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Author Topic: The "operator" i is more complex than that!  (Read 13494 times)
Description: numbers and things as roots of unity and complex factors
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« Reply #30 on: April 30, 2014, 11:29:07 AM »

This potted history gives a good grounding into the use and importance of i but is of course extremely sanitized!

The actual details of the furore over the complex numbers are very instructive and highlight the weakness or human failings of Mathematics and Mathematicians.

Things ascribed to Gauss often originate with other less well known mathematicians or less well known aspects of the work of famous mathematicians, like Euler.

Both e and i while not unique to Euler derived their modern  meaning and use from his presentations and thinking. In particular i meaning infinity was used regularly by Euler, but only once for imaginary in his work on the calculus of arcs, an institution presented for a doctorate at some stage in his career.

It is interesting to see that the constant value has been placed at a point of infinity in the projective geometries, and associated through the logarithm to the concept of angle?

<a href="http://www.youtube.com/v/5Ui5W_BDa8s&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/5Ui5W_BDa8s&rel=1&fs=1&hd=1</a>
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« Reply #31 on: September 24, 2014, 02:10:50 PM »

<a href="http://www.youtube.com/v/MTddX4Qo-io&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/MTddX4Qo-io&rel=1&fs=1&hd=1</a>

Normans complexions!
If you will give up the idea of number you may start here to understand what i is as a formal entity.
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« Reply #32 on: December 04, 2014, 04:09:29 PM »

I have struggled to understand over the years what i represented. It was confusing and confused, and even Euler could not explain it clearly. He did say that he glimpsed a way through which is why he proceeded in his investigation of \sqrt(-1) and relabelled it i.

I am sure that he imagined it was some infinite value that went out to infinity and returned on the negative side of the counting numbers some how.

To model this he took the unit circle , showing how the sine line segment went up to a maximum then returned while the radius went up with it and came down with it to be evaluated at 1.

Thus the arc length was integral to this evaluation and if i was a magnitude at infinity achieved by a quarter turn then i2 somehow had to be 2 quarter turns.

Thinking of i as \sin (\frac{\pi}{2}\theta) where theta is the arc length scalar with 1 being the tally count for quarter arc segments, then   i2 will be \sin(\pi) and the evaluation will be \cos(\pi)

This was and is confusing and Cauchy and Argand struggled with the notion of Durection cosines  as a way of grappling with this curious magnitude. It was Wessels clear identification of direction with these products that encouraged Gauss to reveal his thinking on the matter. Thus was born the complex plane, which was removed from our reality by design, but impinged on it in terms of rotation.

In light of this , Hermann Grassmanns exposition of the relationship between the complex formulas of Euler and the circular arc segments was a breath of fresh and illuminating air, but few actually understood what he was saying even till now.

We do not need to just remove the arc segments to the complex plane , we need to remove geometry to the reale expertises and all our imaginings to the formal ones! Then we can approach space in the right manner,
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« Reply #33 on: December 04, 2014, 05:00:34 PM »

It also occurs that Euler may well have been thinking in terms of the secant and cosecant forms of the trig ratios in which case a combination of sec + tan is an interesting candidate.
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« Reply #34 on: January 21, 2015, 09:48:26 AM »

X2 +1 = 0

Is the equation that caused the problem.( so to speak). At the time Cardano was learning how to solve cubic Equations. He was working from a mental image of a cube but expressing it in what we now call an affine Geometry of a single line!

Well a cube is not a line, but from ancient Pythgorean times all proportions had been reduced to a single line segment, itself segmented in the proportions. This was easily achieved by rotating the proportions onto the largest segment of the form.

A carpenters rule allows measurements in any orientation relative to another, but ultimately the measurement or proportions can be collapsed onto a single carpenters rule!

Let X be 1 half of the rule and Y the other half . If I measure X in one direction but the object is a square then I have to set Y equal to the X measurement and rotate , thus measuring XY or X2. So now what is XY+1?

It turns out I can add a single unit square to XY right on the vertex , but if I do that I add 1Y and 1X to form an enlarged square. So basically I cannot add a unit square without adding these extra parts and So I have no real solution for a carpenter in the plane.

