The Square Root of -1

Mathematics and Neurology

The square root of -1 is the integer 3 in 10-circle, but it is not an integer in 9-circle, where -1 is 8. It is an integer in 5-circle where -1 is 4, and we could go through the numbers, choosing an array of circles that have integral -1 square roots.

Figure 1

In the counting line of counting circles -1 is a counting line to infinity. Thus it is 0 in 1‑circle, because one place anticlockwise from 0 brings us back to 0. It is 1 in 2‑circle, 2 in 3-circle, 3 in 4-circle, 4 in 5‑circle etc., as shown below.

Figure 2

The square roots of these -1 numbers again constitute a counting line to infinity, but one interrupted by non-integer square roots as shown in the following sequence:

Circle -1 in the circle Square root of -1

0-circle             Being without determination, 0-circle has no numerical elements.
1-circle             0                     0
2-circle             1                     1
3-circle             2                     1.4142135623730950488016887242096980785696…
4-circle             3                     1.7320508075688772935274463415058723669428…
5-circle             4                     2
6-circle             5                     2.2360679774997896964091736687312767354406…
7-circle             6                     2.4494897427831780981972840747058913919659…
8-circle             7                     2.6457513110645905905016157536392604257102…
9-circle             8                     2.8284271247461900976033774484193961571393…
10-circle           9                     3
11-circle          10[i]                   3.162227…     Six, or 3*2 more irrationals follow here

before we reach 4, the next perfect square.

List of -1 Square Roots in Sequential Circles

The irrational square roots, which shade off in what we could call infinite imbalance, are as precise in achieving their purpose as any rational configuration. They constitute the background against which the rational stands out. Taken in conjunction with the neural organization of the brain, the contrast between the perfect and irrational results represents the foundation of difference, aligned by functioning nervous systems to create the wealth of meaningful identities that confront life. On this contrast the intelligence in living creatures builds.

The integer counting line, from which, in the square root of -1 the rational/ irrational pattern arises, read into the depth of all bases, is the mathematical foundation upon which the neural determination of known entities, so our knowledge of the world arises. The irrationals provide the supporting background in consciousness for this to occur. From this cloth the known world is fashioned. We usually think that the brain is only interested in the pursuit of order in a rational world, but without the contrasting irrationals there would be no canvas to take the picture.

The rational/ irrational pattern is that, each integral root (0 1 2 3…), is followed by double that number of non-integer results:

The 0 ‘sq rt. -1’ in 1-cir. is followed by 0 irrationals, and 0+0=0

The 1 ‘sq rt. -1’ in 2-cir. is followed by 2 irrationals, and 1+1=2

The 2 ‘sq rt. -1’ in 5-cir. is followed by 4 irrationals, and 2+2=4

The 3 ‘sq rt. -1’ in 10-cir. is followed by 6 irrationals, and 3+3=6...

and so on to infinity. What is happening here?

The answer is that ‘i’, the square root of ‑1, is switching the whole of mathematics from objectivity into subjectivity, translating the numerical pattern from the external or worldly domain into its internal counterpart, the mental space which we call subjectivity or time. We have that relation before us here. The generative pattern is genetic. Its root taps a chromosomal stem. It is not the expression of brain-dependent learning. Projected into the world, and returned as an external ‘sense’, it becomes the comprehension of mathematics, which we again invest in externality in symbolic form, the ‘math’ we know.

This ‘math we know’, to which sensible things like chairs and tables belong, is thus the externalization of an internal (mathematical) pattern that underpins mind. Conventional mathematics, as taught in schools is the externalized reflection of this internal order. We call the internal domain subjectivity, when we wish to refer to our sense of a world around, and time when our reference is to its nature (as distinct from the outer domain of space); i.e., when only the ‘inner’ or ‘outer’ of the domain is in reference. Time, with its present past and future is the stem. Space is its externalized expression.

