An Alternative Interpretation of the Exponent: Part One
Written by D. and S. Birks
Introduction
When analyzing the validity of Special Relativity/General Relativity, perhaps all that needs to be brought into question
is the use and interpretation of the exponent. In equations such as E= mc2, is the exponent being used correctly? Is
Relativity the end result of a simple misinterpretation of the exponent?
Different Applications of the Exponent
Though, as a mathematical symbol, the exponent would seem to be general in its use; in application the exponent falls
into separate and distinct categories.
For instance, in one category, Scientific Notation, the exponent indicates position of the decimal and represents
number of, but not dimension of, unit assigned: e.g.,
3 x 102 meters represents 300 meters, not 300 square meters, and
3 x 103 meters represents 3000 meters, not 3000 cubic meters.
In a second category, the Calculation of Square and Cubic Dimension, the application of the exponent is dimensional,
with a2 and b3 indicating, respectively, the multiplication of a side of a square or a cube by itself to calculate and
represent the number of square or cubic units composing a given area or volume: e.g.,
(3 meters)2 represents 9 meters2, i.e., 9 square meters, and
(3 meters)3 represents 27 meters3 , i.e., 27 cubic meters.
The difference between these first two categories of the exponent is unmistakable, one is non dimensional and the
other dimensional; and being written and applied differently, both can be used in the same equation without confusion.
However, there is another category of the exponent where the distinction is not as clear.
Des Cartes’ Exponential Calculation of Line
In the 1600s, Rene Des Cartes put forward the concept of a separate category of the exponent within geometric
algebra, one not for the calculation of square and cubic dimension, but a category in which the exponent is used to
calculate (and designate) simple line and proportion of line.
Des Cartes described this concept with the following words:
Often it is not necessary thus to draw the lines on paper, but it is sufficient to designate each
by a single letter. Thus, to add lines ... I call one a and the other b, and write a + b. Then a - b
will indicate that b is subtracted from a; ab that a is multiplied by b; a/b that a is divided by b;
aa or a2 that a is multiplied by itself; a3 that this result is multiplied by a, and so on,
indefinitely .... Here it must be observed that by a 2, b3, and similar expressions, I ordinarily
mean only simple lines, which, however, I name squares, cubes, etc., so that I may make use of
the terms employed in algebra.
These words of Des Cartes are significant. In formulating a system of mathematics for calculating line, in which every
procedure results in a line, Des Cartes opened the door to a linear interpretation of the exponent.
With the words, "Here it must be observed that by a2, b3, and similar expressions, I ordinarily mean only simple
lines ," Descartes established a category of the exponent where line multiplied by itself, e.g., a2, b3, does not produce
a plane or a solid but rather a line. This is important as, customarily, line multiplied by line, ab, and line multiplied by
itself, a2, b3, would be calculated to produce an area of square units, or a volume of cubic units (e.g., a meter multiplied
by a meter would result in a square meter). But surprisingly, in Des Cartes’ system for calculation of line (where adding,
subtracting, multiplying, dividing, and exponentially calculating line produce only line and linear units), a meter
multiplied by a meter would result, not in a square meter, but in a linear meter!
Dimensional Analysis
Essentially, Des Cartes’ system for calculating units of length is analogous to the calculation of apples. As
exponentially calculating number of apples does not result in "square," "cubic," or any other dimension of apple, Des
Cartes’ Exponential Calculation of Line results only in line and linear units.
Imagine an exponential calculation of units of measure where
1.
the value of the exponent is not an indicator of dimension,
2.
the exponent indicates only the number of times line is multiplied (or divided) by itself to
produce a length of line, e.g., multiplying 20 linear units by 20 linear units results in 400 linear units (not
400 square units), and
3.
every exponential calculation, regardless of exponential value (a1/2, b1/3, c1/4, ... d1/n ; a2, b3,
c4, ... dn), results in a line and linear units!
Very simply, Des Cartes’ linear use of the exponent is unconventional and requires its own manner of calculation and
dimensional analysis.
For a comparison of dimensional analysis:
In common practice, exponential calculations of units of length are treated as the Calculation of Square and Cubic
Dimension; that is, the value of the exponent is associated with dimension, and dimensions (exponential values) are
added in multiplication and subtracted in division: e.g.,
length multiplied by length produces a result of square units,
[Length1] [Length1] = [Length2] = Area,
and area divided by length equals length,
[Length2] / [Length1] = [Length1].
