- Mach's principle in physics and its meaning for mathematics.
- Vectors as complex numbers.
- Imaginary numbers as averages of their dimensional ranges.
- Means and dispersions of complex numbers.
- Mach's principle in maths applied to the Interval, and a Caldera cosmos.
- Foot-note.

Ernst Mach was a distinguished physicist. He was also a philosopher of science, who formulated the doctrine of Positivism. I briefly discussed this and Max Planck's taking exception to it, in the context of social science. (The reference is my web page: The Moral Sciences as the Ethics of Scientific Method.) Crudely speaking, Mach believed that scientific concepts should be tied closely to reality.

Planck believed that sometimes a concept could be useful to explaining a process, even if you couldnt identify something to which it corresponded. In 1900, Planck didnt believe in the reality of his quanta. But in 1905, Einstein refered to light quanta or photons, as they are now called, in explaining the photo-electric effect.

One can see that Mach has the experimentalist's point of view and Planck has more of a theorist's view-point. This Positivist emphasis in Mach's thought is evident also in Mach's principle. This was a program for physics, that aimed to remove preconceived ways of looking at the physical universe.

The classic case is Newton's making a picture frame of absolute space and time, within which to see the universe in motion. This was the best technique available at the time. But a complete picture of the universe has no outside picture gazer. It is like a Velazquez, where you see a small image of the painter reflected in a mirror in the painting.

By definition, the universe contains everything within itself. The whole picture shows the universe entirely in terms of its own motions, and not with respect to some outwardly imposed frame. After all, anyone can come up with their own point of view. The theory of Relativity relates different points of view so that all observers have a common basis for precisely agreeing in their measurements of any event.

Julian Barbour's book, The End Of Time (a magnum opus, much reviewed on this web site) describes his collaborative work to fully apply Mach's principle to modern physics. In fact, he found that General Relativity is consistent with Mach's principle. Einstein was influenced by the principle but he didnt know that he had succeeded in incorporating it. That just goes to show how involved is his theory.

Even Barbour's gifts, for illustrating mathematical structures, are severely tried on the mathematics of General Relativity. This page, however, sets ourselves a much simpler problem, indeed a childishly simple problem. That is because the mathematics, to follow, comes from the so-called New Mathematics taught to primary school children. This page treats the new math in a newer way. But this Newer Mathematics merely applies very simple forms of statistics to the foundation course.

Some elementary math, I was doing (the only sort I know how to do) made me realise that mathematics itself can be treated as self-representative. I mean that numbers can represent each other. This compares to Mach's principle, in which he envisages any given mass in motion as gravitationly measurable solely in relation to the distribution of all other masses in the universe.

In particular, some elementary statistical treatment shows that each imaginary number can be regarded as an average, or representative item, of a range of imaginary numbers. The function of statistics is to represent or measure typical items, in terms of suitable averages of any given range of items. In other words, statistics is self-representative mathematics. And if you can apply such statistics to abstract numbers, then this shows that mathematics itself is self-representative. Saying mathematics is itself statistical effectively applies Mach's principle to mathematics, quite apart from Mach's principle of a self-representative physical universe.

A statistical meaning to imaginary numbers is just about enough to demonstrate that mathematics is self-representative, in the statistical sense. This is because it has been demonstrated that, after imaginary numbers, no more extensions to the number system are needed. At least, algebraic equations are always solvable within the complex number system, of real numbers combined with imaginary numbers. This was the fundamental theorem of algebra, proved by Carl Friedrich Gauss.

Without going into any detail, real numbers can be represented on a measurement scale, like two rulers, end to end, at a point marked zero. From zero, one ruler is conventionly assigned to be going in the direction of the positive numbers, +1, +2, +3,... and the other ruler is marked off as going in an equal and opposite direction, -1, -2, -3,... Mathematicians have proved that rational numbers, ratios of whole numbers or fractions, can be placed on this scale. And so also can be irrational numbers, which do not have an exact fraction. For example, the square root of two roughly equals 7/5. The ancient Greeks proved it does not have an exact fraction.

Mathematicians thought all numbers were accounted for by the real number scale, until the emergence of so-called imaginary numbers.

In trying to solve algebraic equations, mathematicians sometimes came
across solutions in terms of the square root of minus one. This didnt make
sense as a number, so it was called an imaginary number and was given the
symbol *i*.

Thus i² = -1. So, this imaginary number, to the power of four, must equal minus one squared, which equals plus one. i, to the sixth power, equals minus one again; i, to the eighth power, equals plus one. In short, powers of i produce a cycle of alternating negative ones and positive ones.