Some further observations

We are using decimal to show the pattern in the list of ‑1 Square Roots because it is the familiar base we know, but it occurs in all bases and we could show it in base 2, base 3, base 4 notation… and so on. In circle math we can work in any base, though we will usually choose decimal as our base of expression. The circle algorithms then call on other, we could say, parallel bases, to recruit their results. The circular method is thus properly described as transbase in operation, coming in this a step closer to the function of the brain, which as deep to mind is uniformly transbase.

We should note that 0 is the identity element for adding and subtracting. 1 is the same for multiplying and dividing (adding 0, or multiplying by 1 does not change a number). To multiply by -1 therefore changes only the sign of the result. The ‘1’ in the ‘square root of -1’ is thus a dummy indicator of sign change, whose reality is an internalized counting line to infinity against a background array of fractional results, as shown in the list of ‑1 roots. Internal here means ‘in mind’ and external means ‘in world’, so that ‘ordinary’ arithmetic, with its straight counting line to infinity belongs to the world. Circle math, with its circles rules and results belongs to the mind.

We work from our knowledge of our ordinary or external math, which is objective to us. The internal circle math of the mind is however the parent and projector of the outer result. The ‘internalized counting line’, as in the above paragraph, is given in the square roots of 0 1 4 9 16 25… The ‘background array of fractional results’ that seems to act as a spoiler is the tidy sequence of untidy square roots of 2 3 (two) 5 6 7 8 (four) 10 11 12 13 14 15 (six)… in the -1 non-integer circle series, and the whole is the neurological setting from which developed consciousness springs.

To think, the mind needs continuity and contrast and here it is, continuity in the mathematical sequences and contrast in these very same sequences. The circular system then comes forward as a coherent fine-grain mathematical approach to the subject matter, “how do we think, what is the mind's mechanism and operational principle?

The square root problem

In conventional mathematics there is no real number x, such that

x^² = -1

This is because +x times +x comes to a positive result, and ‑x times ‑x also comes to a positive result, and there are no other options for x^².

Unable to assign a value to x satisfying the equation conventional mathematics stipulates that there is such a value, and calls it i for imaginary. It has no idea what ‘i’ is, other than the square root of ‑1 as expressed in the equation

i^² = ‑1

Given this, for it an impossible equation, it assigns the ‘no real number x’ to the imagination, calling the square root of -1 ‘i’. It then has a term for this ghost in the mind, supported not by understanding, but the fact that it integrates perfectly into mathematical operations and equations. The cost is that mathematics advances as a ritual of symbolic relations, comprehended only as the formalistic integration of signs and results.

‘i’, as the square root of ‑1 is currently a blind spot in our mathematical understanding, walled off and bypassed. The task is to explain it, and in the process turn it from a block to a bond and bridge uniting in-world (linear) and in-mind (circular) math. The imaginary number serves a purpose, making an advance possible, shifting math from its ordinary domain to its higher expression. The further insight needed is to see that this ‘higher’ math is really the internalized circular math we call the mind.

Interlude

In case this needs further explanation, the contrast between perfect and irrational results as set out above, is easy enough to understand. It is there before us and it stands alongside the ‘ordinary’ counting line to infinity, when these numbers, turned into circles (figure 2) yield another counting line of whole numbers, padded out with irrationals.

Now, the point is that these irrationals are rational in their irrationality. Every single digit in every irrational number, and there are countless billions of them, is exactly determined and cannot be other than what it is. Hence we can say, “The irrationals are rational in their irrationality.”

The jump to neurology rests in the fact that this overall pattern maps to the understanding in mind. To give the idea, when we say of two different sense-mediated patterns, “this is a window and this is a goldfish bowl,” that comprehended distinction is carried in the mind by the mathematical distinction we have just considered, between the whole numbers and the irrationals. In other words, the rational/ irrational relation, as found in mathematics is the foundation of difference as found in the sensible world, between this object and that. Making this observation takes us across the math-to-mind boundary. Interpreting ‘i’ within a mathematically circular expression will complete the picture.

