fourier analysis it would not have been possible to describe the square wave clock signal that underlies all computer processing.

Finally the mightily misunderstood Brahmagupta. I believe Wallis understood him quite well. Brahmagupta dealt with Shunaya, And while some chose zero Wallis chose infinity. WE fail to listen to Wallis at our peril. After all he was Newton's  tutor!

Brahmagupta, above all believed in the perpetual motion in the universe, and sought to measure it not by the perpetual motion machine in the form of a wheel, but by the Babylonian, Egyptian and Greek wheel of the heavens, otherwise known to us as the sine tables!
From his deep religious belief he derived 1-1=shunaya/infinity! Thus Wallis's belief that the negative numbers were greater than infinity, an idea ahead of his time and frankly beyond his peers to this day!

The specific properties of √-1 however relate to the finding of the geometric mean, a Euclidean stalwart based on earlier greek geometers geometrical solutions and heavily involving the gnomon and the rotation thetough a semicircle based on the arithmetic mean as a radius!. This curious bit of neusis has proved the most mystifying important construction in all of mathematics to date! Who knows what else will be found by studying the trigonometric relations in the Euclidean and Ptolemaic geometries.

The geometric mean construction (http://aleph0.clarku.edu/~djoyce/java/elements/bookII/propII14.html) is proportional to the ratio of the sides of the given rectangle but the square root of the area of the rectangle is constant, clearly. Thus the longer and thinner the rectangle the greater the circumference of construction, and the greater the enclosing circular area. the proportion of the fixed area to the circular area is an indication of why rotation takes place at all scales, and why self similarity exists at all scales.

The rotation in the geometric mean exists for a square as for any rectilinear form with the same area, thus the rotation along the line of the diameter of a circle is proportional to the area of the square at the centre, but clearly "interferes" with other rectilinear forms with the same area OR the same proportionate area to the circle area.

Thus a seemingly straightforward construction has really complex underlying dynamics, and that includes radially.

The rotation is always a rotation of the diameter or a section  of the diameter through π/2 radians about a point on the diameter.

I believe we really need to restrict + and - to addittion and subtraction and use some other notation for direction.


 

Equally clockwise


or even more directly and familiarly Left Right Up Down et al. !

Finally, exactly these relations exist between =-1,

    and the more general

But this of course leads to a recursive definition of √-1.

The only sound basis in my opinion is the construction via the geometric mean, to which direction/ orientation signifiers are appended.

Inshallah! √-1 (http://www.islamic-dictionary.com/index.php?word=inshallah)

'h¶h cause you to feel gratitude!

Why the flip algorithm explains the subjective nature of the perception of orientation/direction, and how Bombelli captured an essential algorithm which we later used to model mirrors in geometry.

How Geometry is a symbiotic relationship between external structures and our perception of the relativistic relationships.

This is the general form of a trigonometric planar measure.
The a and b are natural "numbers" that is numeral names of integer scalars.
The cos0 is the directed magnitude with orientation 0 and magnitude 1 and the sinπ/2 is the directed magnitude with orientation π/2 and magnitude 1, both these "magnitudes" are in fact ratios of magnitudes, and at  a stroke we have a recursive notion of magnitude at the heart of directed "magnitudes".

The + is an aggregation symbol denoting the two measures are related and brought together by some measuring device into this structure. The measuring device is the cartesian coordinate system.

So far so good , but one wonders why this arrangement when we could use another invention/convention the ordered pair?

I have discussed linear combinations in my blog , and this linear combination is a relic from the days when the interest was in finding the roots to polynomial of quadratic and up to quintic form. The problems were written down in rhetoric not notation, and the geometrical forms they related to were clear to all geometers by diagram or direct description.

Thus "are"as always involved "ar"ithmoi which involve "multiplying" or stacking by the 2 orthogonal sidees of the gnomon. This lead to rhetoric which carefully detailed the method of proceeding through the calculation to the solution,

Of course notation inevitably crept in to shorten the tedium of repetively writing common phrases , but the practice was and is the beginning of removing individuals from the understanding of the iterative , rhythmical nature of mathematical exploration and solution. The link to the fractal foundation of mathematics was being erased by notation and abstraction.

So the "plus" gate arose as a shortening of a physical and mental process of association/aggregation/sequential relation.

