Print Page - The "operator" i is more complex than that!

Title: The "operator" i is more complex than that!
Post by: jehovajah on October 13, 2008, 01:12:05 AM

The operator is i

Title: Re: The operator i
Post by: cKleinhuis on October 13, 2008, 09:12:31 AM

very good statement :D ???

the operator i stands for a number whiches square root is -1 O0

a complex number is made upon a number $x_1 \in \mathbb{R}$ and a number $x_2 \in \mathbb{R}$ times the virtual number $i$ , those two numbers are written $(x_1+x_2i)$ this is a complex number

http://en.wikipedia.org/wiki/Complex_number

Title: Re: The operator i
Post by: lycium on October 14, 2008, 09:01:24 AM

Quote from: Trifox on October 13, 2008, 09:12:31 AM

the operator i stands for a number whiches square root is -1 O0

:police:

Title: Re: The operator i
Post by: jehovajah on November 19, 2008, 11:55:26 PM

The statement itself arose out of being pressed for time but i wanted to intimate the distinction between the concept of number and the concept of operator. Number is a quasi mystical concept with a long philosophical tradition. At each historical advance along the way to our number system the operators have been minimised in favour of the concept number. However the very methods that were being codified or explained were introducing a complex or systematic set of operations on either the numerals or the variables or the shapes and now even on the regions of space. The operations are what are overlooked. Nowadays we call these complexes of operations algorithms. Your algorithm for constructing a complex number is not rigorous enough and only delineates the form. The uperator +i rotates the numeral it is next to through 90 degrees to the number line. This spatial movement is represented in the x,i plane. This is mathematically no different to the x,y plane. So the so called complex number is formally equivalent to a number pair or more rigorously a numeral pair. The properies of the operator i are therfore crucial for determining the actions on the numeral pairs. All these operations/ properties can be represented in the matrix notation making even clearer the operative nature of i.

Title: Re: The operator i
Post by: jehovajah on October 25, 2009, 04:52:54 AM

$i$ i was taught arose out of being unable to solve for the roots $x$ ² $+ 1 = 0$ . However it seems that they arose in an algebraic context while the solution for the roots of quadratic, cubic and quartic polynomials as we call them were being sought. They were a pure invention whose use was not fully appreciated until euler, and later. Euler recognised their rotational value but not that they are an operator. In field and ring theory today they are still treat as a number. Euler's identity $e$ ^ipi $+ 1 = 0$ allows a number value to be arrived at in certain instances but his formula clearly shows that i is an infinitely termed operator equivalent to the two angle operators sin and cos in an algorithm.

The operator i acts on the numeral pairs $(x,y)$ transforming them to $(-y,x)$ a rotation of the plane through $pi/2$ . Using the Euler formula we can specify the angle of rotation more precisely. Thanks to euler the role of the operator i can be specified and it is not part of the numeral system. $R$ x $R$ is a better description of the field of application of i and the field in which any fundamental theorem of polynomials needs to be established.

To extend this operator to $R$ x $R$ x $R$ we could define it as operating on the plane (x,y) followed by on the plane (y,z). This can be extended to higher surfaces.

Title: Re: The operator i
Post by: Cyclops on November 30, 2009, 02:19:10 AM

Quote from: Trifox on October 13, 2008, 09:12:31 AM

very good statement :D ???

the operator i stands for a number whiches square root is -1 O0

a complex number is made upon a number <Quoted Image Removed>and a number <Quoted Image Removed> times the virtual number<Quoted Image Removed> , those two numbers are written<Quoted Image Removed> this is a complex number

http://en.wikipedia.org/wiki/Complex_number

Fox,what do those symbols € and R represent in your statement?(I cant replicate what you typed on my phone)

Title: Re: The operator i
Post by: cKleinhuis on November 30, 2009, 02:25:46 AM

Quote

Fox,what do those symbols € and R represent in your statement?(I cant replicate what you typed on my phone)

this is standard math notation, "R" stands for the Real Numbers ( 3.14.. ) , there also exists "I" for integers, or Q for rational numbers ( 1/2, 1/3...n/m )
so, and the € ( ;) ) stands for is element of

in the stament above i just wanted to say that x1 and x2 can be real numbers ...

