The simplicity of the unity of Dynamic Directed magnitudes

[jehovajah]

I would not want anyone to go away with the notion that the complex numbers are inherently complex as they are not.
Firstly they are not numbers but trigonometric ratios written in a polynomial form.
Secondly they are not complex but they are applied to geometrical situations that look simple but turn out to be complex!

In fact what we have in the past termed complex is better described by thr notion "fractal" as commonly used in these times.

I know some would like a strict definition of fractal, but the genie is out of the bag and normal language development of meaning now applies.That more often than not is a good thing as it indicates  spreading of the idea to the general consciousness, and providing confusion is headed off at the pass can make communication simpler.

Most of us are familiar with negative numbers but maybe not understanding that their proper name is directed numbers, as they have a geometrical application beyond double entry book keeping. As a consequence we may be familiar with the rotation of the number line to describe negative values. We however are misdirected precisely at this point in our understanding.

We may then come across the trigonometric ratios and the importance of direction and quadrant without ever making the connection. We may then be introduced to the weird and wonderful world of complex arithmetic through the imaginary "route", exploring briefly and confusingly the roots of x^2+1=0. Again no understanding or explanation that makes sense is usually given, and the relation to trigonometry is totally void.

Finally vectors may be thrust upon you with the trigonometric conversions and no understanding of where you are, where you have just been, and where this is all leading you toward!

I hope that fairly explains an initial interaction with the so called complex numbers.

In my experience one never quite gets over the √-1!

So lets get over it together.

First of all directed numbers are a special case as they only address opposite directions! What if a directed number is not in an opposite direction, how do we notate that?

If you keep this question in mind then everything else becomes real simple.

We chose, confusingly i think, + and - for the opposite directions, but what did we choose for other directions?
Bombelli attempted to set up a notation for the other directions but was cunfused himself. He chose +√- (piu di meno) and -√- (meno di meno).

This was quite a complex sign which for a long while had no commonly accepted geometrical meaning, but which for Bombelli and Descartes always meant at right angles to!

Therefore as geometers they always looked and moved at right angles to the +- direction in their geometry.

When it became popular to use their system by fixing the ordinate and coordinate in the page then the right angles direction became known as y. Thus historically y came before i  but they both represent the same thing, the orthogonal direction to the +- direction you define on the page.

To say this became obscured is to make an understatement.


Most of us have heard the rule of thumb: the square root of a number is that number you have to square to get the number you start with

Some are familiar with its origin in the geometric mean a*b=x^2 for a pair of crossed lines divided in the ratios a:b and x:x.
These in turn are magnitudes of similar triangles where a:x=x:b for corresponding sides. Thus the geometrical construction of finding the geometrical mean underlies the expression "to square root"

When directed magnitudes are introduced into the geometry what we find is that square rooting involves a rotation through π/2 from the positive direction, in either  an anticlockwise or clockwise rotation.

Thus, very straightforwardly and without any artifice the square root of a magnitude in the negative direction involves rotating clockwise or anticlockwise by π/2

We simply have to set up some convention for these rotations and the simplicity is made secure.

So for example √-1 may be defined as a clockwise rotation by π/2 and -√-1 as an anticlockwise rotation or vice versa. However The Bombelli operator does the job perfectly and should be retained as the standard.

So the rule of thumb then becomes: the square root of a directed magnitude is the rotation of the direction by π/2 and the geometrical mean of the magnitude, under the rules of operation described by Bombelli.

We need only now to realise that √-1 is a trigonometric ratio called if anticlockwise rotation is defined or if clockwise.

The definition of √-1 is implicit in the trig identity:

  geometric mean of=*

From which we derive as a directed magnituce with direction radial in the unit circle and magnitude unity from the unit circle.

It is to be called the fourth root of unity, thereby recasting all numeral schemes as scalars of roots of unity. Our understanding of measure is to be derived from this basis, making -1 simply a measurement in the direction - of 1 unit magnitude, and allowing such measurements in an infinity of directions denoted by radial directions in the unit sphere, with a special subset in the unit circle.

In the unit sphere we define the eighth root of unity for the sphere as

In 3 dimensional spherical  

geometric mean of(geometric mean of(,0,0) *(,0,))
Where the magnitudes of the vectors/ tensors are used.

From this we see that a matrix or tensor format is desirable over a linear form, but a linear form can be written with the additional conventions that the axis are represented by orthogonal vectors, and the aggregation is an algorithm for defining a position in 3 space.