However, Des Cartes’ Exponential Calculation of Line does not follow this common practice. In his system, both
calculation and result are of linear dimension; thus, dimensions are neither added in multiplication, nor subtracted in
division - dimensions remain consistently linear throughout: e.g.,
length multiplied by length produces length,
[Length1] [Length1] = [Length1],
length divided by length produces length,
[Length1] / [Length1] = [Length1],
and, extracting the root of length produces length,
[Length1]1/2 = [Length1].
At this point, I would ask the reader’s indulgence. In order to describe Descartes’ use of the exponent for calculating
simple line (which does not, as yet, have a manner for being expressed dimensionally), I must take license with
existing mathematical language. Obviously, this linear dimensional analysis does not follow standard form; however, in
its own way, it is consistent with the Fundamental Property of Equality, which states:
The quantities on both sides of an equation (an equality) must have the same dimensions.
In other words, the equals sign in an equation denotes equivalence: Equality means "is identical to." In mathematics, A
= B indicates that A and B are two names for the same thing, i.e., in the equation, A = B, A and B must be of equal
dimension.
Therefore, when applying the Fundamental Property of Equality to Descartes’ Exponential Calculation of Line, as the
expressions a2, b3, c4, ... dn are defined as representing "only simple lines" and are of linear dimension, to maintain
dimensional equivalence, both sides of an equation must be of linear dimension: e.g.,
a2 = x, with a2 defined as linear, x must be linear,
b3 = y, with b3 defined as linear, y must be linear;
thus,
d n = z, with dn defined as linear, z must be linear.
Accordingly, the linear exponential calculation of measured quantities would result in equations such as
(1 linear meter)1/2 = 1 linear meter,
(1 linear meter)2 = 1 linear meter, and
(1 linear meter)3 + (1 linear meter)n = 2 linear meters.
Once again it must be stressed, this alternative form of exponential calculation - one of line as opposed to square and
cubic dimension - may, at first, appear dimensionally incorrect. However, to reiterate, the relationship in an equation
(an equality) can be correct only if the dimensions on both sides of the equation are the same. Thus, regardless of the
value of the exponent, with the quantities a1/2, b1/3, c1/4, ... d1/n ; a2 , b3 , c4, ... dn , defined as linear, the results will be
linear. That is, to maintain the Fundamental Property of Equality, Des Cartes’ Exponential Calculation of Line maintains
the dimensional form of length equals length,
[Length] = [Length].
Linear Exponential Conversion
Now for an even more interesting twist .... consider how the result of Des Cartes’ Exponential Calculation of Line would
be converted. To address this, look again to the Fundamental Property of Equality. Specifically, maintaining
dimensional equality on both sides of an equation also applies to conversion:
The conversion factor must correspond to the dimension being calculated.
So, to make a comparison:
In the Calculation of Square and Cubic Dimension
(as the values of the exponent in a2 , b3 indicate, respectively, calculation of square and cubic units),
dimensional equality is maintained by keeping the values of the exponent equal on both sides of the
equation, e.g.,
(3 ft)2 = 9 ft2,
(3 ft)3 = 27 ft3, and
9 ft2 + 9 ft2 = 18 ft2,
and by having the conversion factor correspond to the dimension being calculated: e.g.,
square feet are converted with square inches, with a2 equal to 1 square foot,
a2 = 1 ft2 = 1 ft2 x 144 in2 / ft2 = 144 in2, and
cubic feet are converted with cubic inches, with b3 equal to 1 cubic foot,
b3 = 1 ft3 = 1 ft3 x 1728 in3 / ft3 = 1728 in3 .
In contrast to this, in Descartes’ Exponential Calculation of Line
(as the exponent, in the terms a1/2, b1/3, c1/4, ... d1/n ; a2 , b3 , c4, ... dn is employed to indicate the
multiplication of line by itself to calculate line), regardless of the value of the exponent, to maintain the
Fundamental Property of Equality, not only must both sides of an equation be of linear dimension, e.g.,
(1 linear ft)n = 1 linear ft,
(1 linear ft)n/(1 linear ft)n = 1 linear ft
(1 linear ft)1/n = 1 linear ft
but also, conversion must be of linear dimension.