Title: Re: The operator i
Post by: David Makin on November 30, 2009, 02:27:54 AM

Quote from: Cyclops on November 30, 2009, 02:19:10 AM

Quote from: Trifox on October 13, 2008, 09:12:31 AM

Fox,what do those symbols € and R represent in your statement?(I cant replicate what you typed on my phone)

"a complex number is made upon a number x_1 \in \mathbb{R} and a number x_2 \in \mathbb{R} times the virtual number i , those two numbers are written (x_1+x_2i) this is a complex number"

x_1 € R == x_1 is a member of the set of real numbers
x_2 € R == x_2 is a member of the set of real numbers

At least that's what it means using the correct symbols - I couldn't work out how to get them either :)

Title: Re: The operator i
Post by: cKleinhuis on November 30, 2009, 02:41:36 AM

the laTex notation for those is:

x_1 \in \mathbb{R} = $x_1 \in \mathbb{R}$
x_1 \in \mathbb{Q} = $x_1 \in \mathbb{Q}$
x_1 \in \mathbb{N} = $x_1 \in \mathbb{N}$
use the latex button to insert the <tex> tags in your posting

the "_" sign stands for sub writing,
the "\in" standsfor "is Element of, or is In Set"
the "\mathbb{}" statement does some magic to encode the symbol you want to display in double dashed characters
"HELLOWORLD = \mathbb{R}"= $HELLOWORLD = \mathbb{R}$ <- define a set
"x_{downunder} \in \mathbb{HELLOWORLD}" = $x_{downunder} \in \mathbb{HELLOWORLD}$

this is how the above lines look in sourcecode of a posting

Code:

"HELLOWORLD = \mathbb{R}"=[tex]HELLOWORLD = \mathbb{R} [/tex] <- define a set
"x_{downunder} \in \mathbb{HELLOWORLD}"  =  [tex] x_{downunder} \in \mathbb{HELLOWORLD} [/tex]

Title: Re: The operator i
Post by: kram1032 on November 30, 2009, 04:16:35 PM

$sqrt(x)=-1 |^2 x = (-1)^2 x = 1 sqrt(1) = -1 $
as lyc said: :police: ;)
$ -1 = i*i = sqrt(-1) * sqrt(-1) = sqrt((-1)*(-1)) = sqrt(1) = 1 $
be careful, that notation leads to a problem, proofing wrong all the previous maths.
So you can't say $sqrt(-1)=i$ because that would lead to the caluclation above.
Instead, i is defined as the number/value to give -1 when squared
$i^2=-1$

Title: Re: The operator i
Post by: Cyclops on November 30, 2009, 05:05:08 PM

Quote from: kram1032 on November 30, 2009, 04:16:35 PM

<Quoted Image Removed>
x = (-1)^2

x = 1

sqrt(1) = -1
" border="0" align="absmiddle" />
as lyc said: :police: ;)
<Quoted Image Removed>-1 = i*i = sqrt(-1) * sqrt(-1) = sqrt((-1)*(-1)) = sqrt(1) = 1
" border="0" align="absmiddle" />
be careful, that notation leads to a problem, proofing wrong all the previous maths.
So you can't say <Quoted Image Removed> because that would lead to the caluclation above.
Instead, i is defined as the number/value to give -1 when squared
<Quoted Image Removed>

Quote from: kram1032 on November 30, 2009, 04:16:35 PM

I'm afraid this is all gobbledeegook to me! I'm fascinated by numbers but am uselsess at math and never did algebra at school-you might as well be typing in Swahili! :S

Title: Re: The operator i
Post by: jehovajah on December 05, 2009, 03:23:08 AM

The problem here is not you cyclops but the notion that maths is these symbols. I bet this would look great in swahili and probably more inviting!

You are fascinated by number you say. so were the Pythagoreans, the Indians, the numerologists etc. Modern math has tended to downplay this great driving force engaging individuals in the playfulness of math.

However, math is just thinking about the experience you have of what is going on around and inside you paying particular attention to iterative features of that experience. It is the iteration that drives the interest in quantification, boundarisation, geometrication, pattern and syntax.

In a straightforward way iteration makes "numbers" so fascinating. It is in exploring this that i have come to realise that certain things are not numbers no matter how many times they are called numbers. Here i found a link to a discussion about the operator i and its development
http://i-is-no-longer-imaginary.gmxhome.de/ (http://i-is-no-longer-imaginary.gmxhome.de/) .

