The Strange Mathematics of Electrical Engineering. Applied mathematics of Rotation.

Ben Campbell Biezanek1*

1 Distinguished Researcher, Shropshire hill country England, United Kingdom

*Corresponding Author:Ben Campbell Biezanek, Distinguished Researcher, Shropshire hill country England, United Kingdom; E-mail:

Citation: Ben Campbell Biezanek. (2023) The Strange Mathematics of Electrical Engineering. Applied mathematics of Rotation. Nano Technol & Nano Sci J 5: 146.

Received: April 18, 2023; Accepted: April 25, 2023; Published: May 2, 2023.


Imagine that you are walking into town making direct steps towards the center of town. Then, you remember that you left your umbrella at home, and it is soon going to rain. So, although you have already walked 1,267 steps towards town, you bite the bullet, stop, rotate through 180 degrees and start making direct steps back towards your home. However, with respect to walking directly into town, you are now taking inverse steps towards town. As Carl Friedrich Gauss pointed out in paragraph #24 of his second letter to the Royal Society in 1831, calling our numbers positive and negative is very silly, we should rather call them direct and inverse. So, perhaps there is no such number as minus-one (take away one), but there really is such a number as inverse-one, the rotational inverse of direct-one?

The Arithmetic of Rotation

Let αrω be taken to mean a magnitude of α rotated by the angle ω. Let the value of +1 be assigned to the value 1r0. Now, if the natural whole units of rotation are such that there are four whole quadrant units of rotation per completed rotation, then the value of inverse-one (-1) is 1r2 and the square-root of that, which pure mathematicians call i, has a value of 1r1. The value which the pure mathematicians call -i has the value of -1r1, which could also be described as 1r3. However, as we can also describe -1 as 1r (-2), the square root of inverse-one (-1) has two values, those are 1r1 and 1r (-1), alternatively, just to try and humour the pure mathematicians for tiny bit longer, I could write i and -i.

Clearly, our present understanding of i and -i leaves a lot to be desired. The character i represents the rotational operator, it must always be followed by the angle to rotate by. When a pure mathematician writes i, he ought to write 1i1. Pure mathematicians think that i means the square-root of minus-one, an impossible number. Of course, if a person thinks that the vital rotational operator of the real Universe that he actually lives in, is actually an impossible number, then he will think deranged thoughts and teach a lot of nonsense, but that nonsense has no value, at least to an electrical engineer. My symbol r, as employed above, meaning “times a rotation of”, is what the poor, sorry pure, mathematicians heavily abuse as i.

The Arithmetic of Rotation in Exponential Units.

Fortunately, in exponential counting units, the apparently questionable nature of the orthogonal complex-numerical-plane simply vanishes.Any valid number must be able to be assigned to a natural exponential value of (α + iω). Here, we no longer need to worry about -i as that is taken care of by the polarity of ω. As I can describe counting on the (natural, e-based) exponential-rotational-manifold as counting in the actual exponential ratios of nature, then I need a new name for the natural antilogarithm of that manifold. I will call the resultant anti-logarithmic device the flat-rotational-plane.

In order to interpret what is meant by the exponential number (α + iω), when moved onto the flat-rotational-plane, we need to take the natural antilogarithm of (α + iω). That is found as e (α + iω).

Here I employ a new convention of showing the exponential base as a subscript merely to emphasise that (α + iω) is the exponential number of interest and e is merely the correct numerical environment for decoding that number. Therefore:

e (α + iω) means exactly the same as e (α + iω), it is just that the old convention placed far too great an emphasis on the (natural) decoding number and far too little emphasis on the actual exponential number. This unfortunate habit of ours has greatly confused virtually all students for far too long; for example, it seems to have even confused the great Leonhard Euler himself (see footnote 2).

For myself, I would be far too troubled by the evaluation of e (α + iω) when stated in that formal and correct form. Fortunately, I can break the evaluation down into two parts by stating that –

e (α + iω) = e(α). e(iω)

and it is axiomatic that -

e(iω) = iω

This is because the exponential-rotational-manifold and the flat-rotational-plane share the same units of rotation. In other words, the rotational polarity of the number is unaffected by being stated in exponential or flat numerical form. Exponential form only impacts the magnitude component of the number. Therefore, the evaluation of the exponential number (0 + iω) is very simple.

e (0 + iω) = 1. iω

This fundamental identity of the Universe should be called “the circle of unity”. So, for our interpretation of the existing so-called complex numerical plane on the flat- rotational-plane, I can state the following identities: I should regard +1 as 1i0, i as 1i1, -1 as 1i2 and -i as 1i3, except that 1i3 is identical to -1i1. This interpretation ignores reverse rotation and so although it is interesting and very useful to us, this primitive interpretation of the flat-rotational-plane can have no formal validity.

The flat number zero in the exponential-rotational-manifold.

