Since Al Khwarzim there has been a rule in algebra that you cannot or do not sum unlike terms. Terms is a word from the rhetorical days in Arithmetic and stands for a class or a type of unit. Thd unit of course is the monas.
While it is not obvious that a unit has been chosen Arithmetic has been developed entirely on a monas. Hamilton in his work on Couples, entirely develops the arithmetic of integers, rational and reals based on a single monas. He develops the contrapositives at the same time.
The Arithmetic rules of nullity, a unit, inverse or reciprocal, commutativity, associativity, distributitvity the so called field properties. Because the vector base is not made apparent, or the monas construction on which the arithmetic is based, the fact that these field properties actually are structural invariant properties. The relationships based on a fixed structure are naturally invariant with respect to notation order.
So the monas is crucial to establish the arithmetic, but the arithmetic is invariant, independent of the monas chosen.
The monas is thus the area to study, and Euclid, in his stoikeioon tackles this directly as the beginning of the course of study. Euclid studies the construction of monas and the relationships between monas before he lays out the invariant arithmetic.
The idea of monas is a wider idea than the epiphaneia that Euclid starts with. Harmonia is the relative source of the notions in monas, and along with Phusis, the nature of things comprises the dual reductionism of reality into the sciences. However, one without the other leads to an incomprehension of reality.
Monas has many forms, but the most studied ones in greek science were the rectilinear forms that are cyclic, and especially those that are constructible by a comass and a straight edge. The constraint sems quaint to us, and indeed practical artisans did not limit themselves in this way, but the reason for it was in fact "reason".
We, after decades of rational philosophy have lost sight of the meaning of the word reason. It is a geometrical term based on the notions of Kairos and Logos and related to proportinality. In terms of Benoit Mandelbrot we can now refer to it as fractal, with some degree of consensus. We used to use the notion of generality, to describe methods and solutions of a wide applicability, and anlogy for he use in the less strictly defined sense of a method. This stems from the indian approach to the meaning of methods and solutions.
The vedic mathematicians were caredul to only give advice, for they found inso doing that their "resonings" could be made to apply to many analogous settings. This subtlety in thought and insight passed into th algebra of Al Khwarzim and down to algebraiss today.
By restricting the construction process the greatest generality was obtained and the greatest symmetry and applicability, and certainty of invariance,or "idealness".
When algebra took hold in the west it did so after years of "darkness" and many misconceptions were adopted , and many essentials were rejected or not clearly understood. The notion of monas was only really expounded by Leibniz, and it was not understood or accepted. A much simpler notion of units and dimensions was derived,fitting better the growing industrialization and mechanical techniques of the time.
From a botched and failed understanding of he greek -insian Algebra the notion √-1 appeared to haunt the west before accurate translations of Euclid appeared, and the long tenuous link between indian and greek geometry could be fully explored. The Persian spherical geometry and trigonometry was also not well known, and then the calculus was inented on the back of Cartesian geometry. Despite the fact that Newton entirely derives it from Eucclidean geometry, applied to the ideas of Descartes, it was the Leibnizian, European form that set the notation and notions and the geometry was buried in favour of the cartesian Algebra. Wessel then redisxovers the directed nature of measurement, withot the full picture of the monas.
The monas is a directed unit, with rotational symmetries and translation and reflection properties. The monas is a part of a form. If the form is dynamic then the monas is also. The monas aggregates to a form. The dynamic monas rotates. These monads tend to aggregate to circular or closed periodic forms. A monas that does not rotate aggregates by contiguity at the boundary. however a monas that does rotate may be aggregated but only in an amalgamation,
Amalgamations are formed when different monads are aggregated by constructing a new monas out of 2 or more differing monads in such a way that they aggregate to a form.This is mosaic tessellation. In order to achieve this one or more of the monads may need to rotate to different relative positions. the monas thus formed, if it tessellates is an amalgamated or constructed monas.