‘i’ as a square peg in a round hole

‘i’ is impossible by the rules that have governed math for 500 years, yet ‘higher math’ is adamant that it needs it. As imaginary it stands as a real unreal, as enigmatic as a unicorn or phoenix. To understand this situation we need to look at the unyielding sign laws. Children are taught that ‘plus times plus’ is plus. ‘Minus times minus’ is plus, but ‘plus times minus’ (or minus times plus), gives minus.[ii]

There are only two signs, ₊ (plus) and - (minus). An asterisk * means times or multiply. Positive numbers may or may not be signed. Negative numbers are always signed. The sign rules work and they have order, but they are given without reason. They have to be learned.

A + * + = + so +1 * +1 = +1

B + * - = - so +1 * - 1 = -1

C - * + = - so -1 * +1 = -1

D - * - = + so -1 * - 1 = +1

a and d, whose roots are identical in sign and number yield perfect squares. Only b and c give the -1 answer that ‘i’ requires, but the results are not squares because each utilizes non-identical signs (one +, one ‑). The sign laws make no provision for negative squares. The square root of -1, devoid of actuality, is therefore said to be imaginary (i).

This brings ordinary mathematics to its Waterloo. Workers in complex fields saw that the concept of a function, which they took to be a numerical magnitude equivalent to the square root of ‑1, was useful and orderly, but it was impossible to reconcile it with the sign laws. They therefore consigned it to the imagination.

A new protagonist, circle math, now takes the field. It is strictly mathematical and just as precise as ordinary math, but its rules have been nurtured and born not in mathematics but philosophy. In scope it embraces both the straight math of the world and the circular math of the mind. We see immediately (fig. 1) that -1 in 10‑circle is 9, whose square roots satisfy the problem equation, and this is only a beginning.

In the shallows

The observation belongs to the circular system, founded on a counting line, not of numbers but number circles. Children delight in this form of math because its rules are simple and its operations without fault. It comes to them as a number game, but ‘number’ without its intimidating aspect. 0 gives the answer, so for the children we will draw it with a smile (figure 3 below).

Figure 3

For example, if the sum is 2 + 3 = ? the child has but to locate the 2, locate the 3 and join them with a line. A parallel from 0 gives the answer. It also establishes the 5 direction.

There is more power in this construction than meets the eye. For instance it covers every numerical calculation that sums to 5: 1+4, 2+3, 8+7, 9+6, 0+5, 4+1, 3+2, 7+8, 6+9 and 5+0.

Use a horizontal line instead and you have the 0 direction. This gives the answer to combinations that sum to 0. Do not forget that 9+6=5 in ten circle is the digital stepping-stone to 9+6=15, 9+16=25, and so on. Algorithms combine the sub-steps to final answers.

Notice too, that if you join the numbers on an angle you get all the other directions as well. The one figure, easy to visualize and work with, yields every possible cipher adding combination, while similar constructions cover subtraction, multiplication and division.

Each sum has its distinguishing direction. Turning arithmetic into a game children learn their adding subtracting multiplying and dividing tables without noticing that they are doing so. It is not ‘arithmetic without tears’, but arithmetic with joy and delight.

For the child, the circle is a tiny arithmetic pool in preparation for oceanic journeys to come. Nor is the method confined to base ten. It works in every base, for instance on the clock face. Drawing two lines shows that 11+8=7 (11 o'clock + 8 hours comes to 7, easily done, easily remembered).

Applying it to other bases, it becomes a recreation and pursuit as mathematics becomes an interest in its own right. For example:

Every number in a circle squares to the same result in that circle as the square of its complement. For example, the complement of 5 in 12-circle is 7. 5 squares to 25, which counts into 12-circle as 1 (twice round plus 1). 7, its complement squares to 49. This also counts into 12-circle as 1 (four times round plus 1).

This double squaring shows in smaller circles; say 3 in 5‑circle (fig. 1), which is +3 clockwise and -2 anticlockwise.

Square the 2. It comes to 4. Square the 3. It comes to 9, which counts into 5‑circle as 4. 3 squared is the same operation as ‑2 squared, and gives the same result as 2 squared. Children learn their sign laws hands on, and see for themselves why ‘‑ * ‑’ comes to the same result as ‘+ * +’. The rote-learned tables do not help in the least in this respect, but we must now resume our pursuit of the square root of ‑1 at a deeper level.