I have seen the dramatic push this operator has given to the development of modern math, but how mathematicians still have not tidied things up. The complex string a +b*i is no more a number than p and q are numbers. When Al Jabr was published by Muhammad ibn Mūsā al-Khwārizmī it was specified after the indian practice of letting any mark colour or symbol stand for a numerical value But the literal strings formed with operators were always aigorithms. Algorithms are extended operations, and this indian conception was and is a study of the fascinating subject of operators.I have identified the enfolding set of operators as being a set which i call $\mathbb{T}$ the set of transforms.
Alongside this set is the fundamental process for which and from which these transforms were created : Iteration. The greatest achievement of this operator i is to have brought iteration to the fore.

Title: Re: The operator i
Post by: jehovajah on January 19, 2011, 10:38:51 AM

It seems obvious now, but i think it is worth pointing out:

The magnitude denoted by i is a constant. It is a fundamental constant of not only our universe but probably all universes we can apprehend.

I have learned a lot since i started this thread, and i would not agree with myself now. I is a constant magnitude, but not a number and also not an operator! i am hoping to classify it as an orientation, with some help.

Title: Re: The operator i
Post by: ker2x on January 19, 2011, 10:58:04 AM

Quote from: jehovajah on January 19, 2011, 10:38:51 AM

not a number and also not an operator!

Ha ! i'm not confused anymore. thx ;D

Title: Re: The operator i
Post by: jehovajah on January 31, 2011, 12:40:22 PM

I think i can establish now that √-1 was effectively the y axis orientation, before Descarte's coordinate and ordinate system. That When he mentioned this coordinate system in his mathesis, Descartes used the opposite side of a right angled triangle as his coordinate and the adjacent side as an ordinate, and thus subsumed Bombelli's distinction within a scheme of his own. Then by calling √-1 imaginary he dissuaded any close examination of the similarity between him and Bombelli for hundreds of years,

The trigonometric relations were not affected, but because of the confusion, scientists did not know how to connect the y axis and the right angled orientation to this disparaged negative magnitude!
If it were not for Wallis Napier,Newton, Cotes And De Moivre, i think science would have been harmed by this although mathematics so called may have explored imaginary numbers as an oddity in Number theory.

"i" then is no more or less than the unit of the scaled orientation we call y, it is an oriented magnitude, a directed number of magnitude one and direction π/2 and related to the trig of the unit sphere indissolubly .In so far as 1 is commonly called a number it is able to accept that denotation, However, strictly it is a unity, or a unit magnitude after the greek, and a yoked magnitude after Brahmagupta.

A full explanation has been arrived at in the Bombelli operator thread.

Sorry if i have caused any confusion over the years, but it has been quite an investigation and i hope it has cleared things up for any one interested. It certainly has for me.

Title: Re: The operator i
Post by: jehovajah on February 01, 2011, 03:17:03 PM

Finally i can ask the sensible question: what is the square root of a magnitude that is directed in the direction π!

On the face of it there should be no difference to the square root of a magnitude whichever direction it is measured in, but there is and the reason is because the magnitude is rotating!

So the question is : what is the square root of a magnitude that is rotating between 0 and π? we have to take the orientation of the resultant into consideration, because all our magnitudes are in free rotation.

Thus the answer is :half the rotation and the square root of the magnitude!

You can generalise it to any root.

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah on February 02, 2011, 06:50:49 AM

I have found the most wonderful site (http://www.google.com/search?client=opera&rls=en&q=forgottenbooks.org&sourceid=opera&ie=utf-8&oe=utf-8) for any researcher!

Here i have found Riemanns collected works in german! And of course understandings of the foundation of mathematics untrammeled by ignorance.

Good luck in your studies!

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah on February 10, 2011, 09:55:06 AM

Watched a programme on Fermats last theorem, the modern proof. Well Fermat would certainly not have proved it that way!
I suspect Fermat's proof would not satisfy todays rigorous standards. Fermat knew about Diophantine numbers, and i suspect he conjectured his proof based on the relation $(cos\theta)^n+(sin\theta)^n \neq R^n$ the radius of a circle

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah on July 20, 2011, 05:25:43 PM

Complex vector algebra is how i refer to this topic now. i.j,k are orthogonally oriented vectors. The properties of a complex vector space reflect all the properties of so called complex numbers through the product rules of complex vectors. The myth of number is what has caused the difficulty in accepting directed vectors, and the √-1 is a red herring. People hate negative numbers and that hatred passes on to anything to do with them. But everybody thinks they know what number is and yet they cannot explain what it is easily, so how are they going to do with negative numbers or roots of negative numbers.

