#identity element

Beyond the Enlightenment Rationalists:
From imaginary to probable numbers - V

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The four Cartesian quadrants provide the two-dimensional analogue of the number line and its graphic representation in Cartesian coordinate space.  This is the true native habitat of the square and, by implication, of square root.  Because  Enlightenment mathematicians  found fit to define square root in a different context inadvertently  -that of the number line- we will find it necessary to devise a different name for what ought rightly to have been called square root,  but wasn’t.  I propose that we retain the existent definition of tradition and refer to the new relationship between opposite numbers in the square,  that is to say,  opposite vertices through two dimensions or antipodal numbers, as contra-square root.^[1]

Modified from image found here.

Given this fresh context - one of greater dimension than the number line - it soon becomes clear with little effort that a unit number^[2]ofany dimension multiplied by itself gives as result the identity element of that express dimension. For the native two-dimensional context of the square the identity element is OLD YANG,  the bigram composed of two stacked yang (+) Lines,  which corresponds to yang (+1),  the identity element in the  one-dimensional context  of the number line. In a three-dimensional context,  the identity element is the trigram HEAVEN which is composed of three stacked yang (+) Lines.  The crucial idea here is that the identity element differs for each dimensional context,  and whatever that context might be,  it produces no change when in the operation of multiplication it acts as operator on any operand within the stated dimension.^[3]

As a corollary it can be stated that any number in any dimension n composed of  any combination  of  yang Lines (+1) and yin Lines (-1) if multiplied by itself (i.e., squared) produces the identity element for that dimension.  In concrete terms this means, for example, that any bigram multiplied by itself equals the bigram OLD YANG; any of eight trigrams multiplied by itself  equals the trigram HEAVEN;  and  any of the sixty-four hexagrams multiplied by itself  equals the hexagram HEAVEN; etc. (valid for any and all dimensions without exception). Consequently, the number of roots the identity element has in any dimension n is equal to the number 2ⁿ, these all being real roots in that particular dimension.

Similar contextual analysis would show that the inversion element of any dimension n  has  2ⁿ roots of the kind we have agreed to refer to as contra-square roots in deference to the Mathematics Establishment.^[4]

That leads us to the possibly startling conclusion that in every dimension n  there is an  inversion element  that has the same number of roots as the identity elementandall of them are real roots.  For two dimensions the two pairs that satisfy the requirement are bigram pairs

For one dimension there is only a single pair that satisfies. That is (surprise, surprise)  yin(-1)/yang (+1).  What it comes down to is
this:

If we are going to continue to insist on referring to square root
in terms of the one-dimensional number line, then

• +1 has two real roots of the traditional variety, +1 and -1
• -1 has two real roots of the newly defined contra variety,
  +1/-1 and -1/+1

So where do imaginary numbers and quaternions fit in all this? The short answer is they don’t.  Imaginary numbers entered the annals of human thought through error.  There was a pivotal moment^[5]  in the history of mathematics and science, an opportunity to see that there are in every dimension two different kinds of roots - - - what has been called square root and what we are calling contra-square roots.  Enlightenment mathematicians and philosophers  essentially allowed the opportunity to slip through their fingers unnoticed.^[6]

Descartes at least saw through the veil.  He called the whole matter of imaginary numbers ‘preposterous’.  It seems his venerable opinion was overruled though. Isaac Newton had his say in the matter too. He claimed that roots of imaginary numbers “had to occur in pairs.” And yet another great mathematician, philosopher opined.  Gottfried Wilhelm Leibniz,  in 1702 characterized √−1 as  “that amphibian between being and non-being which we call the imaginary root of negative unity.” Had he but preserved such augury conspicuously in mind he might have elaborated the concept of probable numbers in the 18th century.  If only he had truly understood the I Ching,  instead of dismissing it as a primitive articulation of his own binary number system.

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Image: The four quadrants of the Cartesian plane. By convention the quadrants are numbered in a counterclockwise direction.  It is as though two number lines were placed together, one going left-right, and the other going up-down to provide context for the two-dimensional plane. Sourced from Math Is Fun.

Notes

[1] My preference might be for square root to be redefined from the bottom up, but I don’t see that happening in our lifetimes. Then too this way could be better.