[jehovajah]

The versine  and the coversine reveal the general geometric mean definition of the square root . as it relates to the unit circle.

Thus the diameter of the unit circle is coversine +versine

and the area of the arithmoi or rectilinear forms that can be represented is the product coversie*versine

The geometric mean of the coversine*versine is the sine.

Thus we have a definition of the general square root identity for thr unit circle

* =

This is of course relating the geometric mean to the right angled triangle in the unit circle, where pythagoras gives

= .

The analytical use of the geometric mean reveals that if directed magnitudes are applied rather than absolute magnitudes, then complex behaviour results dependent on the sign of the magnitudes.

Since the sign is just shorthand for opposing directions in general, the interplay of the opposing directions in directed magnitudes accounts for the complex behaviour found in the study of complex "numbers".

This reveals that recursive definitions or applications of just the directions which oppose each other have a powerful significance.

Each relation and /or set of relations chosen, then can form a seperate algebraic field, group or ring on the trig ratios which may or may not return useful applications for the attributions we initially gave them.

The relations between these fields etc are admittedly complex, and any arithmetic devised from them will have "complex" geometric associated forms. This does not mean they will look necessarily out of the ordinary, but rather the relation between parts of the form will exhibit complex dynamic relations. These may or may not be useful in describing, cataloguing or displaying physical phenomena in the experiential world.

All in all, i mean to say that the simplicity of the greek dynamic conception of geometry and their development of trigonometry has founded the study of reality on about as sound and useful a foundation as we can humanly devise!

The proper area of mathematical research turns out to be in the area of these trigonometric relations and identities, with of course the biggest trig identity innovation being Napier's Logarithms.

I have found and am finding a euphony of applicability to all measurement in utilising these simple but profound relations ratios and conventions derived from spherical and plane trigonometry and geometry. In addition the general naturalness of spherical and polar measurement systems over cartesian.

The unit sphere as applied by Eudoxian proportional theory found in Euclids elements is indeed the "bees knees", and allows for the development of relative reference frames and Napierian logarithms as the most applicable measure of physical reality and fractal design and demographic description,

For example it immediately becomes apparent that Newtons Laws of gravity are a second "order" approximation of an infinite logarithmic series description of gravitational behaviour!

Another of my favourites is that mass is the logarithmic count of charge particle quantity  

)

 where q is the charge vector  of a single region with magnitude of charge, orientation of the region and 2 spin axes, one in the direction of the orientation and the other orthogonal to the direction of the orientation. I prefer to call them dynamic magnitudes or dynamic tensors.N={0,1,2,...,Avagadros number/constant}.A is advogadro's constant/number.

A vector sum is implied to denote the charge summations which may or may not equal the zero charge vector.

These are just a few random thoughts that arise from the application of the simplicity of the trigonometric identities and relations enhanced by  Napierian Logarithmic trig identities and polar coordinate reference frames.

[jehovajah]

Since the sign is just shorthand for opposing directions in general, the interplay of the opposing directions in directed magnitudes accounts for the complex behaviour found in the study of complex "numbers".

This reveals that recursive definitions or applications of just the directions which oppose each other have a powerful significance.

Each relation and /or set of relations chosen, then can form a seperate algebraic field, group or ring on the trig ratios which may or may not return useful applications for the attributions we initially gave them.

In my blog i elaborate on this vague set of statements. The importance of mirror reflection is what i am discussing here without realising it clearly.
It is a thought that Bombelli came up with his operator rules while looking in a shaving mirror...I wonder? This observation applies equally to Brahmagupta,although a more meditative repose seems more appropriate.

The mirror reflection is why i elsewhere describe "complex" arithmetic as a fix, and why Lewis Carrol sought to use the  paradigm of a looking glass to parody Hamiltonian "mathesis" of the imaginaries.

The mirror is for me a forgotten geometrical tool, but its significance is so much more than that!

Look at this:

general linear trigonometric form for a planar point/vector/polynomial rotation/directed magnitude



cos0 is the directed magnitude in the orientation 0, is the directed magnitude in the orientation

Thus the trig ratios become or are used as directed magnitudes. This in and of itsellf is worthy of the designation "complex"

=


Now here comes the fix, the switch.
The versine coversine identity is written in magnitudes. We need to to rewrite it in directed magnitudes.

The versine becomes , the coversine becomes (nb!)

This change results in the covert "mirror reflection" of the versine into the coversine.