Numbers have a place but not at the foundation of mathematics. Euclid placed "numbers"about 2/3rd of the way through his elements, because they actually derive from geometry.

Trig ratios are fundamental, and vectors are representations of trig ratios.

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah on September 03, 2011, 09:35:37 PM

i have done a lot of work on generalising the notion of sign without realising that i was in fact generalising it to unit vectors!

Thus i can entirely replace sign by the notion of a unit vector. However to reference the orientation of a unit vector i used the convention of radials emanating from a spherical centre with the radian measure specifying the direction from a pole and a perpendicular rod along the great circles.

Thus spherical geometry had the answer along with everything else.

Now here is the trick, by prefixing or postfixing a radial to a magnitude we create a vector.

The radial clearly records rotational orientation , thus the notion of a vector is linked to rotation, clock arithmetics and moduli. The notion of reflection in a centre of rotation is therefore to be distinguished from the notion of radial orientation .

We do not need to worry about addition subtraction and general aggregation structures, the familiar rules apply, but with consistent significance. We do not get the notion of the √-1 we instead get the notion of the √L1 if L is the left pointing unit vector.

We only get i if we understand that √ is notation for the geometric mean, and in that case we expect not only√1 but also the U vector giving U√1.

We then need to define U•U=r•R•r•R =L that is some aggregation operation produces L from r•R aggregated against r•R. "r" is some action on vectors. (confused? i know i am! And that is the point. The definitions of i are in confusion!! instead of using the notion of "multiplication" we in fact just rotate. The action r is in fact "rotate to vector radial + π/2", and we do not use i as a number but as an instruction)

These unit vectors when employed as gnomons for aggregation have to produce a consistent result. The gnomon may be a curved gnomon and it acts on a vector.

When looked at in this way the complex decisions and aggregation structures that are being chosen are revealed, but they are all understandable and not mysterious or imaginary.

This also seems to be done in isolation but in fact it is done in analogy with a fractal scale of other relations. This is the genius of Hamilton's exposition of complex algebra.

Thus i replace sign by a vector and the sign rules by a set of operators or actions that give the required results . Unit Vectors with a set of actions replace the complex number notion.

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah on January 01, 2012, 11:19:04 AM

I have recently posted some links to material that makes intelligible the actual proceses involved in apprehending√-1, and places Hamilton and Grassmann in Context.

http://jehovajah.wordpress.com/jehovajah/blog/the-ideas-that-led-to-quaternions

Although the material contains mathematical notation, the sense is well explained without needing to understand the notation. By this you may come to realise the struggle that took place to find an intelligible notation for the field of "sets" , and how expediency has led to your mathematical education being a melange of half baked ideas.

In any case enjoy and deepen your knowledge of where fractals "come" from in space.

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah on May 13, 2012, 10:15:24 AM

My latest thoughts.

http://jehovajah.wordpress.com/jehovajah/blog/2012/05/04/the-shunyasutras-and-1

Title: Re: The "operator" i is more complex than that!
Post by: hgjf2 on May 13, 2012, 05:03:34 PM

Quote from: jehovajah on May 13, 2012, 10:15:24 AM

My latest thoughts.

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

...

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah on June 02, 2012, 03:14:11 AM

Quote from: hgjf2 on May 13, 2012, 05:03:34 PM

...

:D
You know me too well!

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah on March 08, 2013, 11:53:11 AM

I was in conversation with a research scientist just the other day and he was bemoaning $sqrt( -1)$
And I thought
" Haven't I dealt with this subject in one of my posts?"

Well I have revisited it here thinking hmm I have forgotten so much!
The confusion created by this particular piece of conventional notation is profound. It turns the world of mathematics upside down and sets it into a flat spin!
I refuse to engage with this notation nowadays. Instead I view it as a procedural call in a computational language. The procedure or algorithm it specifically calls depends on the argument of the call ie the context.

Now, when I say I refuse to engage with this notation I do not mean I do not note and value it's significance. It is an invitation to engage in what Hamilton called the mathesis of the imaginaries. This mathesis is hidden away in algebra, as a subject boundary and particularly in abstract algebras. If we take Algebra to be a blind persons geometry then the symbol becomes a braille mark alerting the reader to this singular fact. The plane is not the real world! One cannot model a 3d world in a 2d plane! We have to go off piste!( pun also intended).