[2] By the term unit number,  I intend any number of a given dimension that consists entirely of variant elements of the number one (1) in either its positive or negative manifestation.  Stated differently,  these are vectors having various different directions within the dimension,  but all of scalar value -1 (yin) or +1 (yang). All emblems of I Ching symbolic logic satisfy this requirement. These include the Line, bigram, trigram, tetragram, and hexagram.  In any dimension n there exist 2ⁿ such emblems.  In sum, for our purposes here, a unit number is any of the set of numbers, within any dimension n, which when self-multiplied (squared) produces the multiplicative identity of that dimension which is itself, of course, a member of the set.

ADDENDUM (01 MAY 2016): I’ve since learned that mathematics has a much simpler way of describing this. It calls all these unit vectors. Simple, yes?

[3] I think it fair to presume that this might well have physical correlates in terms of quantum mechanical states or numbers. Here’s a thought: why would it be necessary that all subatomic particles exist in the same dimension at all times given that they have a playing field of multiple dimensions, - some of them near certainly beyond the three with which we are familiar? And why would it not be possible for two different particles to be stable and unchanging in their different dimensions,  yet become reactive and interact with one another when both enter the same dimension or same amplitude of dimension?

[4] Since in any contra-pair (antipodal opposites) of any dimension, either member of the pair must be regarded  once as operator  and  once as operand. So for the two-dimensional square, for example, there are two antipodal pairs (diagonals) and either vertex of each can be either operator or operand.  So in this case, 2 x 2 = 4.  For trigrams there are four antipodal pairs, and 2 x 4 = 8. For hexagrams there are thirty-two antipodal pairs and 2 x 32 = 64. In general, for any dimension n there are 2 x 2ⁿ/2 = 2ⁿ antipodal pairs or contra-roots.

[5] Actually lasting several centuries, from about the 16th to the 19th century. Long enough,  assuredly,  for the error  to have been  discovered and corrected. Instead,  the 20th century dawned with error still in place,  and physicists eager to explain the newly discovered bewildering quantum phenomena compounded the error  by latching onto  √−1 and quaternions  to assuage their confusion and discomfiture.  This probably took place in the early days of quantum mechanics when the Bohr model of the atom still featured electrons as traveling in circular orbits around the nucleus or soon thereafter, visions of minuscule solar systems still fresh in the mind. At that time rotations detailed by imaginary numbers and quaternions may have still made some sense. Such are the vagaries of history.

[6] I think an important point to consider is that imaginary and complex numbers were, -to mathematicians and physicists alike,- new toys of a sort that  enabled them  to  accomplish certain things  they could not otherwise. They were basically tools of empowerment which allowed manipulation of numbers and points on a graph more easily or conveniently.  They provided
their controllers a longed for power over symbols, if not over the real world itself. In the modern world ever more of what we humans do and want to do involves manipulation of symbols. Herein,  I think,  lies the rationale for our continued fascination with and dependence on these tools of the trade. They don’t need to actually apply to the world of nature,  the noumenal world,  so long as they satisfy human desire for domination  over the world of symbols it has created for itself and in which it increasingly dwells, to a considerable degree apart from the natural world’s sometimes seemingly too harsh laws.

© 2016 Martin Hauser

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Beyond the Enlightenment Rationalists:
From imaginary to probable numbers - II

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When a geometric interpretation of imaginary numbers was at last proposed,  long after they were invented,  it was as though accomplished by central committee. The upshot was easily enough understood but also simplistic. In broad brushstroke here is what seems to have gone down.

The 3 dimensions of Descartes’ coordinate system-a number already deficient from the perspective of mandalic geometry-were reduced to just one.  Of the real number axes then  only the x-axis remained.  This meant from the get-go  that  any  geometric figure that ensued  could encompass only linearity in terms of real numbers and dimensions.  It was applicable only to a line segment,  so the complex plane that resulted  could describe just one real dimension and one imaginary dimension.  It consecrated the number line in a single dimension, to exclusion of its proper habitation in two others besides. Strike one for imaginary numbers.^[1]

With that as background let’s look now at the rotations described by this geometric interpretation of imaginary numbers in the context of the complex plane.

i in the complex or cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis By Loadmaster (David R. Tribble) (Own work) [CC BY-SA 3.0orGFDL],via Wikimedia Commons