If a structure existed in part or was constructed from parts the "plus" gate eventually replaced this description, and we lost the geometrical relation of the objects or parts through the notation. In particular we lost the dynamic relationships through the static , fixed notation. thus our solutions only applied to items in static equilibrium, and with a "fudge" to items in dynamic equilibrium, the fudge being we ignore or remove the dynamism from the interpretation, we factor it out , we do a modulo dynamism arithmetic!

Thus the directional nature of the "plus" gate is ignored as is the directed nature of the "minus" gate: aggregation and disaggregation as processes are hidden from our view, the shattering of a plate into a million shards looses its triking analogy to subtraction and division.

 So i was rhetorically correct to "write" a 3d or 2d object in a linear form, because that was how writing was done, and the syntax of oral and written language was utilised to describe a dynamic apprehension of a 3d structure.

However, visual geometers, such as Descartes found an economy in notaton that was "diagramatic". That is to say ABCE could represent a diagram, be a code for a diagram which enabled with a little practice the ability to think visually around a form simply by reference to the "order" of letters. Thus geometers were able to reduce diagrams to notation providing a few syntax rules were observed. Slowly the order of writing rhetoric or tracing /drawing diagrams "on the page" became important, and then significant.

Many early mathematicians were prodigies of one sort or another, and many were visual and possesed eidetic memeories, an image was remembered throughout a calculation or exposition so a reference to it was hardly necessary! To them a "proof" was a guided tour round a geometrical form from which the conclusion became obvious!

Some mathematicians were not so visual and they utilised symbol or rhythm. Indian mathematicians culturally favoured rhythm in their sutras, even though they were every bit as visual as any Greek geometer. Chinese artisans revered the symbolic brush strokes. Each culture showing a different appreciation of the representation in a different form of the structural geometry within their experience.

Thus By Descartes the ordered pair came to represent a position on a plane, and rather awkwardly and strangely a solid form could be described by a sequence of datum points, again reducing the dynamic apprehension to a static form!

Now the sequence of data became important as did the order within the ordered pair. The whole page became a "table" of points, every position on the table being significant. Thus the notion of "tableau" or matrices became important and a 3 dimensional form was no longer apprehended it was read from a page! Or so it appeared.

In fact those who understood the subject in hand still played with the 3d model, still described it rhetorically, and interpreted between the symbols and the notation. Why? because they were taught to behave in that schizophrenic, mad as a hatter way!

The notation and codification of mathematics meant that it became a closed book to nearly all but a few. And when a book is closed it may as well be thrown in the rubbish pile!

So we end up with these archaic form rubbing cheek by jowl together and we are left wondering...

The olynomial form x³ is a cube in geometry . x³+3x²+3x+1 is another cube related to the first by additional areas(x²) and lengths(x) and a constant 1, which is the unit cube.

But how can that be? Avolume equates to a volume surely? Indeed it does, and i by sleight of word have misled you, and also by convention and notation!

x² is in fact 1*x² a volume not an area
x is in fact 1*1*x a volume not a length.

Our mathematical conventions confuse us, mislead us and lie to us, and all right in plain sight.!

The polynomial form is therefore a misleading notation unless we add that all terms are of degree three! or rather represent volumes!
We do not add this, we in fact strenuously encourage the notion of the form, the notation and not the geometry.

I have written on the history of why we do this, and it is no just economy of writing,it is an absurd arrogance that some of our pedagogues have displayed which has come down to us as de rigeur, the fashion, or in modern speech, this is how we roll!

Well stop rolling and start making sense you mathematicians!

So the linear form i started off with in fact describes an area on a plane in terms of directed trig magnitudes.

Well now so does a+ib.

So why is a+ib ≠a cos0+sinπ/2*b ?


Well the answer is due to another fudge Euler made

Bombelli observed that√-1*√-1= -1 as expected, but he skipped over
√-1*√-1=√(-1*-1)=√1=1

What happens when you show a mathematician this? He/she gets cross, tells you off and speaks to you as if you were some numskull!
Bombelli avoided this because he was high on greek juice! He knew that he anted the "symmetry" not the notation to be right. There was no notation for this in his day so he defined it, and he defined it geometrically in terms of symmetry. But what symmetry?

He defined it in terms of mirror symmetry. This is what i have more generally called the flip algorithm,because once he defined it for opposites it has to be defined for all directions.