Now at



And similarly
 at



Thus through this covert mirror reflection we can apply a geometric mean(GM) identity without dropping down into the direct GM!

Thus we now get

=

Which gives a result in the general directed magnitude form



I wanted to show you this as it reveals the source of rotation (GM) , mirror reflection(the directed magnitudes) and the fundamental basis in trig ratios of the "complex numbers".

I do not suppose that we would have easily located these sets of identities and relations if we had not had Bombelli's operator and insight. As it is, even with his insight and the minds of some great mathematicians we still managed to not quite put it all together until now. 600 years of fumbling in the dark!

Of course this is only one of an infinite set of identities and relations amongst the trig ratios. This makes the trig ratios a rich vein to mine.

But it does not end there. The trig ratios are formed on a closed boundary. We can extend the trig ratios to an open boundary, specifically a spiral. The spiral i choose is Theodorus spiral. It probably adequately repesents a Hilbert space and links in to quaternions.

`i wonder if hyperbolic functions are doing this already?

Before i leave complex trig ratios in directed magnitudes there is an extension into 3d that follows the geometric mean to find the cube root.

[jehovajah]

I finally went on to the realisation that it is not mirror reflection, reflection in a plane that is fundamental, but rather reflection in a centre of rotation. The difference is subtle but profound.

[jehovajah]

My latest thoughts on this process.

http://my.opera.com/jehovajah/blog/2012/05/04/the-shunyasutras-and-1
Jehovajah the shunyasutras and at Wordpress.com.

Google it.

[jehovajah]

As I am currently looking at Quaternions in some detail it is just a note to say that the master of the imaginsries, Sir William Rowan Hamilton, says the equivalent of what I have stated here.
He defined quaternions as the quotient of directed lines in 3d space. He later called these directed lines vectors. Thus quaternions are the quotient of 3d vectors.

Now a quotient is the correct term used for the resul from performing a fraction operation, and a fraction system is a ratio specifically defined as a number system. Thus we are looking at the ratios of vectors when considering quaternions. However this is a bit confusing until you realise Hamilton established the foundation of group theory, and what he is saying is that the quaternions form a group structure with 2 operations commensurate with addition and multiplication, and they have a division algebra. Thus the quaternions were the first group algebra with a ring structure which also supported division. They lack one property which is commutativity, but that in general is a special property of. Indirect reference.

Of course vectors derive from Hamilton himself, and they are a trigonometric combination structure. The gauss combination of vectors relate directly to this trigonometric form. The use of numbers and the Cartesian framework tends to obscure all this detail about the fundamental nature of vectors.

Now since vectors are lines, they cannot be operators. This clears up the confusion about I which is a vector not an operator in this group structure. The operator is and always has been the rotation motion which has never had a proper symbol to identify it. We have only ever identified the resultant of it. In polynomial rotations I dealt specifically with that issue without ever fully understanding it!

[jehovajah]

The Stoikeioon book 5, in which Eudoxus introduces the Logos analogos concept is the root cause of this difficulty experienced by Cardano and Tartaglia. They had inherited analogos duality without realising it was different to isos duality. Bombelli, by sticking closely to the Stoikeioon, or rather the works of Proculus and Pappus , at least sensed the difference and created appropriate notation.

Brahmagupta was fully aware of the difference and used the logos analogos methods to develop his concepts of Shunya. Within this concept, the notion of yoked magnitudes is developed to explain fortune and misfortune, both from Shunya. Thus to conclude that Shunya does not mean everything is one of the grossest errors of translation ever made! Traduce trattore !

Simply, in isos duality quadrature naturally was based on neusis, that is on gonu or rotation around the circle. In book one Euclid demonstrated a construction that showed how to transform any rectilineal form into a square that was its isos! This naturally involved Pythagoras theorem, which naturally is in regard to an Ortho gonu, a rotation of a right angle!

However, when coded in the Logos Analogos duality, all neusis has to be done prior to the encoding. Thus it is hidden within the code. It only shows up due to the rule that different kinds cannot be compared. Thus  a magnitude in a different direction or orientation is a different kind of magnitude to one in the standard direction.

This fact, because it is a subjective distinction , was appreciated by the Pythagorean school where Eudoxus learned of it and developed his full theory of logos analogos duality. The myth that created a crisis among the Pythagorean leadership tells us more about the geopolitical situation in Italy than it does about the truth! Pythagoras faced down an attempt to wrest control of his commune by some politically motivated " devotees" who wanted to destroy his growing influence in the area where he gave Kairos!