However if I add one other turn into the third dimension then I can add a single unit cube within the square without adding X and Y. The result is I get X with 2 "prongs " orthogonal to it attached. These prongs are in the X,Y,Z directions but they do not extend X in the  X direction, but in the Y and Z directions , similarly they do not extend Y in the Y direction but in the X, Z direction. When the carpenters rule is opened out again by a Quarter turn, these prongs then construct the single unit cube.

X and Y Also appear but not in the XY plane!

I realised that the common way of writing a degree n polynomial is misleading, because dimensionally it does not make sense. Dimensionally a cube should be xyz not xxx, by ignoring orientation we lose contact with the rotation and we therefore do not understand the difference between X and Y in the XY plane to X in the XZ plane and Y in the YZ plane!
So preserving these distinctions how does this help understand the solution?
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« Reply #35 on: August 13, 2016, 09:12:15 AM »

The question is no longer
X2 + 1 = 0
It becomes
XY.1 + 1.1.1 = 0
Because we are ow dealing with Pythagorean solid Arithmoi ( vide Stoikeia book7)
In this case it becomes clear that to make the undefined cube ( XY.1) the " negative of the unit cube the X or the Y has to be a unit magnitude and a negative magnitude.
It then has 7 possible resolutions in terms of a old cube being tucked away into a cube sized hole. They all can be described as rotations , or translation rotations , but the most direct involves rotating the unit cube by a quarter turn.

The seven solutions is why one cannot simply write the answer as an extension along the carpenters rule . The solution is in fact a construction process in 3 d space and involves rotation , extension and rotation of extension.

The concept of number lost so much of hat is natural to the senses of a skilled artisan , and wrongly elevated the algebraic expression above the physical process it was depicting as the processor( thinker) devised the resolution to the question.
Bombelli's carpenters rule gave him the simplest insight into what was actually being solved by Cardano and Taglia . Wessels surveying intuitively gave him the same insight, direction and rotation must not be ignored !
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« Reply #36 on: August 13, 2016, 10:10:07 AM »

The use of notational devices is very often not explained, and so is mystified!
In comparing or depicting magnitude all we can do is Factor  and count. Katameetresee is in fact doing just that . We can jazz it up a bit, make it into a song and dance make it more rhythmical and fun but it is still Factoring and counting. By the way I m ll for jazzing it up.
 So now if I have a factor i then how do I count?
2i should be two of them right ? But why not i2?

We adopt conventions to depict ubtlety differences.  They both count the number of factors but they depict different processes. The first depicts the sum process. At its most basic that is just sticking or placing factors contiguous to the neighbouring one.

The second depictsbth arranging process! It is more complicated. The factors are arranged in a way that shows Dependency. Thus the order of the depiction conveys which symbol is dependent on which. It also istinguish es the different semantic value of each symbol so for example one may be a scalar the other tomth left or right a vector etc.
 In basicvrithmetic the binary depiction separates the factor nd the count of that factor.

When factorisation was organised to be taught something called multipliction was invented and another concept called division was cooped into the role of the inverse of multipliction!!

I digress.

Thus i2 was a shorthand to depict a way of arranging the factor on the Left of a binary depiction by the symbol on the right( or vice versa)

The basic arrangement of factors in these Arithmoi mosaics could sways be associated to a specific shape. And by using a standard Metron the factor could be arranged not by the contiguous connection of the factor only, but by the oniguous connection of a " part" of the Metron ( say a side for example ). So the count symbol could count the sides arranged along a contiguous line while the factor was counted in its sum position.

The concepts of Area derive from theses Arithmoi. Volume  isvalsona further derivation

That being understood how does That " square" with the circular arc magnitude?
Given i is a quarter arc what is i2? It dawned on me recently while considering curvilineal forms(Shunyasutras) that it should represent an area formed by arranging the quarter arc along a cooniguous quarter arc.
While 2i depict the half turn or the semicircular arc i2 depicts that area formed by taking the semicircular part of a disc and placing it on the top of 2 quarter discs of the same radius arranged to be in kissing contact.
This curious mushroom like shape is one that often complicates geometric comparisons of measurent schemes for area of circular forms
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« Reply #37 on: August 19, 2016, 06:12:16 AM »