Euler, unlike Newton made he mistake frequently of √-1*√-1=√(-1*-1)=√1=1 even after he tried to avoid it by changing notation to

i*i=i²=-1 (he in fact defined i=-1/i and 1/-i)

Bombelli attempted to define a geometrical relationship, observed when solving equations by neusis. He observed a conjugate"reflection" as in a mirror while using his carpenters square to find "roots". He also observed a π/2 rotation inherent in his use of a carpenter's set square.

Using directed magnitudes "flipped" things around! Finding the "square root" was not something Bomelli would have understood as it is a description derived much later from ExQuadrature, meaning "the making/measuring of a square". What Bombelli did was find the geometric mean.

The geometric mean of √-x was geometrically obvious: it was the mirror image of √x!

This is where Bombelli got his inspiration for his ditty "piu di meno"

piu di meno was taken to mean the "square root of minus", but not by Euler. Euler took it to be an imaginary magnitude, possibly infinitely large with behaviours akin to Brahnmagupta's shunya. However i believe what Bombelli meant was Radice- the finding of the root by geometrical mean!(GM)

Thus the positive GM of the negative by the way of the positive GM of the negative makes a negative!

This symmetry left him only to Guess what the positive GM of a negative by the way of the negative GM of the negative makes!

Symmetrically he had no choice, his mirror flipped it back into the positive realm!

It was Bernoulli who suggested the circle diagram to Euler, and it was Wallis who suggested the idea to Bernoulli years prior to Euler exploiting it in his famous formula. However, Roger Cotes had precede him by decades due to Wallis and Newton's influence. The Wallis school did not seem to have the problems Bernoulli and Leibniz had with imaginary magnitudes or negative logarithms because they were greek geometers to the core!

The reason why √-1*√-1=√(-1*-1)=√1=1 does not work is not because it or you is wrong, but because it is divorced from the underlying geometry and symmetry involved in the finding of the geometric mean.

So why is a+ib ≠a cos0+sinπ/2*b ?

Well in fact is is equal to it, because both involve finding the geometric mean of a directed magnitude and the trigonometric form makes explicit which directed magnitudes are involved"

√cosπ = sinπ/2 and sin3π/2 because the geometric mean of

cosπ*cos0=sinπ/2 * sin3π/2

=>cosπ*1=sinπ/2 * sin3π/2
=>cosπ=sinπ/2 * sin-π/2
=>cosπ=sinπ/2 * -sinπ/2
=>cosπ=-(sinπ/2 * sinπ/2)

So what do mathematicians do? They hide the  1! By this i mean, clearly by current notions

cosπ=-(sinπ/2 ) and sin3π/2 and sin-π/2

but √cosπ ≠-(sinπ/2 ) or sin3π/2 or sin-π/2 or sinπ/2
But of course it does by the geometric mean which is only apparent when you show the one.

But the treatment also shows the inherent reflection in the definition of √-1 and the inherent rotation in the squaring of the geometrical mean.

So Descartes and Euler missed the geometrical significance of the imaginary magnitudes. Descartes viewed them only in terms of solving geometrical equations , Euler recognised their use in extensive periodic series, but the final twist of the reflection in a mirror escaped them both.

Now there are many natural forms that dynamically move in a symmetry that is explicable only in terms of a mirror. We have to extend our notions of the trigonometric arithmetic to allow us to reflect this. In particular

√cosπ = sinπ/2 and sin3π/2

Which we can extend to

√cos(π+ø)= sin(π/2+ø) and sin(3π/2+ø)

In the mirror world we can easily see that geometric structures and constructions have there counterpart, so we know that the magnitude V-1 exists but we cannot get at it. We can measure it by measuring it's counterpart in the real world.

Now the fundamental nature of the orthogonal mirror can be established, the orthogonal mirror has a reflection in the mirror called by me O. Now the reflection of my actions in the mirror as reflected in O are linked to the real world by rotation. Thus we see the necessary link between orthogonality, reflection and rotation.

Euler in fact wrote down the relationship between what we now should call the 4 roots of unity, but it was De Moivre And Cotes who generalised it decades before him.

Where I am at the moment, I see all the pieces underlying the notion of complex magnitudes, and I realise that they have been cobbled together with builders cement to have a shape they cannot according to their nature have.

It has taken 600 years to make this hybrid and to make it work, but it is too smooth and manicured to be anything other than a fix.

In another thread i come to some peaceful resolution of this topic. However, my appreciation of the fitting of properties to suit, and the interferenc in the flow of computation actually highlighted something in my mind that lead to further investigations. I skimmed through Hamilton's work and decided to study couples. From that encounter i formulated a concept called by me a motion sequent.