There are monas in physics which are formed from dynamic monads, the parts(monads) dynamically spinning relative to other parts (Monads), and the structure being combined as a new monas. In chemitry these are called elements or compounds, and the relationships between the monads are described in a more detailed convoluted geometry, fractally linked to Euclids. The set of relations are more extensive involving notions not present in Euclid, but evident in Phusis. Many geometrical, monadic constructions are therefore at the agency of phusis, and this is why Phusis is crucial to an apprehension of reality.
If you are not familiar with the the metaphysics, be aware that Phusis and Harmonia are greek goddesses who typify the order and structuring of reality.Scientist may pay only lip service to the notions, but that is the formal structure of their art.
Metaphysically we may replace Phusis and Harmonia with a fractal description of Reality and its structure and nature, but we would be remiss not to know what it was we were replacing.
Hence the construction of monas is what phusis has been doing and requiring of Harmonia a suitable structure. Harmonia has supplied in Euclids Stokeioon the fundamantals, which have been added too by Appolonius, and Archimedes, and of late Mandlbrot.
We needed a notation of the actions the agent took in constructing a monad. Euclid had done this rhetorically, but the fashion was, how can this be done algebraically.
Gauss was the one who set out the first altered arithmetic monas as a+√-1*b . In fact he copied Bombelli in doing so, although he does not i think give him credit. This was an amalgam, not an aggregation. But because Gauss et all still thought of them as numbers, they applied the arithmetic rules of distributivity in multiplication.
This makes no sense, and in fact it is nonesense, but it lead to correct answers. The reason for this was actually shown by De moivre, geometrically, before Wessel, and it was based on rigonometry and logarithms as discovered by Napier. De moivre was able to use Napiers construction to show their were an infininite class or trigonometric relations that solved certain types of equations. He worked closely with Cotes to establish this fact. These trigomometric relations relied on √-1, not as an ordinary fixed "number", but as an intermittent "number" one that appeared at the right time to make an equation work, otherwise it dropped out. De Moivre and Cotes even explained the rotational cycle of its operation. De moivre went on to use it in his work on probability theory.
Thus these relationships were important, they were related to rotation and they used what was then called an imaginary magnitude. Now i would call it a vector because the notion that it was impossible or not real was more a comment on the profound misconception that the renaissance west had of Euclidean geometry, than it is on the mysteriousness of this notation appearing in a set of conventions. The set of conventions were misconceived due to lack of knowledge of spherical trigonometry, and the general darkness in the west due to religious intolerance.
Now, i say that the notion of a vector is a misconception of a monas, because the vector attempts to construct an algera based on rithmetic, rather than on Euclids monas constructions and other relationships.
Such a development i hoped that Grassmann was pursuing, and i have yet to determine if it be so.
To develop an algebra on an arithmetic is to misconstrue arithmetic. Arithmetic is the algebra of a vector monas, and the monas aggregates to a geometrical form. Thus the spaciometry /geometry contains and constrains the vector algebra, and the vector algebra contains and constrains the scalar arithmetic of the monas vector.
In my thread calle polynomial rotations i actually came face to fac with the fudge that Gauss and others have foisted on the concept of vectors, vector a;gebras and numbers. The only way to resolve it was to fudge the results, over and over again. Now we have cmputers that can do that it makes it seem as if it is right, but it is still very wrong to use a+b√-1 as a linear form, a vector or a complex number. Each of these area n longer requires it, having replaced its notions with more consistent definitions, but still carrying the misconceptions of number.
Grassmann and Hamilton took it as read that this form was the de facto definition and derived alternates to it, but they did not see the need to expunge its inconsistencies.
De Moivre and Cotes were the first to explore the consequences of including the form in their geometrics, and they resolved it into trigonometrics. They could get no farther because o them there was not a problem, and in the context of trig ratios many issues are resolved, requiring only the vector notaion to bring it to the modern form. However from the point of view of the monas the issue is what has been amalgamated?