This Shunyasutra would have area calculable as 2 r2 and have one pair of arc lengths that form a semicircular half perimeter. Its equation or definition as a magnitude of -1 makes no sense as a spatial " absence" of a square of that magnitude, but as a prime number that is a proto Arithmos it designates a Shunyasutra whose sides are associated to a trochoidal form that emphasises rotation around a circle by. Quarter arcs .
The negative sign or any symbol or superscript conveys only what the defined/ designer specifies , but the fundamental topology of the Pythagorean school asserts a formal basis that is countable and arrange able into a mosaic. In book 2 of the Stoikeia Euclid establishes the straight or good line segment as factorable/ sectionable into contiguous parts. At the contiguous joint rotation is expected in order to create or capture the Parallelogram form. This form is then used for factoring larger forms by which we measure space as we count.

The circular arc is sectioned into 2 quarter arcs. It is the semicircular form within which we define the constancy of the quarter arc for any triangle based on the diameter and emanating from a " point" on the hemi perimeter. The dynamic relationship between a right triangle form and circular rotation defines our algebraic expression of a circle.. This dynamic processes of all of nature all appear to us to be trochoidal, to acknowledge and express that by some harmonising measure is the religious duty (theurge) of any Pythagorean whose contract( sunthemata) with his special muse required a personal symbol( Sumbola) of his/ her devotion.

Eulers i, Bombellis meno, Brahmaguptas misfortunate numeral algebra all convey the same thing
Shunya is everything, and all things being  dynamic or still come from Shunya with a balancing opposite by which we may count, measure and interact mentally and physically with Shunya .
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« Reply #38 on: January 21, 2017, 11:21:25 AM »

I had some further thought on the quarter arc magnitude called by most i xx
NjWildberger will soon post some videos on the complex numbers so called in terms of their algebraic impact, but for me before Algebra is Arithmetic.

Now by Arithmetic I mean the study of Arithmoi, or specific mosaics. It is these Mosaics that givevrisevto so called Geometry as well as the associated Arithmetic.

The Stoikeia as an introductory course in Pythagorean philosophical thinking spends 6 detailed books preparing the student for Arithmoi! The first 3 of those books are often called geometry , but in fact book 2 is quite a sophisticated Algebra!

In my mind an algebra is a symbolic arithmetic, but in point of fact it is the fundamental thought patterns derived from combinatorial considerations of physical objects which objects then stand as themselves or asvrepresentationsvof other specified objects. In this sense algebra is more fundamental than arithmetic, but only if you accept the conventions surrounding subject boundaries.
The Stoikeia demonstrates the interwoven sequences involved in the synthesis of the Pythagorean view of the interaction between mind and nature.

So the sphere and the circle are fundamental to analysing Nature and our imposition of Law on Nature.

Whatever the universe is, it is certainly not what we think it is! But fortunately we can derive regularities through measurements which themselves are dynamic activities we engage in correspondingly xxx

So when it comes to the quarter turn/rotation magnitude  it is clearly independent of the extension of the perimeter of the circle involved. What we find is a quarter turn is a perceptible magnitude best measured against a circle.. But going on fom there we can note that circles with differing diameters have this identical property and the connection between these measure, the duametervandcthe semi perimeter is very robust. It is assumed to be perfect.

Then it can be noticed that this quantified measure applies to all measures within the compared circles. Before you know it we are at the basis ofbBook5 in the Stiikeia, the book that deals with parts of a magnitude and the comoarison( Logis) of those parts. And the comparison is of dynamic growing situations. Within that comparative study certain distinctions are defined:Logos,Analogos, diplassioi( squared multipleforms) triplassioi( cubed) and so on in an inductive sequence, inverse, perturbed and several other comparative relationships between logoi. Proportion or strictly Analogos is only one of a number of identified comparables in these circular magnitudes .

So when the square multipleform is defined, it is defined by an Analogos or a proportion which then carries over into any object used as a part to form a measure of a dynamic sketchable form. By induction the cube multiple form and so on are defined.

That these definitions may apply to lineal ,areal and voluminal objects is by design . That we can not see beyond 3 dimensions is by ignorance of the inductive process these definitions lie within. 


We have wrongly identified orthogonality as representing reality ! It is merely the most convenient arrangement of 3 lineal magnitudes to depict spatial position . However other spatial and structural properties may well be best handled with more than 3 magnitudes. It turned out that rotation in space was one of these .