I skimmed through Maxwell and from that learned that physicists were reaching out for a 4d algebra that could be applied, Hamilton being a scientist promised to fulfill that need, but eventually his work was passed over in favour of someone else's.

For a time it looked like a concerted movement toward a vector algebra based on other less partisan notions of a vector, but then i skimmed through William Clifford and found that he had actually superseded Hamilton! What a shame! to spend your life betting on a horse to find it is beaten at the post by a horse from your blood stock!

For completeness sake i skimmed Grassmann's work. The Ausdehnungslehre i found was a mistreated masterpiece. In fact, it was the real source of why Hamilton was superseded. Through Clifford and Gibbs and Peano Grassmann stretched out a long arm over the future course of mathematical physics and mathematical geometry, and he was totally obscure.

I have made some extensive sketches of his importance in my blog  (http://my.opera.com/jehovajah/blog/2011/07/19/modern-notation-algebras-subalgebra-space-and-subspaces-in-grassmann-hamilton) .

A treatment of complex vector algebra inspired by Grassmann

I start with 3 generators of the space which will be a plane. I do not define a plane but observe that the 3 generators are straight line unit extensions of arbitrarily small regions and they can be arranged to form an equilateral "triangle of  unit extensions" as the 3 arbitrarily small regions are non identical. As an aside, my spider friends told me to understand how 3 straight lines that intersect or meet define a plane!

To arrange them to form an equilateral triangle i have to make certain relative translations of the extensions, and this is done by extending the extensions until they cross, and then rotating the whole extended "lines" relative to each of the points of intersection, that is rotating around an arbitrarily small region of intersection which i will call a centre of rotation, until the conditions of the radian measures are met for an equilateral triangle. Then i shrink the extensions back to their unit size .

You hopefully will recognize the actions of affine transformations in my description. This appears to be Euclidean, but it is not how Euclid would have expressed it and indeed no one else until Grassmann.

Consequently i can write
e₀,e₁,e₂ as the unit extensions of p₀,p₁,p₂ the 3 arbitrarily small regions,
and r₀,r₁,r₂ as 3 relative radian extensions, that is circle arcs between the ends of the unit extensions.

So r₀ is a rotation between e₀,e₁; r₁is a rotation between e₁,e₂;and r₂ is a rotation between e₂,e₀.

The condition is that the unit extensions form a equilateral triangle.

This is now my reference frame in the plane of the equilateral triangle, and i am not dealing with a static situation but an elastic or stretchable one. It is a dynamic geometry as it should have been recognised from the outset. Certainly the Greeks know it was dynamic. Neither are the regions of intersection (or meeting) points nor fixed. Also i am not limited to one triangle in the plane. I can establish a net of triangles to cover the plane with relative reference frames and so provide a reference frame system that is relativistic.

i make the following transformations: call the intersection of the extensions of  e₀,e₁ the origin/centre of rotation and shrink e₀ so the intersection of the extensions  e₂,e₀ achieves the centre of rotation or origin.

This moves e₂ to the centre or origin without leaving the plane and makes p₀,p₁,p₂ the 3 arbitrarily small regions all intersect at this origin. They are defined in this case as coincident. but as you can see it is no coincidence that they are in this arrangement!.

Observe that no rotation extension has been actioned, and this is therefore a parallel translation  along e₀ extension or equivalent to it, as i will  re-extend e₀ to its unit length.

I can notate this using  l as a variable length such that l<1  tending to 0 gives an extension

l#e₀. (the # is to avoid confusion with ordinary multiplication at this stage)

The e₀ is an extension of unit length that is l=1.
 l is the modulus of the extension ®, and of course ® =e₀ when l= 1.

Thus ® represents any extension in the orientation of e₀ and if the extension is dynamically active it is said to be directional or a direction. So e₀ is a pure orientation in the plane but translations in the plane are effected by a variable modulus function and these are generally called directions.

Similarly for radian extensions we have a modulus function and these extensions are generally called rotations. The unit radian extension is simply called a radian. We will see later how it can be made equivalent to infinitesimal variations in orientation while travelling in an infinitesimal direction!

