It is unsatisfactory, all the attempted explanations of √-1. As you may know, i have tried to fathom it. But i finally admit that i have not done so until now.
V-1 is a process.
I am working on the process algebra that underpins all algebras and particularly the vector algebra. Because i am free of the tyranny of number, i can look at the main elements of the issue. The main foundation of the issue is the introduction of the imaginary numbers by Brahmagupta. That's right, the "imaginary" numbers which Brahmagupta called the misfortunate numbers! This is where the problems start.
The problems do start for Brahmagupta. His critics cannot understand what he is going on about. They think it is absurd that he could say that you can thinks of numbers that are cut away fron shunya. Only his students were schooled in his thinking. Gradually misfortunate numbers found a natural home as "debt collectors".
In China, these numbers were represented by black rods. They took the idea from the indian sages and modified it chinese style.. Thus the misfortunate numbers were an underground secretive system that popped up in different places under different guises. At each flowering it met with stiff and vitriolic opposition. It was hated and despised by many eminent geometricians, feared by many superstitious artisans, and the cause of stupefaction among many common people.
The numbers therefore were not considered real or workable except in one place, book keeping. It led to the development of double entry bookkeeping and accounting.
Algebra kept the misfortunate numbers under wraps. They only crept into open use when some geometricians realised they helped to solve certain equations and roots, and revealed a kind of symmetry. But even there they were regarded as a sophistication for the more arcane and abstruse thinkers.
These imaginary numbers grudgingly gained a place in the "mathematicians" toolbox, although geometricians saw no real use for them until Bombelli. The fruit of Brahamaguptas mind now started to take seed and grow, but not until Tartelli and Cardano competed to find the cube root formulas and beyond.
With the innovation of Brahmagupta comes the first contrasting process. The processes of combination, counting, comparing and construction of duality are organised into a superarching series of sequences of duality whose combinatorial aim is a method of solution that gives a numerical result.
A process algebra is a duality sequence algebra that consists of subsequences of duality processes combined in series that support a deductive or inductive conclusion through a duality sequence.
What tends to happen is that the process algebra is instanced and that produces a certain terminological structure or schematic, from which a notation is instanced by a further application of the process algebra. Finally the terminological expression is processed and a numerical duality or ratio is produced as a comparison.
Of course we do not know this as children, nor are we taught this as adults. The nearest we may get to this is in instruction on problem solving or Heuristics , to give it its grand name. In fact, the best method of learning about this process is in learning computer programming.
In the early days of programming insruction, flow charts and graphs were used to visualise the construction process of a programme code. The process irself waslooked at and refined to produce more natural language structures and better machine code and low level language code for coding. Today we have major languages that code in object oriented styles and Hrml which codes around an object to finesse a screen presentation. We probably do not see the same types of flow charts that we once did that make the process Algebra explicit schematically.
Because of blinkers, i did not see that computer programming was expanding my algebraic toolbox. In the early days computers were a dirty word for mathematicians: they were not to be trusted. I well remember my Number theory lecturer pointing out the "error" in computed solutions, and how the whole "numerical solution" paradigm was unreliable at best.
To be fair, computing was routinely represented as number crunching. However, computers never have, nor will they ever crunvh numbers. This is the first clue.
I have been bold enough to say that in fact the problems we have in mathematics is due to the concept number. We cannot even go back to Pythagoras, because he did not use numbers either. Arithmoi are not nmbers but special,styled quantities . And the style of the quantities is important, for they must flow in a rhythm into each other, they must make visual poetry, and soundless music!. With each arithmos, pythagoras associated a dot or a seemeioon, and the dots were counted, given sequential names, in a rgythm, in a dance, in rhyme and music. And with each dot was a shape, and with each coun a shape was apprehended, step by step, verse by verse: combined in the minds eye into a whole.
And the wholes flowed, they transformed and comined, seeking to return to the Monad from which they ere divided.
Thus to Pythagoras, the Arithmoi were before music and gematria and thus geometry. They were never mre"numbers" or "numerals". If they were "alive" it was only as parts of the living whole, the monad from which they descended through the Henads, and to which they strived to return.
Today we consider these transformative characteristics in the mathematic concept of topology, inthe concepts of differential and integral equations of transforming bodies or motions, and these notions do not derive from number, but from quantity or magnitude, and experience of the dynamic reality of space.
Thus our process algebra instances processes. In computing we may similarly call these processes functions, sub routines, procedures, methods, and the structures within which they sit we may call classes, objects or programming structures. Within these objects we will define quantities as variables of different sorts, and these variables can take any quantitative value because they are nothing more than "patterns" of a specific quantity, a level of voltage and magnetism.
We have processes for combining hese patterns o produce more patterns which we can then use to process relationship betweenany kinds of quantity. WE do this though the sunthemaa and the sumbola. The sunthemata fixes by agreement the relationships between patterns and us, and how those patterns will be interpreted by the sumbola, the symbols we use to represent them.