Using the Quarter turn magnitude panders to orthogonality . It produces some specifically odd correspondences, but like any measurement scheme expertise copes wuth these oddities.

Problems arise with the quarter turn magnitude when we do not apprehend it. So we know that we can add quarter turns, but what does that mean?
The other question is what is quarter turning? For example is it an object, a point a region, us as the observer?
The next question is about what is this quarter turning happening?  So we can have many centres around which a specific quarter turn might happen.

In tandem with this deep consideration of the circle we have the other ratio( Logis) which we call the sine ratio. all these logoi somehow are related and it took a while to find a consistent and robust relationship between them which we confusingly call imaginary even today  xxx
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« Reply #39 on: January 24, 2017, 10:22:18 AM »

http://m.youtube.com/watch?v=OqxYLyGLqcs

Whatever NJWildberger says about circles is always a development of these ideas.

In his latest video he introduces a complex linear combination a + bi based on the rational parametrization of the circle. (1-t2/1+t2,2t/1+t2)

This I correct to
e(it):= 1-t2/1+t2 + i(2t/1+t2)

As it is supposedly in the " complex" plane and I want to draw attention to the Euler -Cotes version based on the trigonometric ratios, or the trig functions and the exponential function.

ei:=cos + isin

or as Cotes wrote it in the natural logarithm version
ix:=ln(cos+isin)

http://m.youtube.com/watch?v=Snq7UJT8EWg

So in this light my discussion of the quarter turn/rotation magnitude i relates to how we think about curvilineal segments specifically in this instance quarter arcs of any general circle contrasted to how we think about the favoured but intensely special straight line segment!

Just as sketched quarter arcs how may we combine them?
The simplest way is to join them at the meet poit where one such arc finishes and the other begins.

The shunyasutra curvilineal segment we obtain depends on whether we rotate the arcs at these meet point or not!

If I use the same arc and duplicate it many times, and carry out the process as described, but insist on no rotation, I draw a curvilineal segment which lies on a 45 slope across the page connecting all the meet points!
Because of this  particular slope the axis i usually drawn in the " complex" plane is in 1-1 correspondence with each meet point on the curvilineal segment or shunyasutra.

Thus 2i,3i etc corresponds to the combining ofb2 and 3 etc quarter arcs without rotation. We therefore ignore the translation across the plane in this type of representation.

Now consider what happens when I combine a quarter arc to a preceding quarter arc specifying a strict quarter rotation at the meet point. The resulting shunyasutra is a semicircle.
If I continue this rule I end up covering the circle as many times as I complete 4 such combinations.

These combinations however correspond not to lineal addition but to lineal multiplication.
Thus i,i2 represents the combining of 2 quarter arcs notationally, or as it is also called composition of the operator i.

We can see the comparison with x,x2 when we set out the general calculation for the area of a square. In this light the area of the shunyasutra would be definable as i2
If you follow the usual formulae through the quarter arc has defined length Pi x r/2 giving Pi2 x r2/4 as an area for the shape that looks like a kind of mushroom . Compare with 2r2.

The identification of i2 with -1 is clearly not a magnitudinal one but a curvilineal translation or rotational one. This is why the square root sign has a different interpretation when dealing with this process or composition because we are not determining a magnitude of space but a magnitude of rotation .
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« Reply #40 on: January 25, 2017, 08:54:12 AM »

The issue of quantification is intrinsically tied up with the imaginary but better pythagorean Arithmoi.

The introduction of a formal definition of the term number really screws things up!  

We start with space that is topos. This is not an abstract space like the complex plane or vector plane, it is an experience of dynamic regionality.
within any dynamic region I appreciate other types of regions some more dynamic than others and some more similar, symmetric, or identical than others in distinguishable ways and comparable cognisances or perceptions.

I may respond by singing, dancing in dynamic correspondence or relationship to these nested Topoi. And I may be energised to draw or sketch or mark representations of these experiences or even more elaborate and aesthetic representations. I may be entrained by their dynamic to create and develop a language that references each and all Topoi and topological experiences.

I might be inspired to make mosaic representations and records of these dynamic experiences and perceptions, and when I do, particularly through a systematic process of analysis and subsequent synthesis I might refer to this whole mosaic using process as quantification.