Starting from the above set of definitions and conditions i have to establish a set of rules that if followed give the behaviour of the complex magnitudes. in so doing it allows me to then redefine complex magnitudes as complex vectors and break the association with number , establishing the association with algebraic structures, relations and conditions. this is my goal because fundamentally confusion arises from the misplaces and inappropriate use of "number" which in fact refers to a collection of adjectives, and has adjectival significance. You can already see that the concept of number is scarcely used so far and where it is used it is used as an adjective, in a nominal sense. Although vector is not ideal it serves better in this context. The real idea is a tensor, but i will come to that by and by.

To define a complex vector arithmetic i utilise the 3 vectors/extensions in the plane and the radian vectors/extensions.

e₀,e₁,e₂, r₀,r₁,r₂. The layout is as described in the previous post.

There are many fundamental euclidean geometrical results i am taking for granted at this stage, and of course rigour requires that i give demonstration of their validity in this system. Fortunately Grassmann has done all this hard work.

The relationships i will define are product relations
e₀•r₀=e₁
e₁•r₁=e₂
e₂•r₂=e₀

By which we see the modularity of the system, and thus the inate periodicity, rotational symmetry etc.

e₀•e₁=cos(c#r₀¬k#r₂)#e₂+sin(k#r₀)#e₁
e₁•e₂=cos(c#r₁¬k#r₀)#e₀+sin(k#r₁)#e₂
e₂•e₀=cos(c#r₂¬k#r₁)#e₁+sin(k#r₂)#e₀



where c and k are the modulii of the radian vectors and + is a vector sum, so this product is a vector. ¬ is a radian extension/vector sum.

As you can see i need to be clear what is being aggregated, and the use of + for everything will only be introduced where it has been shown not to obscure. This is in fact one criticism of Grassmann's use of notation. However he explained the conceptions clearly enough to redact the notation to additional use.

The products are commutative by set up. That means that whatever the order of the vectors in the product the set up is not changed and so the referent is clear. This shows that commutativity is related to what the product refers to. If the geometry does not show a distinction then the product will be commutative.

The radian vectors are dependent on the extensions and so they cannot relate to any other extension, so again the product is commutative.

For the radian vector we can write the unit radian relative to an extension as r_u1  etc and then r₁ can be written in terms of unit radians.
eg r₁= c#r_u1

We define the sine and cosine functions as acting on the modulus of the radian vectors to evaluate a vector magnitude. The vector it is applied to is given in the definitions.

The first thing to set up for the complex vectors is the perpendicular basis vectors.

e₁,e₂.

Then rotate this pair until e₁ coincide with e₀ depending on the product. Thus:
e₁•e₂ => set e₀ to the initial setting:
but
 
e₁•e₁ => set  e₀ to  coincides with  e₁:
 
and
 
e₂•e₂ => set e₀ to coincides with e₂.

What this indicates is how the basis vectors are fixed relative to each other but are dynamic relative to the third. The grid reference frame rotates under the complex vector operation

Now writing a general plane vector as
c=a#e₁ + b#e₂  and
 
g=d#e₁ + f#e₂

we can work out c•g.

This may seem to be a particularly arcane way of going about things, but in reality it formalises the dynamism in euclidean geometry and makes explicit what is being done or what action is happening, the Neusis, extension and rotation that occurs throughout Euclidean geometry is what is made explicit here.

The product is designed to model complex vector behaviour and makes it explicit.Because this set of behaviours is useful in solving equations it is kept. Other products can be designed and their usefulness explored.

If you have been following my construction/analysis of the complex vector algebra, you should be advised: this is my very first attempt at applying Grassmann to this issue. So i admit i have made mistakes even if i do not know where yet!

There a 3 special conditions i have not written down only discussed and i will ammend the above post accordingly.

I considered what i had found out through the exercise and thre conclusion is that Grassmann highlights the "fudge " factor. Officially it is a fix!

You kinda know it has to be, but no one is letting on, and everyone is saying, "The emperor has "new" clothes!".

So what comes from that observation? Well surprisingly a grudging admiration for Grassmann. Grassmann was a genius linguist, and if anyone was going to overthrow the "poncey" mathematicians it was going to be a linguist, because of GRAMMAR!

Although i do not argue for a language origin to mathematics , but rather both being the product of the Logos Summetria Response, it is language that has "formalised itself" around rules of grammar including syntax and allowable constructions, ie good grammar and bad grammar, sentence structure word order etc. etc. Grassmann simply says "These are the rules of the game". So a fudge becomes a paradigm. More importantly Grassmann so efficiently strips bare the bones and flesh of mathematical systems that it become possible to see how to construct other mathematical systems. And this is the genius of Grassmann's approach: lay out the rules and follow them . If they don't work change them until they do.