Consider this cycle in polar co-ordinates. Going from plus one to minus one and back involves two consecutive turns of 180 degrees or pi (the Greek letter for pi being the usual symbol). The square of i is in effect two repeated turns by 90 degrees or pi/2. And the meaning of i is one ninety degree turn. i³ means a 270 degree turn. And so forth.

Imaginary numbers are not really imaginary. This is just a historical name that has stuck. They are used in algebraic representations of vectors. A vector like wind velocity has speed magnitude and compass direction. Wind vectors might be represented on a map by length of arrows for their strength, and pointed with reference to global co-ordinates, for their direction.

A complex number is a familiar or 'real' number and an imaginary number. The way they are added together on a graph ( see figure 1 ) is like the way two vectors, as arrows with their lengths and orientations, add up to a combined length and orientation.

Take the complex number, z = x + iy. Most simply this could be: 1 + i1, that is 1 + i. The meaning of this number z is in the operation it performs. Refering to figure 1, suppose a marker, z is at position P, on the x-axis of a co-ordinate system. The operation 1, on z, leaves it where it is. ( If x were 2, the marker would be sent twice its present position along the x-axis. ) The operation i, on z, turns it thru 90 degrees, to position N on the y-axis. If x were zero, so that only the latter operation were to be considered, then z would simply be moved from its original position on the real axis to the position N, at 1 on the imaginary axis.

This two-part operation of a complex number is performed like the addition rule for vectors. The imaginary number i is like a vertical arrow that must be added to the end of the horizontal arrow on the real axis, resulting in the marker's new position at S.

The diagonal line R - we'll use small r - corresponds to the so-called
resultant vector, OS, of OP and ON, the two vectors' combined effect. So far,
the complex variable has been described in rectilinear co-ordinates. It can
be put in polar co-ordinates of angle Q and radius r. Here, the term 'radius
vector' would apply. The equivalent values are x = r cos Q and y = r sin Q.

r is given, in terms of x and y, by Pythagoras' theorem.

In polar co-ordinates, the stretching ( or shrinking ) and turning, of complex number operations, is given to the radius and the angle, respectively.

In rectilinear co-ordinates, any complex number is given the convention in
algebra of z = x +iy.

z does not mean a third co-ordinate of three traditional quantities or real
numbers. z is any position on the plane marked out by the two x and y
ordinates, where y is refered to as the imaginary axis, as distinct from the
real axis, x.

This differs from traditional geometry. Suppose there is a second complex variable, w = u + iv. Traditional relationships between variables, in effect, make y and v equal zero, so that we have only to deal with functions of real variables, which here would be u as a function of x. On a graph, the real function can be two correlated sets of values plotted from the linear vertical axis for the dependent variable in relation to the linear horizontal axis as the independent variable.

But a complex function involves the correlation of two planar functions on two distinct graphs.

The complex variable z = x + iy can have a change of sign into Z = x - iy. On a graph, see figure 2, the latter is the mirror image, across the x-axis, of the former. The mirror variable, on the negative side of the y-axis, is called the 'complex conjugate,' of the complex variable on the positive side of the y-axis. (It has its own special symbol in the text-books: the letter z with a tilde or wavy bar on top. Here we have to use "Z".) The multiplication, of a complex number by its conjugate, is: (x+iy)(x-iy) = x² + y² = r².

There seems to be no obvious reason why any real number cannot always be an average that is representative of other numbers. Rather it is the obviousness of this assertion that makes it seem trivial and unimportant. Less immediately obvious is that any imaginary number can always be considered as an average of other imaginary numbers.

In figure 2, the four axes can be represented in terms of imaginary numbers, not just the positive and negative y-axes. Thus the x-axis at 1 becomes i^0, which reads: the imaginary number i to the power of zero. This equals one. Turning 90 degrees or pi/2 degrees, we come to the positive y-axis, already designated i, which is short for i^1, or i to the power of one. Ninety degrees further, on the negative x-axis, -1 = i² ( or i^2, in the previous notation). Another pi/2 degrees turn brings the negative y-axis at -i = i^3.

Turning another pi/2 degrees brings us back to where we started, the positive x-axis. But to signify that we have been thru 4 turns of the circle, the imaginary number becomes i^4. This is the same position as i^0. The difference is that i^0 means that there have been zero turns of the pi/2 or 90 degree turn operation. When the letter i is used to signify this operation, turn pi/2 degrees, it is sometimes called the operator i.