It is the process of quantifying dynamic regions along with synthesising them in material elements and substances that duplicate the perceived properties both static and dynamic that we call mechanical model making or Mechanics.

And what Newton, Cotes and DeMoivre found was an ancient practice: using the ideal of the circle, derived from the sphere, both pragmatic perfected notions, all quantification can be harmonised!

And the expertise is to know and understand what mosaics are necessary to build whatever type of dynamic me banish and to duplicate that mechanism as often as desired. .

Thus the quantities of area and volume have no meaning if it is not anchored into a matrial mosaic construction purpose.

The quarter turn magnitude is a quantification of a dynamic , and only one of many possible rotational quantifications. The fact that it came to be denoted as roots of unity is an interesting story in its own right, but it is an enduring and beautiful quantification scheme inspired by Newton but delivered by DeMoivre and Cotes.

The expertise is not to apply the scheme to what it was never devised for! .

This quantification method allows experts to quantify dynamics in such a way that complex rotational systems can be established to describe virtually any motion or topological dynamic in a quantified way. And from these constructed quantification schemes quantities like area and volume  may be appropriately defined and utilised to topologically encapsulate a particular dynamic mosaic or mobile that we may call a mechanism or a mechanical device.

But beyond this I may construct theoretical models of invisible or insensible dynamics that to a natural philosopher, with expertise gives insight into a familiar topological dynamic as that dynamic is closely analysed repeatedly to test if certain regularities and similarities hold at each technological scale.

Those regularities, similarities and symmetries are often called "Laws", but they are of human perceptual origin an  imposition upon Nature.

The universe remains a place where the possibilities are infinite!
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« Reply #41 on: July 14, 2017, 10:12:42 AM »

I have for a long time abnoned the concept of number that promotes real and imaginary quantities. In that regard I have been schooled a little by NJ Wildberge. But I have Ben ken to promote the fundamental topological basis of our comprehension, both language and mathematically conceive. I reintroduce the term spaciometry to avoid entangling notions of geometry with my perspective.
Thus after a long while I could perceive the quarter circle arc as a known quantity ignored by modern lgebra and mystified into the imaginary quantity.
In a similar way vectors were mystified.
But I found that the Grassmnns patiently explained all.
The quarter rc is naturally a plane curve in a planar circle. Like ll constructible forms it is a vector/ Trger. Associated ith this circular arc vector is a straight line vector called the diameter and the half diameter vector that traces round the arc perimeter from the circle centre. Thus the diameter vector is used to define the orientation of the circular arc nd the radius vector to describe the phase or beginning nd ending phases of the circular arc vector.
The uarter turn arc is the fundamental arc vector because of Thsles And Pythagoras theorems  to do with the semi circle nd the quarter arc.
We may represent this quarter arc vector by an exponential form in which the diameter orientation is used to define a radial rotation magnitude in radians and the scalar factor represents the quantity of the radius or half diameter .
If the symbols used for these quantities are dynamic thn we are ble o represent rotationl dynamics from straight lines through spirals , trochoids and into " tangled" complex planar motions.

The Quaternion format allows us to do this in 3 dimensional space .
The Ausdehnungslehre actually Naples us to represent many dimensions and collapse them don to 3 . In none of these forms do we require the so called imaginary quantity. In factbHemsn  Grassmann consistently used an angle notation with his exponrntil form.
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« Reply #42 on: July 15, 2017, 08:29:37 AM »

Fox,what do those symbols and R represent in your statement?(I cant replicate what you typed on my phone)
Again having images transfered on the russian site PHOTOBUCKET
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« Reply #43 on: July 15, 2017, 08:50:55 AM »

Again having images stolen by russians on the russian site PHOTOBUCKET

Photobucket - Wikipedia
Photobucket is an American image hosting and video hosting website
On June 28, 2017, Photobucket changed its Terms of Use regarding free accounts and third party hosting (hosting on forums, eBay, etc).
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« Reply #44 on: July 15, 2017, 09:16:51 AM »

I believed that PHOTOBUCKET would be made in Russia, due the logo of the site PHOTOBUCKET look like a traditionaly russian fur hat allthrough is photo camera.
 A peacock Wow
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