In the depths

Typing ‘square root of minus 1’ into a search engine can find Zhenming Zhia, who writes under ‘Square Root of -1 as a Consciousness factor’, July14 1996, that “In Einstein's special theory of relativity there is a Lorentz transformation that leads to Minkowski's four-dimensional space. But the fourth dimension is obtained by replacing time, t, with the imaginary [square root of -1 multiplied by the speed of light, c, and by t itself]. After this, the temporal dimension becomes totally symmetrical with all other three dimensions of space…”

Zhia continues: “That means that time is itself one more dimension of space but is perceived by our consciousness as different and uniquely temporal. The modification with the imaginary square root of -1 therefore corresponds to the unique involvement of consciousness in the process.”

He adds that, “Surprisingly, in Quantum Mechanics, the Schrodinger's wave function also involves the same square root of ‑1 when the spatial locality breaks down and a conscious observer gets involved in the process of measurement.”

“This apparent coincidence indicates something extremely significant for understanding the universe and consciousness. It suggests that human consciousness may be a dimension of anti-space that merges into the fourth dimension of negative space so that we cannot see it as spatial any more. It appears as one dimensional time instead.”

“Therefore, the square root -1 is the Consciousness factor. Working out its implications might lead to a true revolution in human understanding.” Quotation ends.

In circle terms factoring i into t imports the equation into subjective time, where subjectivity has not the meaning of opinion, but the inner mathematical realm that in circularity grounds consciousness, objective in its very subjectivity.

Two mathematical systems

For centuries mathematics on a stormy path, fractured internally by dispute has played an autocratic role in education in a civilization torn by dissent and war. In its circular form, now being presented, it gains a subjective level, complementary to its familiar objective expression. We now see that in its developed form it is accessible to children as well as having an application in higher math and physics.

Gradually across the years the square root of ‑1 that divides math and maps to the schism between ideality and reality, has come forward as the kernel of the theoretical problem. Its resolution, whose completion is critical, will mark the turning point between that past and a brighter future. In this realization, the difficulty in the imaginary number, proving itself to be imaginary, will vanish, and people will wonder what it was about. Validating Zhenming Zhia's prediction, mathematics, as an internalized discipline, is the logos upon which consciousness stands and the sensed world resides. The square root of ‑1 is the logical switch between that sensed world and our internal world of understanding.

Definition

In a circular setting ‘i’ is a function (like Narcissus looking into a pool), that reflects mathematics from its external presence into the depth of its in-mind source, but we must examine some existing definitions, starting with the fact that ‘negative number’ is still taught as a ‘quantity less that zero’.[iii]

In ‘wordiq.com’ we can read, “a negative number is a number that is less than zero, such as ‑3. A positive number is a number that is greater than zero, such as 3. Zero is itself neither negative nor positive. The non-negative numbers are the positive numbers together with zero. Note that some numbers are neither negative nor non-negative, for example the imaginary unit i.

The following discussion will make a frontal assault upon this position in order to create a sense of neutrality in the matter, upon which a new system of expression can arise. Mathematics is an integral part of all knowledge. It is not a castle to itself, and it cannot have a different logos from that of language. To be a discipline it must accept discipline. We now look at this in more detail.

Ordinary math relates number directly to quantity. This is logical where quantity is amount, but it enters another thought alongside, that negative number is a quantity less than zero. This is illogical because quantity is a specified amount or number. If we start with a quantity of apples and remove them one by one, the quantity diminishes. When it reaches zero we can remove no more. The concept of a number less than zero is therefore false, and false in the world, false in mathematics.

The problem does not occur in circle math whose counting lines are closed. Positive or add means clockwise in the circle. Negative or subtract means anticlockwise. The confusion vanishes. The math remains. Thus, in 10-circle ‑1 is 9, which squares to 1, and has 3 and 7 as its roots. The wordiq.com statement (as above) nails the error to the masthead by defining positive numbers as greater than zero, and negative numbers by the illogical expression, ‘less than zero’.