The degree, of the power of operator i, signifies how many 90 degree turns it goes thru. To show that an imaginary number can always be represented as an average, we remember that an average or mean represents, or is a typical item in, a range of values. We can show that each operator is the operational mean of the range of its adjacent 90 degree turns.

The mean we use to show this relationship is the geometric mean. For the range, we take the ends of either the x or y axis. Thus, a dimension has a statistical interpretation as a range. It doesnt matter how many number of turns are taken. But we'll start at the beginning with x-axis range from i^0 to i^2, that is from one to minus one in real number terms. (We will actually be using an imaginary number as the average of a real number scale. Except that the real numbers have been transformed into turn operators along with the imaginary number dimension.)

When powers are multiplied, an addition rule applies to their indices. Multiplying i^0 by i^2, you simply add indices zero and two. This gives their multiple as i^2. Or, i^0 x i^2 = i^2. Taking the square root of two multiplied end-points of a range gives the geometric mean of the range. The square root of i^2 or -1 is i.

In fact, this interpretation modifies the famous expression that the imaginary number is the square root of minus one. Its statistical meaning is that the imaginary number is the geometric mean of the range from minus one to one. The geometric mean imaginary number is found by applying the arithmetic mean to the indices or powers of the range values. In the above example, take zero plus two, then divide by two (for the two range values) which gives a power of two over two equals one. And i to the power of one is just i.

To take the next example, if the y-axis is the range of values from i (or i^1) to -i (or i^3), simply add the powers one and three, which equals four, and divide by two, which results in two. This gives the geometric mean of the y-axis to be i to the power of two, or i^2 = -1.

In principle, one neednt stop at end-points of two number ranges. For instance, a geometric mean of a range of three imaginary numbers would be the cubic root of the three numbers multiplied.

Weve considered imaginary numbers as rotation operators but what of complex numbers, that combine real numbers with imaginary numbers? How are they to be considered statisticly, as averages with ranges?

We can find the geometric mean of a complex number by multiplying a range of complex numbers and taking the root corresponding to the number of items in the range. For a range of two complex numbers, we take the square root. With regard to figure 2, there are four quadrants, each with their own complex number variable. Upper right quadrant is the complex number z = x + iy. This maps any position in that quadrant of the two-dimensional plane.

Lower right quadrant is the complex conjugate of z, usually symbolised as z with tilde (or wavy line on top), here as Z = x - iy. The top left quadrant is -x + iy = -(x - iy) = -Z. That leaves bottom left quadrant, -x - iy = -(x + iy) = -z.

We can try all the combinations of z, Z, -Z and -z, to find their respective geometric means (GM). The GM of z(-z) = iz = i(x+iy). The GM of Z(-Z) = iZ = i(x-iy).

We now multiply, in turn, the adjacent complex numbers for each quadrant. This gives us a result that can be equated, by Pythagoras' theorem, in terms of the radius squared. The square root gives the geometric mean, which is in terms of the radius, r. The GM of zZ = (x²+y²)^1/2 = r. The GM of z(-Z) = i(x²+y²)^1/2 = ir. The GM of (-z)(-Z) = -r. The GM of (-z)Z looks the same as of z(-Z) but we assign it the value -ir. This answers to the possibility that the square root of minus the radius squared can be plus or minus ir.

But this assignment does require a general sign convention regarding the non-equivalence of z(-Z), whose square root equals ir, and (-z)Z, whose square root equals -ir. The same applies to the next step, where (-z){-(-Z)} is also not equivalent to {-(-z)}(-Z). Likewise for successive negations, with the complex number or its conjugate having one more negative sign, before it, than the other.

To use the jargon, the previous paragraph is a non-commutation rule, where the two multiples have the same number of the same signs but their different positions in the brackets render a different result, when their respective roots are taken.

We can now relate each geometric mean to the two bounds of its range. I assumed, without really knowing, that this can be done for a complex number range, as it is done for a real number range. For example, this involved taking GM, iz, of -zz, then subtract lower bound -z from iz: -z - iz = -z(1+i). Now subtract the upper bound, so: z - iz = z(1-i). If you add the two differences from the mean, you should get the total dispersion (that the GM represents as an average or typical item): -2iz.

However, by trial and error, I found it was slightly simpler, to find the
dispersion about the geometric mean, by subtracting each bound from the
geometric mean, and then adding each difference, for the over-all dispersion.
Changing the way round the difference is taken, between the mean and its
bounds, changes the signs. The dispersion, about iz, changes from -2iz to
2iz. The dispersion about iZ, GM of -ZZ, is similar: 2iZ.

