Bombelli operator

[jehovajah]

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?.

[jehovajah]

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.

[jehovajah]

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?

[jehovajah]

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

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 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 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 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.

would be nice, with

 

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.

[jehovajah]

Inshallah! √-1

'h¶h cause you to feel gratitude!

[jehovajah]

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.

[jehovajah]

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)

[jehovajah]

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.

[jehovajah]

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.

[jehovajah]

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 .

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.

[jehovajah]

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.

[jehovajah]

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.

[jehovajah]

Calmness ensues Having settled my mind on certain issues i have been able to review the impact of the great Rafael Bombelli.

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.

[jehovajah]

I have written 2 more posts on this issue

the-shunyasutras-and-1

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.  

[jehovajah]

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.