The number 1 is the multiplicative identity element for real numbers and the number -1  is the  reflection inversion element  for real numbers.  Put another way, the number one times any number equals that number;  the number -1 times any number is  a negative of that number  or  the inverse number through a reference point, usually taken as zero. Multiplying by 1 then leaves 1, -1, i and -i all unchanged. Multiplying by -1  changes  -1 to 1, 1 to -1, i to -i, and -i to i.  In terms of rotations in the complex plane, these changes  all involve a rotation through 180 degrees.  Multiplication of the number 1 by i changes it to i; i by i changes it to -1; -1 by i to -i; and -i by i to 1.  These changes all involve rotations through 90 degrees.  And finally, multiplication of 1 by -i changes it to -i; -i by -i changes it to -1; -1 by -i to i; and i by -i to 1: all changes involving rotations through -90 degrees.

The figure below shows another way to interpret these rotations that amounts to the same tbing: i¹ = i; i² =-1; i³ = -i; i⁴ = 1.  Click to enlarge.

Four numbers on the real line multiplied by integer powers of the imaginary unit, which corresponds to rotations by multiples of the right angle. By Keφr [CC0],via Wikimedia Commons

I think a committee of some sort must have come up with this resplendent plan. For certain it was an Academy of Mathematics and Sciences that endorsed and enthroned it. All bow to central authority.

I had planned to include a comparison of imaginary numbers and probable numbers in this post as well but because that is a long discussion itself, it will have to wait till the next post.  I might add it should prove well worth the wait.

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Image: A drawing of the first four dimensions. On the left is zero dimensions (a point) and on the right is four dimensions  (A tesseract).  There is an axis and labels on the right and which level of dimensions it is on the bottom. The arrows alongside the shapes indicate the direction of extrusion. By NerdBoy1392 (Own work) [CC BY-SA 3.0orGFDL],via Wikimedia Commons

Notes

[1] Mathematician William Rowan Hamilton  subsequently addressed this deficiency in 1843 with his  quaternions,  a  number system  that  extends the complex numbers to three-dimensional space.  Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space  or,  in other words, as the quotient of two vectors.  This complicated matters even more by introducing a non-commutative multiplication operation to the system, though to be fair the quaternion coordinate system has found some useful applications mainly for calculations involving 3-dimensional rotations,  as in 3-dimensional computer graphics,computer vision, and crystallographic texture analysis. Still it becomes problematic when theoretical physics attempts use of quaternions in calculations pertaining to  atomic and subatomic spaces  where rotations do not actually take place.  The conclusion to be drawn here is that quaternions can be usefully,  if somewhat clumsily,  applied to 3-dimensional macro-spaces but are inapproriate for accurate description of higher dimensional spaces. What is here unfortunate and misleading  is that quaternions apparently do describe outcomes of events in the quantum realm to some partial degree,  if not the mechanisms of the events themselves.  Physicists would not long tolerate them were that not so.

[ADDENDUM (24 APRIL, 2016)
Since writing this I’ve learned that quaternions are not currently used in quantum physics nor were they ever, to any great degree, in the past.]

In other words, sometimes  the right answer  can be reached by a wrong method. In the case under discussion here, we should note that it is possible for a rotation to mimic inversion (reflection through a point). A 90° rotation in two dimensions can mimic a single inversion in a single plane through an edge of a square, and a 180° rotation in two dimensions can mimic a single inversion through a diagonal of a square  or  two successive inversions  through  two perpendicular edges of a square.  A 180° rotation in three dimensions  can mimic three inversions through three mutually perpendicular edges of a square;  a combination of  one inversion through a diagonal of a square  and another through an edge perpendicular to the plane of the first inversion;  or a single inversion through a diagonal of the cube. Subatomic paricles exist as discrete or quantized entities and would follow such methods of transformation rather than rotations through a continuous space.  Of course, transformations involving a diagonal would require more transformative energy than one involving a single edge.

Such patterns of relationship and transformation could no doubt be described in terms of quantum states and quantum numbers without too much difficulty by a knowledgeable theoretical physicist.  Surely doing so could be no more difficult than using quaternions,  which may give a correct answer while also misleading and limiting knowledge of the the true workings of the quantum realm by using an incorrect mechanism, one non-commutative to boot. Nature doesn’t approve of hat tricks like that.