Thus, in both cases of the geometric means, iz and iZ, their dispersions are
twice their magnitude.

The meaning of a complex number multiplied by an imaginary number is that the x and y axes are inter-changed: iz = i(x+iy) = -y+ix = -(y-ix). This is like -Z = -(x+iy), except the axes have been changed round. And iZ = i(x-iy) = y+ix. This is like z = x+iy, except that the axes have been changed round.

The above two cases are just a complex number multiplied by itself, with just a difference of sign between them. More revealing are the cases of multiples of complex numbers by their conjugates, including under various signs. The GM, of zZ, is r. Its dispersion is the sum of the two bound differences, from the mean, of: r - (x+iy) and r - (x-iy). Adding the two sides of the dispersion gives: 2r-2x = 2(r-x).

The two sides of the dispersion about the geometric mean are unequal. We would expect this from the geometric mean on a real number range. But this is a complex number range and there is a twist, that when you add the two parts of the dispersion, their mutual cancellations are such that each side is left with an equal contribution to the dispersion.

This turns out to be the case for all the other dispersions. In table 1, the dispersions are all multiples of 2, for the two sides of the dispersion making an equal contribution to the over-all dispersion. (See foot-note.)

Table 1 is for the geometric means and their dispersions of the successive multiple pairs of complex numbers, where each complex number maps a quadrant of a circle.

Complex multiple | Geometric mean | Dispersion |

zZ | r = r.i^0 | 2(r-x) |

z(-Z) | ir = r.i^1 | 2i(r-y) |

-z(-Z) | -r = r.i² | -2(r-x) |

(-z)Z | -ir = r.i^3 | -2i(r-y) |

-(-z){-(-Z)} | r = r.i^4 | 2(r-x) |

-(-z){-(-(-Z))} | ir = r.i^5 | 2i(r-y) |

-(-(-z)){-(-(-Z))} | -r = r.i^6 | -2(r-x) |

-(-z)(-Z) | -ir = r.i^7 | -2i(r-y) |

The geometric means of complex numbers multiplied by their conjugates, under various signatures, all appear to be geometric means of rotation of the radius. The GM of zZ equals r, or (i^0)r. This stands for a zero turn of the radius line, resting on the positive x-axis. ir would be the radius line turned pi/2 degrees or 90 degrees to rest on the y-axis as the imaginary axis. i².r is then two 90 degree turns to the negative x-axis, and -ir = i^3.r is three such 90 degree turns.

i^4.r is four such turns, starting to repeat the cycle all over again.

The previous section showed how imaginary numbers can be geometric means. The only difference there was that their magnitude was implicitly unitary. That is they were all set at r = 1, such as: i^0 = 1.i^0 = 1.1 = 1; i = i^1 = 1.1^1; i² = i²1 etc. By replacing 1 with r, any positive number, all weve done here is to lift the restriction on the magnitude of the imaginary number. These imaginary numbers represented in increasing powers, the successive axes that divide the circle into quadrants. Multiplying the imaginary numbers by r, merely represents the circle as not being confined to a radius of unity, in considering rotation operations.

The dispersions from geometric means of complex numbers multiplied by their conjugates all relate to the difference between the radius r of a circle and values on the x or y axes, as formed by dropping perpendicular lines, from where the radius line meets the circumference, to the x or y axis. That is as in figure 1, if the radius r were inscribed in a circle.

Table 1 further signifies not only a circle of radius r, but an outer circle of twice the radius or 2r. The inner circle of radius r, represents the geometric mean. The distances, from zero to r and from r to 2r, represent the respective dispersions with their equal distance contributions to the over-all dispersion. The dispersion varies, according to the values of x or y. The maximum values of x or y, at equal to r, reduce the dispersion to zero. Minimum values of x or y, at zero, maximise the size of the dispersion to 2r. That is the first row on table 1, similarly for corresponding positions on the outer circle, 2ir, -2r, etc, shown in the successive rows of the table.

Minkowski's Interval, unifies different view-points of an event, approaching light speed, into one shared space-time observation. It is essentially the circular function just described. The Interval, I, relates to the radius, r, which is fixed, like the shared space-time observation.

Below is the equation of the Interval, I. Light speed is c. Two observers of an event, measure different local times, t and t', and local velocities, v and v'. The velocities can be, but doesnt have to be, considered in just one dimension.