Hamilton initiated this approach to mathematics in the modern sense by his ground breaking treatment of Couples or conjugate functions, that is complex numbers in common speak. All mathematicians of repute knew of the fundamental significance of Hamilton's work on couples. Hamilton also suggested a Grassmann approach in this work. Despite Sartre, only Grassmann could have pulled it off.

So the game is up. The rules we play by are set by a board of judges who opine about this and that and try to give the maximum generality to the system. But it is a system. It is constructed, and it is based on good old Euclidean Geometry.

Start with Euclid and vary the Rules, then invent a whole bunch of gobbledy gook words and "Hey, presto!", you are a professor of Mathematics!

Like Jakob Steiner and other artisans know in their gut, the advice is: Found everything on experiential "geometry", that which i call spaciometry.

So why does every body say complex numbers are essential for physics? That is the propaganda.The relationships within and among the rules are what are foundational to all subjects.Relationships are what are important not number. We are moving on to a more radical set of relationships in physics and science in general.

Calmness ensues Having settled my mind on certain issues i have been able to review the impact of the great Rafael Bombelli.  (http://my.opera.com/jehovajah/blog/2011/08/12/when-younger-i-was-introduced-to-algebra)

http://jehovajah.wordpress.com/jehovajah/blog/

To understand his significance one has to place him in a line of thinkers starting with Pyythagoras, and including Euclid, Brahmagupta and then the dissolute and despondent Cardano.

The natural elegance of the man is why he became an engineer, and why he contributed so much to Algebra. He introduced a free flowing vector algebra which could not be understood or realised until Hamilton and Grassman. No one had the word vector to describe what Bombelli held in his hand when he solved difficult equations. As snobbery goes that takes the biscuit! A common artisans tool was the implement to solve all the conundrums  of polynomial mathematics.

Bombelli's vector, is Pythagoras' mystical law of "arithmoi", and Grassman's fundamental exterior and inner products in his Ausdehnungslehre. Without Brahmagupta and then Bombelli we would not have the dynamic descriptors of reality in a consistent aggregational geometric structure, which Pythagoras opined was a fractal model of reality based on Harmonised ratio scales.

We would call it standardised ratio scales but the principle is clear: whatever monad or unit we choose as a standard there is a ratioed representation of it at infinitesimally smaller and infinitely larger scales.

I have written 2 more posts on this issue

the-shunyasutras-and-1 (http://jehovajah.wordpress.com/jehovajah/blog/2012/05/04/the-shunyasutras-and-1)

completing-the-square-by-the-gnomon (http://jehovajah.wordpress.com/jehovajah/blog/2012/05/08/completing-the-square-by-the-gnomon)

The second of which i believe lays bare Bombelli's method as he derived it from Euclid. The process algebra which i make mention of is not yet complete and may never be completed by me, but nevertheless it gives me something to chip away at.  ;D

The fruition of Bombelli's dream has taken a while to come about. However I think that Norman has a presentation that begins to make complete sense, bar the reliance on the number concept.

This playlist brings together most of the important players in the story.

http://m.youtube.com/playlist?list=PL8677C1577A250211

I place it here because not many acknowledge Bombellis crucial innovation in this topic, nor that of Brahmagupta.

The progress made by Bombelli came on the backs of Tataglia and Cardno, both of whom had rediscovered Eudxs theory of proportions and the Logos Analogos framework. Thus Bombelli is comparing different kinds of magnitudes in his pui di meno mnemonic. It is clear that the concepts of Apollonius, as recorded in Stoikeioon book 3 motivated Bombelli's solutions even up to degree 6 in polynomials.

This is an exposition of the thesis.
http://jehovajah.wordpress.com/jehovajah/blog/2013/08/15/eulers-i

The interesting thing to me is how confusion reigns and obscures what after all are simple and obvious "truths".

When I thought of Bombellis carpenters set square or rule as a vector, I was being imaginitive! Turns out that given a clear definition of a point field then a carpenters rule adjusted to the point space lineal structure is in fact a representation of a vecto!

Grassmanns law of 3 points in hich he defines the law of 2 Strecken, is the component definition of a vector in a parallelepiped point field.

Norman shows how Bombelli's vector makes the situation accessible, but the framework required to justify it goes back. Long way!
http://youtu.be/W9-NmJEE3ao