In the Logos Analogos system, because of its rigour, rules or formal procedural statements are fundamentally necessary. It is by following these rules that agreement with the more natural, veinal, fibrous duality is obtained. The logos analogos duality is a subjective internal judgement deliberately linked to the mosaic patterning on the floor of the pythagorean mousaeion . Consequently it represented a generalisation of the Isos duality, which enabled Pythagoras and his school to link projective geometries together.

The 3 main projective geometries are the parallel projection, the point projection and the circular projection. These were all projections onto a line. Or in 3 dimensions( Mekos, plates, bathos) onto an epiphaneia, especially an epipedos( mosaic) os speripedos surface.

The introduction of negative numbers was some invention that although traceable back to Brahmagupta has a lot more to do with the misconceptions of later Mathematicians. It was heavily resisted and resented when first misconceived. The fact that Cardano and Tartaglia resorted to them was considered highly inauspicious!

The confusion they created was deliberate. It was the custom to claim intellectual primacy by sending coded messages to publishers, societies etc, and it was the custom to keep secret what one had discovered. Both Hooke and Newton are examples of this tradition.

Descartes thus never fully understood Bombelli's prior Art in his "ars del Algebra", declaring the complex roots as "imaginary" in a perjorative sense. He did tend to big up his own ideas in La Geometrie which is an algebraic text . This annoyed Newton, who had no problem with the , because he understood the logos analogos duality, and utilised it throughout his Principia.

It is therefore less remarkable tht his student De Moivre , along with Cotes fully explored and explained the Cotes De Moivre theorem on the so called roots of Unity. These notions were heavily linked to the ongoing work on the tables of the Sines and the newer Napierian and Briggs logarithmic tables. All these works were the centuries long computation of the logos analogos ratios which underpin our notions of numbers and mathematics today, and our empirical metrics and data analysis and synthesis.

[jehovajah]

I discuss how the Circle theorems in Stoikeioon 3 miror Apollonius' work on circles, and how it provides the framework for the whole of "modern mathematics", including the confusion about complex magnitudes. The problem arose due to a poor understanding of Eudoxus' theory of proportions.(Logos Analogos)

http://my.opera.com/jehovajah/blog/2013/08/05/factorisation-and-the-sphere
<a href="http://www.youtube.com/v/qNMFNk6o_xY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/qNMFNk6o_xY&rel=1&fs=1&hd=1</a>

http://jehovajah.wordpress.com/2013/08/02/eudoxus-on-logos/

<a href="http://www.youtube.com/v/YAdEfQsIGt8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/YAdEfQsIGt8&rel=1&fs=1&hd=1</a>

While Norman gives a modern overview, it is a modern interpretation of the greek in the Stoikeioon, so unfortunately it has to be ignored. I show it to highlight the confusion created by the misconception we call number.

[jehovajah]

The concept of number arose in the Renaissance, as individuals rediscovered and translated ancient texts, principally Indo Greek texts as preserved in Arabic, Aramaic and Persian.

The introduction of negative magnitudes was an innovation That philosophers have never recovered from. They were resisted for the better part of their history, but were finally bludgeoned into our consciousness by determined Pedagogues. The consequence of numbers and then negative numbers leads inexorably to imaginary numbers.

While Descartes is often opinionated, his etiphet "imaginary" was apt and not at all derogatory initially. He did however deride those who attempted to make such magnitudes representative of "real" magnitudes in heaven or on earth! However, his argument was always on shaky ground , since he accepted the notion of numbers per se.

We do not need numbers. The concepts, algorithms, tricks and shortcuts they provide are based on sensory and kinaesthetic experiences which are all utilised in many other ways, and with many other meanings.

the ingenuity and inventiveness of the human mind is astonishing, but due to these ideas we overlook the similar and profound genius in those other animates we share this planet with.

At the level of cells we find an even greater similarity, and truly at the level of so called microbes, every so called uniquely human or animate or plant trait is evident in abundance. We marvel at the computational power of our processors while ignoring the astonishing computational power of the organisms we call enzymes, which in the end are proteomes.

Are crude molecular theories simplify and conceal an incredible complexity. while seeming to give us god like powers of transmutation. None of this is due to us or our ingenuity. We tap into an incredibly dynamic universe which exists without us, creates without us and destroys without us.

We are here for the ride!