I² = (ct)² - v²t² = (ct')² - v'²t'².

Suppose we subtract the below average dispersion (which is the Interval, as the geometric mean, minus the lower dispersion bound) from the above average dispersion (which is the upper dispersion bound minus the Interval):

I - ( ct - vt ) - {(ct + vt ) - I} = 2I - 2ct.

This difference between the below (Interval) average and above (Interval) average dispersion, for any given observer, may be compared to classical relative motion which subtracts the velocities of two observers going in the same direction.

Thus statistical special relativity seems to re-create a statistical version of Galileo's relativity principle in classical physics. The sum total dispersion of the below average dispersion plus the above average dispersion may likewise correspond to the classical relative motion for two observers moving in opposite directions, so that they have to add their respective velocities to get their relative motion to each other.

The Interval's dispersion difference, 2(I - ct), is like the first row of
table 1. Conversely, the circular function table values were derived by
adding dispersions but with complex number bounds.

[Also conversely, the circular function's dispersion difference is: r -
(x-iy) - {r - (x+iy)} = 2iy. While, this corresponds to the Interval
dispersion's sum total of:

I - ( ct - vt ) + {(ct + vt ) - I} = 2vt.]

From table 1, the radius becomes the Interval; ct is on the x-axis; vt = iy, where y is distance, is on the y-axis. In special relativity, using Minkowski's Interval, there is an imaginary fourth dimension, usually related to time as a dimension.

The Interval is the constant observation of an event by all observers. This can be represented by the constant radius, r, making a perfect circle. Alternatively, re-arrange the Interval in terms of light speed constant, c:

c² = I²/t² + v² = I²/t'² + v'² = I²/t² - (iv)² = I²/t'² - (iv')² .

The geometric mean is then the speed of light, with a dispersion between bounds, (I/t - iv) and (I/t + iv). Subtracting each bound from the GM, which is c, and adding the two differences gives: 2c - 2I/t = 2(c - I/t). That is for one of the observers. The same would apply to an other observer with the indexed velocities and times, t' and v'.

This consideration leads us to a statistical definition of different observers, in special relativity. Their respective observations measure different velocity dispersions about the same geometric mean velocity of light.

Looking at the total dispersion, 2(c - I/t), we have a theoretical upper bound of twice light speed and a lower bound of zero speed, as I/t varies from zero to c.

The speed of light is constant and this would be represented by the constant radius, r, making a perfect circle. One side of the dispersion is from the zero-valued origin of the circle to the circumference. The other side of the dispersion is from an outer circle, twice the radius of the geometric mean radius of the light speed up to the inner circle of radius, r. The shape of these two sides of the dispersion might look like a caldera.

That is a volcano with increasingly steep inner and outer walls - actually two walls that never quite meet. A further difference is that for the light speed model, the walls would be of infinite height that no body could ever reach. You have to imagine someone climbing the ever steepening inner or outer sides, finding it tougher and tougher going with every step up. And never reaching a summit.

The inner wall represents the climb of objects towards the speed of light. The outer wall represents the climb of faster than light objects (tachyons) never quite going slow enough to reach light speed. Tachyons are moving inwards from the outer to the inner circle, on a graph of space and time co-ordinates, so they are moving backward in time.

Slower-than-light objects are sometimes called tardyons. Lisa Randall's book, Warped Passages, says that recent findings reveal tachyons as instabilities, in equations, to be resolved. This suggests that tachyons are a mathematical figment rather than a reality.

One physicist said of the failed attempts to detect tachyons, that they are no longer thought of as possible real objects but as fancies, like unicorns. However, the caldera model implies that tachyons will not be observed in our tardyonic cosmos. So, there is no conflict there.

But the model does imply the necessary existence of a complementary tachyonic cosmos to our tardyonic cosmos. This is because a statistical interpretation of light speed as a geometric mean speed implies it is the mean of a range of values, one of whose two dispersions is tachyonic, the other tardyonic.

Tachyons are unobservable by standards of classical and relativistic mechanics. But it cannot be ruled out that tachyons may be implied by quantum mechanical observations. Feynman diagrams may offer a first intimation of the possibility of tachyons. Sometimes the diagrams have an ambiguous interpretation, that allows a particle, in a sub-atomic interaction of particles, to be considered as briefly moving backwards in time, in effect faster than light.

One might fancifully imagine a tachyonic Feynman, on the outer side of a caldera cosmos, showing a diagram, in which an interaction involves a particle, that might be thought of as moving forward in time.