In reality $i$ is a label as is $\pi$ and they both alert to the rotational dynamic in Spaciometry. The procedural call is to some process algebra of rotation.
$sqrt( -1)$ is also a functional symbol that has to hang about in process algebras awaiting it's sequential call. In these cases it highlights a meta algebra, a meta mathematics which more prosaically I identify with any sufficiently high level computer programming language that has a mathematical and statistical sub routine library.
This is fair, and in fact invigorating. It means that we can dispense with mathematics as a singular discipline and place the content securely within the contexts of computational science and philosophy.

The only recognised person, after Newton to attempt such a transformation is Hermann Grassmann in his Ausdehnungslehre of 1844. The version his brother Robert masterminded in 1862 is far removed from Hermanns original conception. That he was able to draw attention to the similarities and developments of his original ideas is perhaps what illustrates the applicability of his original notions.

You see, by my meanderings, where this topic leads and always must lead away from mathematics so called to a broader process algebra, even beyond group and ring theoretical analyses.

That we can watch a 3d animation is a consequence of the actual meaning of $sqrt( -1)$ !

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah on May 23, 2013, 09:55:17 AM

The nature of i is discussed in this thread
http://www.fractalforums.com/new-theories-and-research/geometric-algebra-geometric-calculus/msg61550/#msg61550
in which the symbolic complexity is demonstrated, as it represents the geometrical form and trans-form-ation.

I think i have already made the statement that i and √-1 shhould now be clearly distinguished. One is algebraic and a constant proportion, the other is a complex label that acts as a mnemonic during actual arithmetical calculations.

Title: Re: The "operator" i is more complex than that!
Post by: Roquen on July 11, 2013, 02:24:02 PM

I always advise completely ignoring 'i' except when (in)formally demonstrating the product. Once that's done, you never need it again. It's nothing more than a comma or period past that point...or a typographic element to denote which of the two linearly independent 'parts' one's talking about. i = sqrt(-1) is at best very misleading and at worst completely wrong (depending on how you interpret the expression).

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah on July 16, 2013, 01:41:04 AM

Euclids Stoikeioon book 5 helps to explain where the confusion derives from..

The Greeks and the Indians had no concept of negative magnitudes. There was also no concept of multiplication. They also had two concepts of duality.
http://jehovajah.wordpress.com/jehovajah/blog/2013/07/14/isos-and-analogos-different-notions-of-duality

Because of these precursory facts, one has to understand Brahmagupta and Bombelli exactly as they wrote and thought. Quantity was factorised into multiples, and they were aggregated or disintegrated.

Brahmagupta did not introduce negative magnitudes, rather he demonstrated how Shunya was at the back of everything. If we subtract what we have we can easily see Shunya. And what we have to subtract is multiple if what we have is multiple. In explaining this he utilised analogous duality, straight from the Eudoxian concept of analogos and logos.
http://youtu.be/UcZ9HhVck9Y

The question of the rule that a rectangle formed from a pair of magnitudes that are going to be subtracted should be aggregated does not make sense in an isos duality. The issue never arises practically. It is only in an analogos duality that this rule makes sense. This is because of the single rule that magnitudes must be of the same kind to be compared, aggregated or factorised.

Logos analogos constraints , applied rigorously mean that algebraic rules are the foundation of Arithmetic. The creation of natural numerals, and then fractional numerals and finally negative numerals were on the basis of the logos analogos duality. The isos duality was restricted to direct geometrical comparisons. In this case negative quantities are geometrically rotations of magnitudes.

On the page they look no different to any other geometrical construction. However analogos duality requires 4 magnitudes, and a negative magnitude is drawn within a gnomonic form, where the parts to be subtracted actually have relative positions!

This formal presentation of a magnitude with a subtracted magnitude, forces the requirement of an additional magnitude. This rule is expressed as – * – = + in Bombelli's notational rhetoric. But Brahmagupta had explained this in his treatment of analogos duality.

These rules actually arise in the cubic setting, and it is only in that case where 3 pairs of magnitudes are compared and each magnitude has a part subtracted, that the full rule appears. However, to reduce it to a rule of signs requires the creation of negative magnitudes, as opposed to parts to be subtracted in the process.