[jehovajah]

Sometime in the past 5 years, I do not remember when, but I distinctly remember where, I realised that mathematics had never defined the unit quadrant in the plane in the Wallis -Descartes-DeFermat coordinate system.. That is to say that Bombelli had defined it in terms of his carpenters set square , and tried to communicate it through his pui de meno poem. But others had ignored it. And Wallis got close to defining it when he said that Square root –1 must be in the plane of the coordinate system, based on his work on the circle and general Conics.

<a href="http://www.youtube.com/v/EnxV3&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/EnxV3&rel=1&fs=1&hd=1</a>


The problem was that geometers were happy to define the coordinates of a square in positive terms, but. Not in mixed terms. Thus a rotated square falls into 4 quadrants. By the sign rules the positive square exists, the negative square remained undefined, or considered a reflection, not a rotation..

The difference this makes is in the definition of the square root. Geometers were happy for it to be positive, less happy for it to be negative, but they conceded, but they would not have it mixed between the 2! Thus they would not define the other 2 roots of unit square. But by calling it i for for infinite Euler slipped it in as an imaginary quantity or magnitude with strange behaviours.

This was a strange algebraic number notion that was completely bemusing because no one would geometrically define the reflective quadrant as a rotation involving mixed axes.

Once we do allow mixed axes in the definition of the Square root we resolve the problem and return to a methodological consistency.
The method Norman shows can be used to demonstrate that the square root of -1 oscillates between 1 and –1, which is made sense of by defining the reflective square as the negative unit square which has sides which are measured positively in one axis and negatively in the other axis.

Switching to vectors / Strecke enable the rotational effect of the product to be highlighted over the reflection.

It was Cotes who decided that as a magnitude square root –1 was an arc , and this lead him to his famous equation decades before Euler. It was Euler who concluded that in the measurement of Arcs i as he defined it was the magnitude of a quarter turn or a quarter arc of a circle. The valuation 1 and –1 came at the intersection of the circle with the coordinate axes. His famous equation was the first vector treatment of this rotation. Cotes formula was the logarithm and also contained the vector  cosø + isinø, but it was not until Argand, Cauchy and Gauss that this format came to be established. However it was Grassmann that completely reworked the theoretical base of all arithmetical measurements in general, necessary to set out the method soundly.

[jehovajah]

It occurred to me yesterday that while complex numbers is a totally misleading name for these processes, complex ordinals is quite apt!

Most of us know the ordinal numbers only in their lesser role, but in fact ordinals underpin every fundamental aspect of thought. We cannot count without ordinals and we certainly cannot measure without them.

Ordinals are thus suitable for encapsulating the notion of Arithmos, which is a standardised ordinal net . A complex ordinal is thus equivalent to an Arithmos, a net of standard square forms used to order , meaur And rotate in pace!

[jehovajah]

It is a while since I posted here and some links have broken and google has abandoned some platforms!
Wessel gave Gauss the Confidence to publish on imaginary magnitudes, but the French Argand and Cauchy we're steaming ahead with applications and methods, albeit quite involved in layout.
Few really appreciated the work of Newton as expanded by DeMoivre and sir Robert Cotes, not even Bernoulli who handed of the problem to Euler his young protogé.

Like Cotes and DeMoivre Euler studied Newton and focused on the unit circle( that is rafius 1) . This powerful device can be found in Newtons scratch pad, and in the works of his 2 students. The one noteable difference that Euler introduced was the circle perimeter measure or arc segment. However it is reasonably clear that this was what Cotes was about to share with Newton before he died.

The symbol i took on many meanings in Eulers work on the calculus of arcs , but its commonly accepted meaning is .
However it is clear that Euler viewed this as a magnitude of arc! That we have not grasped this is due to an academic blindness to arcs in favour of trigonometric line segments! Newton himself showed how to resolve circular arcs into trig line segments.

The measurement of an arc length is philosophically problematic, but by defining the semi perimeter as pi Euler and Coates provided a consistent measure that was always approximate but always in true ratio to any approximation of the arc segment by a straight line measure .

We do not need a straight line measure, we need the diameter of the appropriate circle to construct the correct circular Arithmoi with which to measure .
The i thus becomes a symbol of arc measurement .

The linear combination of the 2 trig line segments defines i as a quarter arc!

DeMoivre showed that using that arc we can define a symbol for any arc segment in a proportion to the trig line segments .
It is and was a complex notation for the visually obvious arc segment of a circle.