I must now turn to the thorny issue of the number zero on the flat-rotational-plane. When counting upon the exponential-rotational-manifold, I must express the flat number zero as (-∞ + iω). I can evaluate zero   on  the  flat-rotational-plane  as  e(-∞ + iω) and so upon the flat-rotational-plane, regarding zero as if it were a finite number must be a non-starter. That is just as well because zero is the rotational fulcrum of the flat-rotational-plane.

The assignment of revised polarity labels for both rotational forms.

Fortunately for myself, this polarity label reassignment task has already been done for us by Carl Friedrich Gauss in paragraph #24 of his second letter to the Royal Society in 1831. Gauss suggested that in place of our existing polarity labels of positive, negative and imaginary, we might employ the superior polarity names of direct, inverse and lateral. I have always found those Gaussian names to be highly appropriate and most helpful to me.

As far as I can tell, the only reason that Gauss did not write this paper in 1831 and save me all this trouble 192 years later, is that he became stuck, like a rabbit in the headlights, by the absurd and grossly unhelpful radian unit of rotation. There are four natural units or quadrants of rotation in a completed rotation; stating that there are 2π units of rotation in a completed rotation is something to do with the length of an imaginary arc, but what has that got to do with the rotation of numerical polarity? There is not the faintest connection.

All Numbers are Imaginary.

The designation of the number one as being real is absurd. If I say, “one orange”, then that might mean “one real orange”, but please show me a negative orange. If I take away all real units (say oranges) then what is real about +1? That number (1i0) means unity, times an imaginary unit, times a rotation of zero. The numbers 1i0, 1i1, 1i2 (-1) and 1i3 (-1i1) are all equally imaginary. You might say use debits and credits marked in oranges. If you owe your neighbour a debt of three oranges, you could perhaps say that your stock of oranges is minus three. But accountants tell us correctly that you have a liability for three oranges, because there is no such thing as a negative orange. While you owe your neighbour three oranges, he had an asset of the three oranges that you owed him. The liabilities and assets coexist simultaneously, the direction in which the value is owed is relativistic. But at that time, there might be no real oranges in either home; however, the liability and its relativistic direction is real.

Footnote (1)

We can now glance at both Rotational Number types in graphical form.

Our Natural Universe actually works with exponential ratios, so, showing a reconciliation between any kindergarten-arithmetic (or flat-earth-arithmetic) and the actual exponential-rotational-manifold of Nature, must always be an imperative.

Assigning a polarity of positive or negative to rotation is nonsense as rotation has two senses, left and right, aka anticlockwise and clockwise, but rotation is relativistic; for clockwise (or right) to mean anything we also have to define the perspective (in front of or behind the clock face). Both rotational directions have equal validity. The author finds a solution for universal space-time with the conventional direction of rotation (as shown above) which also works perfectly in the opposite direction of rotation. However, unless one wishes to go the way of Gregor Cantor, do not try to make these two solutions work with both rotational directions simultaneously.

Rotational-number-theory shown in a graphical sketch.

Footnote (2)

Leonhard Euler’s great misunderstanding.

All numbers can be expressed in the natural exponential form as (α + iω). Euler’s Identity, in the form that he originally stated it, is invalid. “Invalid” is very polite of me, “complete nonsense” would be a far better description. Euler stated that:

e+ 1 = 0

This is nonsense because iπ is not in the proper exponential number form of (α + iω). On its own like that, iπ is not even a valid exponential number at all. Let us now convert the nonsense of an orphaned iπ into a valid exponential number by the simple expedient of converting iπ into the valid natural exponential number (0 + iπ). So, now I can convert from Euler’s original and understandable confusion into a valid and meaningful identity by stating that:

e (0 + iπ) + 1 = 0

Note that e (0 + iπ) is not an exponential number but rather its antilogarithm, that antilogarithm is a flat number. The exponential number there is just (0 + iπ). This identity can be manipulated into a more comprehensible form as follows:

e0. eiπ = -1

Well, every schoolboy knows that e0 is 1 and everybody ought to be able recognise that e refers to a rotation of π radians. The great confusion for the so-called “mathematicians” of flat and rotationally rigid so-called “number-theory” and all of their poor students over the last 240-years, lies not only in thinking that i (the vital rotational-operator of the Universe) is a number (the square-root of minus-one) but also in thinking that the radian is the natural unit of rotational angle.

This Identity is much easier to understand (in radian rotational units) as -

e0. eiπ = 1. iπ

Note that taking the natural logarithm (or the natural antilogarithm) of iπ (rotate by π radians) makes no change to the rotation (i.e., no change to the rotational polarity of the number). Interpreting 1. iπ is easy, that just means direct-one times a rotation of π radians to inverse-one. Choosing a symbol for the rotational-operator to be i (for imaginary) is the most comical thing that human-beings have ever done. What on Earth is there about the rotation of Earth that is in the slightest way imaginary?