In isos duality the $sqrt(-1)$ is no different to the $sqrt 1$ . All quadrature is done in the circle, using chords crossing the diameter perpendicularly. Only by using rotation orientation as a distinguisher can any distinction be made , but it does not actually uniquely define a value. On the other hand analogous duality does enable a unique representation in notation..

The rise of coordinates and complex combinations as vectors has meant that the imaginary quantities can be distinguished in coordinate notation. The myth created around so called imaginary numbers can be dispensed with and more accessible rules and notation can be used.

We do not, as a rule seek to confuse our colleagues if we make a mathematical discovery in these days. However, in the past it was a recognised way of determining primacy to send out or publish a confusing or coded message.

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah on October 23, 2013, 12:44:04 PM

Believing that I was some mathematical entity, Clifford went on to use it in the further development of his algebras.
http://rsta.royalsocietypublishing.org/content/356/1740/1123.full.pdf

In this regard, I has become a behavioural marker, guaranteeing a predictable mathematical behaviour.

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah on April 30, 2014, 01:28:30 AM

http://www.youtube.com/watch?v=oybzcvv-ZVo
I have placed this here to avoid jamming any ones computer! It is very relevant to the work of Grassman as it occurs historically after he has written and published the Ausdehnungslehre 1844. However it shows little trace of Grassmanns concepts, and consequently it is quite hard to follow . After Riemann's death Mathematicians turned to Grassmanns work to make sense of Riemanns !

The work of Gauss similarly in 1827 the year Justus published a paper on Trigonometry in space, contains some concepts of Euler regarding parameter switching and surface analysis by tangent planes. The linear combination of the map from the parameter plane to the surface, and then from the surface to the unit sphere for the normal to the surface shows ideas that are similar to Hermans but not expressed in terms of vectors, or dot products etc. Thus Hermanns concepts are in keeping with his times, but add that extra insight that makes the difference for the general mathematician.

Gauss may have had a good grasp of what he meant but without the presentation of Hermann Grassmann it was very difficult to follow his argument or disquisition
http://www.youtube.com/watch?v=drOldszOT7I

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah 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?

http://youtube.com/watch?v=5Ui5W_BDa8s

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah on September 24, 2014, 02:10:50 PM

http://www.youtube.com/watch?v=MTddX4Qo-io

Normans complexions!
If you will give up the idea of number you may start here to understand what i is as a formal entity.

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah 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,

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah 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 secø + tanø is an interesting candidate.

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah 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?

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah 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 !

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah 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

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah 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 .

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah 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

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah 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.

e^iø:=cosø + isinø

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

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 .

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah 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!

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah 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/ Trãger. 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.

Title: Re: The operator i
Post by: hgjf2 on July 15, 2017, 08:29:37 AM

Quote from: Cyclops on November 30, 2009, 02:19:10 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

Title: Re: The operator i
Post by: 0Encrypted0 on July 15, 2017, 08:50:55 AM

Quote from: hgjf2 on July 15, 2017, 08:29:37 AM

Again having images stolen by russians on the russian site PHOTOBUCKET

Photobucket - Wikipedia (https://en.wikipedia.org/wiki/Photobucket)
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).

Title: Re: The "operator" i is more complex than that!
Post by: hgjf2 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.
:peacock: :wow:

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah on July 15, 2017, 05:35:29 PM

If I consider the Quaternion form as 3 circular quarter arcs , I need to specify that adding arcs is done by ensuring the diameters are parallel and the arcs add as in vectors head to toe , or the end of one to the beginning of the next.
This in fact produces a step like firm that moves a point diagonally in a plane .
To obtain a circular movement we have to specify that the second quarter arc has its diameter rotated orthogonally to the first before continuing. This is properly and analogously represented by a composition we may consider as multiplication .
Now in addition we have to define the composition of quarter turns which though continuous / contiguous are not in the same plane. .
So I + I is 2I but I x I is I² a quarter turn of the diameter in the plane of the circle .
If J is a quarter circle arc in an orthogonal plane then J x I or I x J requires a quarter turn of the diameter into the second plane.
We need to define the direction of this quarter turn and the order in which the process is carried out.
These quarter turns of the diameter are not notated and this has been the problem historically. By not defining the composition rigorously and clearly no one understood, and indeed were mystified by the concept of multiplication! To be sure multiplication is a Designed process, it always has been, but we have been taught that it was a natural gd given function of numbers! Nothing could be farther from the truth!

