[center][tex]a cos0 + bsinfracpi2[/tex][/center]
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
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 naural 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)