[jehovajah]

A simple line segment and a simple circular arc segment are Things that we call magnitudes. There are other magnitudes  but typically we use one magnitude to represent another, usually a convenient one to stand for a more involved one .
Proprioception enables us to make these kind of links or mappings.
The exploration of our proprioceptive thought patterning as it applies to magnitude was initiated by the Pythagireans. But the Grassmanns took it on board to reexamine these proprioceptions.
Justus along with Abel and others of his era considered the fundamental combinatorial structure of perception and Natures complicity in that. So we perceive Combinatorially because Nature is itself combinatorial?
How can we know?
In point of fact we cannot express how we as individuals know but in general it is an acceptance of associations  .
Sometimes it seems these associations "fit" Combinatorially naturally . But we can never know that is the case, because all we do at a sensor level is compare !

Assuming uniformity is therefore alwaysbourvstarting point. It is the Prime Mover in all our systems of reference or comparison..

But now we can apprehend that uniformity is fractal. That is we fragment topology into uniform tessellating forms of varying complexity at any scale.

The prime mover is itself not moved. That is just a way of saying everything is relative until we accept an absolute such as a Prime Mover, then everything is relative to that Absolute.

The way tjis works out in measurement principles is discussed by Newton in his preamble to The Principles for Astrologers (Mathematikoi). He therefore posits 2 absolutes: time and Space. These are fractal of a uniform structure against which all else is compared and motion is intrinsically revealed in all its forms.

Against these 2 prime Movers Newton reveals the general trochoidal motion in and of space as Tyme is set out as a record of spatial positions!
Thus a series of snapshots records Tyme and spatial position as an indissoluble link or complex. And these snapshots when themselves in motion past the observing Eye play back the dynamics of all they record!!

In this analysis and synthesis the circular arc segment has been greatly mystified because it too is a complex link of 2 magnitudes and dynamics. Nevertheless Newton and His Peers did not shrink from setting it at the foundation of their principles and setting out sound rules to transform between the 2 fundamental extensive magnitudes, the circular arc followed by the subsequent straight line that is dependent on that circular dynamic for definition and realisation!

So I struggle still to explain a line segment as in general a trochoidal line but in particular a combinatorial dynamic of a circular arc and a radial line segment both of which are free to vary independently .

This would seemingly be a recipe for " chaos" we're it not for the remarkable fact that for any given whole circle a right triangle sits upon its diameter with any point chosen on its perimeter!

Thus the unit diameter circle is our most fundamental measuring scheme, and from it we may construct all fractal measures, all Arithmoi.
But more still: all dynamical measures!
Both Cotes and Euler recognised this deep thought in Newtons scrapbook along with DeMoivre. But how to express what Even Newton could not express!

The magnitude i holds the symbolic expression of the quarter arc turn . The amazing identity between the exponential function as an infinite series and the complex sum or rather lineal combination of the sine and cosine infinite series beggars belief,
But when you sit back and realise these are computations of interpolations of the proportions of right triangles sitting on a diameter in a circle it becomes visually clear!

The line segment has no " length" that is intrinsic. It is a magnitude that requires a fractal scale to be designed and assigned to it . Such a scale is what we mean by length. But similarly and simply an arc requires a fractal scale to be designed and assigned to it!  It then becomes clear that the scales are of a different character. They are Not homogenous!
The Pythagoreans stated this in the beginning, Eudoxus explains this in book 5 onwards and yet we still puzzle over the value of !

And , i and the quarter turn are thus indissolubly linked.

So any system of arc and line segments is called a lineal algebra because of the work of Grassmann and Hamilton, but they drew on the works of Newton, Lagrange and Euler, Cotes, DeMoivre and the Pythagoreans to puzzle out the combinatorial patterns we impose though proprioception on our magnitude perception.

[jehovajah]

Why is 2i different to i²?

It is a notational thing. For example  2*2 is different to 2² but they give the same resultant evaluation.
Is i+i therefore the same as -1?
No , because we are adding quarter arcs to get a half arc, but i² is a different process it is a composition of dynamic motions of the plane, not solely an arc magnitude,

Here we come to grips with 2 uses of the label. One identifies an arc magnitude, the other identifies a rotational magnitude , we use the same symbol !

[jehovajah]

Grassmann Hamilton Euler really define I as a planar magnitude, that is a circular arc segment. Thus they defone Planar rotation.
The composition rules for their product designs impose the dilation associated with the rotation. In addition the factorisation of magnitudes imposes constrictions on the way outputs may be geometrically represented .

Learning all these regulations enables an expert to methodically design processes that are notationally simple but procedurally complex!