Bearing ths in mind we can design a Quaternion using 3 orthogonal diameters from which quarter arc vectors are described and a lineal vector to the point of intersection of these 3 diameters. That lineal vector may be used to determine the radius of the quarter arc vectors as well as to provide a rotational Axis to drop peroendiculars onto,
This way we connect a Quaternion to the fundamental lineal orthogonal axes we have accepted as standard.
We o not have to accept that as a standard, but we must agree some conventional standard. .

When I started the thread polynomial rotations I struggled to grasp ths simple process definition, and if course the confusing notation
Now here's.the rub: we do not have to use quarter turns! We can use any fraction of a circular perimeter as a circular arc vector and the planes in which they operate do not need to be orthogonal! .

The Cotes DeMoivre theory of the roots if unity deal with this concept using polynomial depictions , but what I am describing is a straightforward spaciometric construction.

Whether we confine the circular vectors to a sphere or a composition of planar circular arcs we can describe motions of points by lineal or circular arc vectors ?
In addition if I is the arc vector for a unit diameter 2I is the sum of 2 of them but will give a resultant point equivalent to R if that is the arc vector for a diameter of twice the unit diameter.

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah on July 17, 2017, 04:15:43 AM

The addition and multiplication of circular arc vectors generates trochoidal motions or positions, the addition rule can be modified to allow the radial phase vector to be attached to the radial vector of the next circular arc tip to toe. Thus the position or motions trace out the familiar trochoidal patterns.
The usual exponential form of a circular arc vector is based on this type of circular arc addition. Thus the Fourier series in exponential form is a trochoidal line pattern .
Using three Fourier series can capture a 3d trochoidal description of space.
However a Quaternion Fourier series or a Grassmann Twstir depicts a trochoidal surface pattern based on interacting spherical surfaced.

To depict saciometrically a volume dynamic a Quaternion Fourier transform would be naturally descriptive, but the mathematical behaviour of Grassmann twistors is poorly understood and poorly researched.

There is one application of a Quaternion Fourier transform and that is in image compression . So far this work has been pioneered by just one person in the main.

Title: Re: The "operator" i is more complex than that!
Post by: jehovajah on July 17, 2017, 06:30:05 AM