As mentioned on my first page on the subject, A Statistical Basis for Special Relativity (section: constant light speed as a constant average) a wave pulse can be made to appear to move backwards in time but no single photon can be identified as tachyonic.

Observable quantum effects might convey a tachyonic message from the outside of our caldera cosmos. For instance, relativity theory allows no escape of energy from inside a black hole's event horizon. That is the radius within which even light cannot escape the gravitational pull of this super-massive object.

But a quantum effect allows a black hole to give off incredibly minute quantities of energy, called Hawking radiation. The vacuum of space while being empty of matter is also prone to the spontaneous creation and annihilation of matter, within the limits imposed by Heisenberg's uncertainty principle.

For instance, a particle and its anti-particle might be created. Being of zero net energy, energy is conserved. Before they annihilate each other, one particle might be captured by a black hole, releasing the other as Hawking radiation.

After their own fashion, cross-over effects might be detected in the context of a tardyonic-tachyonic caldera wall. The caldera model differs from the black hole model of the universe. A black hole does allow a one-way traffic into it. But the caldera model offers an impassable barrier either way, as far as we know, apart from possible quantum effects.

A sub-atomic phenomenum called "quantum tunneling" allows particles out of a potential well of energy that they could not possibly escape from according to classical laws of motion.

Other weird and wonderful effects are continually being discovered in quantum physics, so it would be unwise to assume that there may be no possible experimental confirmation of a tachyonic outside wall to the tardyonic wall of a caldera cosmos.

A caldera is the crater left by a volcanic explosion. This is literally a big bang. And it makes one think of the Big Bang in cosmology. The "calderic" property of light might be explained as a left-over of the big bang, just as a caldera is the left-over of an eruption.

The calderic "wall" of light is infinitely high. Any massive object experiences an accelerating difficulty in approaching light speed. Tho subject to further revisions, current atronomical data has come round to the view that the universe is set on an infinite expansion, indeed that the expansion is accelerating.

I dont know whether there is any correspondence, in the previous paragraph, to the light caldera and the expanding universe, as such. I am not saying there has to be. But one can't help speculate about a possible correspondence between a light caldera that is not infinitely high and a universe that does not go on infinitely expanding.

Why might there be a correspondence?

Well, the analogy is only the roughest of guides. But in some cases, the
height of a volcano's caldera might be governed by the violence of the
eruption. And indeed, other factors. The analogy isnt to be taken too
literally. It just helps with the drift of ideas, without being applied too
rigidly.

Finally, a warning: Mach's principle applies properly not only to physics
and mathematics but to politics, where the self-referential principle is
representative democracy. Scientific progress is liable to increasingly
destabilise society and the environment, if there is not the required
scientific commitment to democratic progress.

As mentioned above, all the complex geometric means and their ranges show that the two sides of the dispersion are by themselves asymmetric. But when you add them, they make a symmetric contribution to the over-all dispersion. Intuitively, this makes sense, in that the mathematics is that of circular functions. Circles are symmetric, yet they can be described by exponents, which relate to series, or ranges, which are asymmetric about their average, a geometric mean. At least, real number ranges, that can be graphed as an exponential curve, are asymmetric about their most typical average, a geometric mean.

In fact, complex number z equals r(exp)^iQ, where r is the radius of a circle; exp. stands for the infinite number constant called the exponent, of about 2.718..., describing constancy in the rate of change of the rate of change, and so on. The exponent is progressively calculated by successive terms of a series, called the exponential series, the importance of whose successive terms, starting with 1, followed by another 1, then increasingly rapidly falls off. In other words, it is an asymmetric series, unlike the normal distribution, which is a symmetric series. The exponent is taken to the power of imaginary number i, multiplied by an angle Q, made by the radius line above the positive x-axis, within the (upper right) quadrant, mapped by z.

Being an infinite series, the simple exponential series, that sums to 2.718..., lacks a lower bound, with which to calculate a geometric mean. But in practise the first few terms of the series may approximate a pattern in a collection of data, and the geometric mean is taken of those most important terms.

The normal distribution, which is a symmetricised kind of exponential series, lacks upper or lower bounds, but since they are symmetrical, they average out, with the arithmetic mean, at the middle peak of its "bell curve." This shape graphicly shows the magnitude of successive terms, approximated by the binomial series.

*Richard Lung**.
9 March 2007;
slight changes 22 march 2007; 24 & 25 may 2007.*