But hang on a minute! Don't we represent the symbols by those patterns? Yes we do because the communication is both ways, and the ghost, the god is in the machine.
This is of course not the clinical way of discussing the relationship we have with our machines, but it is the "human" way. We cannot but anthropomorphise our relationships with the space around us.
So now Hamilton, in founding the mathesis of the imaginaries used the notion of a conjugate Function to explain √-1 in terms of his notion of The science of Pure Time. He had to move to the level of functionality, or fundamentality to found it because the common vitriol was to insist that it was some impossible and imaginary number.
The new functional notioon allowed notation to obscure the obvious number 1 and address the function of the notayion. When Euler used i, he did not make this step, despite his obscuring the number 1 and showing that it related to a quarter turn in a unit circle (which it emphatically does not!). He related it to a quantity, which it is not. And yet it produces a quantity when squared. Wessel went on to show it was consistent with a rotation of π/2 if used with signs and directed lines. Cauchy and Argand again rehearsed the "vector - like" nature of it. Gauss published his thoughts and metaphysical misgivings on the subject. presentng the linear combination of it as Bombelli had centuries before.
It was Bombelli who highlighted the process of it, how it sat within the geometrical mean process, provided his modified rules, first announced by Brahmagupta for his imaginary numbers, provided these were faithfully used.
Both Bombelli and Hamilton understood the notation as a process or function, not a number!
In the process of programming this type of functionality into a subroutine, so that one may instruct a computer to symbolise and perform the calculations of so called complex numbers, one never presents the computer with the instruction "√-1" this step is usually missed out and supplied by the user. However we do now have parsers that can read the symbolic notation as an instruction to call up the complex number subroutine.. Thus √-1 acts as a function call
Thus it is in plain sight now, that √ is in fact a function call, and so is ()2. In fact all surd notation is a function call and our mistake has been not to recognise this in its plain simplicity.
√-1 is therefore not a number, or a rotation or a vector, it is a function that calls any one of these processes to the application of the duality process that is attaining to a result.
The pursuit of this function has led to many innovations in mathematics, but i go back to Newton, De Moivre and Cotes, who used this function call so intuitively as to establish many astonishing results, while Bernoulli and Euler, and Leibniz were arguing about the correct interpretation of the logarithm of the "negative" numbers.
If we now accept that √ is a function call, we may also accept that "=" is also a function call.
Now the question of "rotation". It was the practice of Euclid to rotatr the sides of a rectangle from a segmented line, and in this way to fom the gnomon by which a multiple form or a form may be produced. Thus our notation xy refers ro aline semented into parts x and y and thus x + y i combined or x*y if producted. Thus x*x as a product contains a quarter rotation. Now also the geometric mean process of a*b = x*x contains a quarter rotation between the line a+b and the line x+x, as well as another quarter rotation each when the segments are respectively producted. Thus it comes to be expected that in finding the root of any square a quarter rotation back to a line is to be expected, and in fact th quarter rotation should take the square line back to the perpendicular, while the square could take it to either the so called positive direction or the negative direction. Now since no one, except Bombelli defined a negative square (below the line a+b) and a positive square(above the line a+b) then it was a mystery as to why we should get a quarter rotation. It is no longer a mystery. The adoption of Cartesian coordinates also explains the relation to couples, or as Hamilton called them conjugate functions.
Finally Wessels directed lines and the unit circle conflate into Hamilton's Vector notions, especially on the Cartesian Plane.
Nor ought we to ignore that the Cartesian system is a development of the ancient greek system of using an point and a directrix . and thus the Greeks had algebraic formulae before Descartes. Descartes benefits from the Renaissance, De Fermat, and the printing press.
Now if √-1 is a function/procedure, a function call in an equation, what is i?
i is what is known as a " pointer" in most high level languages. Thus i does not take a value of the function call,it points at where the function call is stored. Thus i returns a procedure address and a=i*i returns the result of that procedure address squared. Using that result as a pointer to a unique value -1 gives us an implementation of the complex process involved with i. We need only to implement -i and we have the whole complex procedure as another process.
Most languages have subroutines that deal with general complex arithmetic in a formulaic way, and also utilise the trigfunction calls to provide the full range of complex processing.
I return briefly to the yoked unities of shunya. These actually i believed derived from the Yi Ching. Brahmagupta had reason to study chinese philosophy as a result of cultural ties with China. In the Yi Ching these yoked quantities relate to the astrological reckoning of fate, how fate is going to throw you a curve ball! These potentialities represent probabilities of certain outcomes, but they also represent positions of roots of unity on the unit circle. Each root of unity is in fact function calls for different positions on the unit circle and thus conjugate the 2 semicircular peripheries by ratios of a+b to x+x,the diameter to the geometric mean chord. This is what Ptolemy used as his angle measure, and relates probability to these ratios in the unit circle. the versine and the vercosine.