This is a quote from an interview with David Hestenes in 2009

Quote

Cambridge University published “Geometric Algebra for Physicists.” That book arose from more than a decade of GA research at Cambridge that produced many important results, most notably, “Gauge Theory Gravity,” which improves on General Relativity. Now GA is being applied to robotics and there are conferences on GA every year around the world. It is clear now that the whole field will keep growing without my help. My ultimate goal has always been to see GA become a standard, unified language for physics and engineering as well as mathematics. GA is arguably the optimal mathematical language for physics. For example, you can do introductory physics using geometric algebra without using any coordinates. Actually, my Oersted Medal lecture, published in the American Journal of Physics, is an introduction to geometric algebra at an elementary level. So, I’m willing to bet that GA will eventually become the standard language, even in high school. There is a need to integrate high school algebra, geometry, and trigonometry into one coherent system that is also applicable to physics. GA puts it all together in a remarkable way.
T: So is it easy to make sense for . . .?
H: Well, you see, if you’ve already learned a different language, right? A new language looks hard.
T: Yes.
H: No matter what language! However, if you analyze GA in terms of its structure, it can’t be harder than conventional mathematics, because its assumptions are simpler. The geometric interpretation it gives to algebraic operation is more direct and richer than ordinary vector algebra. It includes all the features of ordinary vector algebra, but it’s not limited to three dimensions. It works in space-time, and so you have a vector algebra for space-time, which, as I have noted already, improves on the Dirac algebra. Indeed, it turns out that I discovered something amazing when I reformulated the Dirac equation in terms of space-time algebra, where Dirac’s gammas –the gamma matrices– are now vectors, okay? The gammas become an orthonormal frame of vectors in space-time. But, what about the imaginary unit i in quantum mechanics? Well, it turns out that you don’t need it.
T: You don’t need it?
H: You don’t need it! You don’t need an extra imaginary unit because the frame of orthonormal vectors suffices when multiplication of vectors is defined by the rules of geometric algebra. Of the four vectors in a frame, one is a timelike vector and three are spacelike vectors, right? If you take the product of two spacelike vectors you get a new quantity called a bivector, which generates rotations in a plane of the two vectors, and its square is minus one. As I proved in 1967 (in the Journal of Mathematical Physics) the generator of phase in the Dirac wave function is just
such a bivector. And what is the physical significance of the plane specified by that bivector? Well, that plane determines the direction of the spin. Thus, spin and complex numbers are intimately, indeed, inseparably related in the Dirac equation. You cannot see that in the ordinary matrix formulation, because the geometry is suppressed. Because matrix algebra is not a geometric algebra; it was developed as a purely formal approach to handle systems of linear equations. In contrast, geometric algebra gives the Dirac equation geometric meaning. So, there is a meaning to the imaginary unit i that appears in the Dirac equation. We have seen that it represents the plane of spin. Eventually, I also proved that this property remains when you do the non- relativistic approximation to the Dirac equation, going to the Pauli equation, and then to the Schrödinger equation. Now, it is usually said that the Schrödinger equation describes a particle without spin. But, the fact is, when you do the approximation correctly this i, which generates rotation in a plane in the Dirac equation, remains precisely as the i in the Schrödinger equation. Thus, the i in the Schrödinger equation is generator of rotations in a plane, and the normal to that plane is a spin direction. In other words, the Schrödinger equation is not describing a particle without spin; it is describes a particle in an eigenstate of spin, that is, with a fixed spin direction. Studying the implications of these facts has been a major theme of my research to this day. And more results will be published soon.
T: Great
H: Yeah, so, that keeps me going. .
T: And you’re still excited after forty years?
H: Yeah, that's right. So, if you are interested I tell you a little about what it has all lead to. Have you heard of zitterbewegung?
T: I’m not familiar.
H: That’s a German word meaning “trembling motion.” The term was coined by Schrödinger. He noticed that if you try to make a wave packet with the free particle solutions of the Dirac equation something funny happens. You can’t make a wave packet using only the positive energy solutions. The Dirac equation has troubles because there are both positive and negative energy solutions, and everybody believes that for a free particle the energy has to be positive. And, you need both positive and negative energy solutions to make wave packets, otherwise you don’t have a complete set. When you make a wave packet it has oscillations between positive and negative states that Schrödinger called zitterbewegung. The frequency of these oscillations is twice the de Broglie frequency. Do you know the de Broglie frequency?
T: Hmm!
H: It is mc squared over h-bar.. The zitterbewegung frequency is twice that, okay? Schrödinger

What is important is that i is identified as having a geometric interpretation. Hestenes does not refer to the work of the Grassmans directly but we know that Hermann Grassmann based on Eulers work had already made this identification in the Vorrede of the 1844 Ausdehnungslehre . And of course his use of " vectors" inspired Caleys Matrix representation of many of his designed and defined products.

Now I have identified i as the quarter circle arc, a circle defines a plane, so a circular arc defines a plane of rotation .paulis sigma algebra defines a sphere of rotation as a unit volume which spins in the i pland of rotation
Quantum physics is about trochoidally dynamic surfaces . It differs from so called classical Einsteinian physics as a straight line differs from a curve! On the very large scale and removing unwarranted restrictions like absolute speed or velocity the two descriptions can be made to agree.
On the very small scale a straight line can not approximate a small curve ! Or rather a large curvature. On such a scale classical physics becomes too complex and too slow in computation .
When Newton tried to define a fluid dynamic version of space, he found the computation too onerous. Simplifying assumptions he made and the fundamental assumption that fluids only act resistively led to innacuracies that just were not there in the point mass model.

The point mass model was based on one keen observation. Motion tended to create a point or centre of symmetry. By reducing mass to a point he could ignore all the extraneous motions in his quantity of motion and identify the position of a body. What he could not do is explain the real world fluid dynamic motion of such a body because much of that motion is rotationl and vibrational and exhibits as different forms of energy to put it simply. Thus in his fluid dynamics he failed to account for counter rotations within and around defined bodies, and so there impact on the resultant motion. Le Sage provided an important analysis of the effect of some environmental I pacts, but he did not appreciate the contribution of rotationl dynamics in fluid dynamics, nor the presence of magnetic behaviours in space as we now know.

All these issues can be addressed with modern computational means and new mathematical developments of forms

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Fractal Math, Chaos Theory & Research => Complex Numbers => Topic started by: jehovajah on October 13, 2008, 01:12:05 AM