It then compounds the error by creating a new category; zero as a number that is ‘neither negative nor positive’. The truth it is struggling to realize is that the symbol 0 represents the mind, and the mind, as the viewpoint within which number and math exist, is above, and so indifferent to the objects it contemplates. As mentioned, 0 is not a number at all.

Conventional math has led us so far astray that it is painful to have to point out its absurdities, but it has to be done, for imbued in our consciousness since early childhood they prevent us from seeing the world revealed in science intelligently within the context of reality. Because at the level we teach how to spell CAT we also imply that 0 and 1 are numbers, and at the level we teach “Rover has a ball” we include the thought that zero is a number, ‘negative’ numbers exist, and so on, mathematicians talk about i; science is unable to merge quantum and relativity theory and astronomers talk about the ‘size’ of the universe. We search for the error in the clouds when we should be looking at our feet.

The ancient Greeks pointed out that 0 is not a number because it is nothing, and 1 is not a number because number means more than one. The idea of 1, as opposed to 0, is derived from entity, originally creatures and things. Singularity is the ‘one’, through which we identify kind, and plurality is ‘greater than one.’ Plurality is thus the general term for number, which latter is the specification of plurality by count or estimation.

0 and 1 are functional as the hinge between mind and world. 0 interfaces with the mind, 1 with the world. It is convenient to call 0 and 1 numbers, but strictly speaking they are the agents that, through counting lines and bases, create number, unraveling in the world the order that already exists in the mind.

Once the bases as the gathering ground of the mind are formed, we have to consider ‘i’, the square root of ‑1 as the focus of the mathematical connection between mind and world. To begin its resolution, take the straight counting line into the circle. In our example ‑1 is 9 in 10-circle, with the relations of square and root that then apply. Mathematics comes to a point in ‘i’. From this, expanding from the ‘i’, a world in consciousness opens up. 0 and 1 are the active agents in generating this focal relation, and circlemath puts the whole mechanism in place for us to see.

The one step (switching into the circles) resolves the philosophical ground of math and the ‘square root of -1’, which now falls into place as fully accessible at primary school level. In the circular and straight we now have two systems, one subjective inherent, the other projected and objective or worldly. They map to each other through i, which is not a number but the confluence of mind and world mapped within a mathematical continuum.

We are done with the fallacy of negative numbers, ghosts of the imagination, born with Fibonacci in the 12^th century and matured in the 16^th. The positive sign simply means add, and the negative subtract. This, with the rules of circle math are all we need to carry all mathematical computation, clockwise and anticlockwise across the sea of number and the ocean of bases. The imagination (read mind) is just as mathematical and just as real as any castle of brick and stone.

The fact that ‑1 in 10-circle is 9, whose square root is 3 (and 7) is the touch of death for the mystery sequestered in the square root of -1. This one facet does not explain i, for it is only a fragment of a greater whole, but it is the key. Turn it in the lock and the door will open, for it can be readily understood. Even for those who are far from grasping the pattern of the whole, the illusion of i, the square root of -1 will begin to fade and will continue to melt away until it vanishes.

Circle math emerged from the shadows of ordinary math by systematically eliminating every non-universal relation and mending each gap left behind.[iv] Eventually the entire circular system stood forth, universal and complete. The task was to map mathematics to the pattern of inherent mental function. We now have ‘i’ unraveled, and with it a whole new field of mathematical relations, rich in algorithms accessible to children who take to it like ducklings to water.

Education Departments were not so happy with it. They saw it rather as a threat, much as a hen that has hatched out ducklings will run along the bank in fright as its ‘chicks’ swim and dive in the water. They were not geared for the massive change in learning patterns that it represents.

The circular mathematical system depicts mathematics as it exists in the subjectivity of mind. It arises directly from the neural mechanism that responds to the world, and heralds the end of the defect that medicine knows how every organ in the body works, with but one exception, namely that which makes sense of the whole, the brain; that finally we may know what thinking is, upon which our human being and understanding rest.

Response comment and inquiry welcome.

You can contact me by email:

stetay @ bigpond.net.au

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