This means, astonishingly, that these same equations (exponentially calculating line) would have a
linear conversion factor, i.e., the resultant 1 linear foot would be multiplied by 12 linear inches per 1
linear foot:
(1 linear ft)n = 1 linear ft = 1 linear ft x 12 linear in/linear ft = 12 linear in,
(1 linear ft)n/(1 linear ft)n = 1 linear ft = 1 linear ft x 12 linear in/linear ft = 12 linear in.
(1 linear ft)1/n = 1 linear ft = 1 linear ft x 12 linear in/linear ft = 12 linear in.
Admittedly, this form of conversion is unorthodox - who has ever heard of converting the result of an exponential
calculation of length with a linear conversion factor? However, as the reader will note, these equations (and this article)
are introducing a radical mathematical concept:
In opposition to the generally accepted view that all exponential calculation of units of measure
should be treated as calculations of square and cubic dimension, I submit that (to maintain the
Fundamental Property of Equality) Des Cartes’ Exponential Calculation of Line to produce line
is a separate category of dimensional calculation, a category which requires a separate form of
equation written specifically for Linear Exponential Calculation and Conversion.
Conclusion
The break from using the exponent for something other than area and volume calculations begins with Des Cartes’
linear interpretation of the exponent. But is it being interpreted as Des Cartes intended? The point being, exponential
calculations that are square or cubic are certainly not linear, and vice versa. Taking a step back to Relativity, this of
course begs the question: In relativistic equations such as E= mc2 or (E/m)1/2 = c, how should the exponential
calculation of time, mass, and energy (units expressed in linear scale) be interpreted? Dimensionally speaking, what is
velocity squared or the square root of energy or mass? For example, is the time squared in c2 to be understood as
1.
2.
3.
an "area" of time;
as second per second, a form of acceleration; or,
is it really only dimensionally valid as Des Cartes’ calculation of line (linear scale units)?
To put this into context, with velocity being distance over time, what exactly is being squared in velocity squared?
Specifically, in squaring miles per hour, I can understand a square mile (an area), but what is a square hour, or for that
matter any square unit of time; a square second, a square minute, or a square day? (How many square minutes are
there in a square day?) In turn, this conceptual "area of time calculation" is surely not dimensionally equivalent to
acceleration, a valid second per second calculation (an increase of velocity over time). And, an attempt to exponentially
calculate and convert time as line (as Des Cartes’ linear approach), though perhaps dimensionally valid, would not
provide a valid measurement, as the result would vary as a function of the unit chosen. For example, one week is
equivalent to seven days; however, when these equivalent time intervals are squared (or the root is extracted) the
results are not equal: one week squared equals one week, but seven days squared equals forty-nine days, or seven
weeks. (Also Des Cartes’ linear approach would be the same as a purely numerical approach, as all that is being
squared is the number assigned, not the unit of time itself. The number of hours/days/weeks can be
calculated/squared, but time itself cannot.)
So any way you look at it, is there any valid approach to exponentially calculating time?
Look at this quadratically: if solving for t in the quadratic equation of
0=t2-2t-8
is this to be taken as a geometric quadratic, where an area of time, t 2 , minus an area of time, t , minus an area of
time, 8, is equal to an area of time 0? How ridiculous is this?
Or, is this to be interpreted as an acceleration equation, where for example t 2 , seconds per second, minus t seconds,
minus the number 8 is equivalent to zero.
Or going even further afield, is the quadratic equation,
0=t2-2t-8
supposed to indicate an "area" of time, t 2 , minus a "line" of time, t, minus a number, 8?
Are any of these approaches realistic or logical? In any of these instances, how would each of the separate terms be
expressed and converted? And, what would be the result of subtracting these terms of incompatible dimension?
So to summarize:
Being mathematically founded upon the quadratic and the squaring of time (referring to the equations of Lorentz and
Einstein), is Relativity the result of a simple misinterpretation of the exponent, compounded by trying to calculate
incompatible terms? In Relativity, it appears that all truth of dimension has been lost, all laws of dimensional equality
that govern other equations have been set aside, and linear scale units are being attempted to be calculated and
converted as square units. Again, examine the equations of E = mc2 and (E/m)1/2 = c . How can it be possible to either
square time or take the square root of energy or mass? What is really being expressed, dimensionally, with the
exponent in these relativistic equations?