Scientific Notation, Fractal Numbers and the

Symbolic Formulae Based on Roots

Two Possible Answers for Some Planck Constants


Charles William Johnson


The scientific notation is a form of fractal numbers with the specification of the decimal placement attached after the significant numbers [1.3661 x 1012]. Fractal numbers represent the significant numerical value with reference to the decimal place directly listed in the numerical value [ 0.13661; 1.3661; 13.661; 136.61; etc.]. In the Earth/matriX series of essays, the author employs the fractal numerical values of terms without reference to the decimal place, where all fractal numbers are expressed as 1.0, the decimal place always listed as such. The reasoning behind treating the fractal numbers in this manner is to call attention to the interrelationships of the significant terms, especially for procedures of multiplication and division. At times, the fractal numerical values may be employed for procedures of addition and subtraction, but each particular case requires its own level of discussion and clarity. The author exams the practice of scientific notation and fractal numbers in the symbolic formulae of select fundamental physical constants that contain roots and powers. Illustrations reveal how the symbolic formulae, when square roots and above are employed in the symbolic formulae produce variations for the constant numerical values. Generally, the fundamental physical constants whose symbolic formulae contain roots are said to produce a single correct answer. The different computational levels are laid bare in order to show how the same symbolic formulae that contains root expressions produces variations of answers.

In previous essays, I have discussed extensively why the symbolic formulae of the Planck constants, for example, that are based on square roots produce different possible answers. Generally, it is held that each Planck constant produces a single correct constant numerical value. However, this is incorrect given the nature of square roots.

Except for the number 1, whose square root is 1, there is no number that has only one square root. Even the unit number 1, when expressed with decimal places either to the left or right of the unit 1 value, has different roots, depending precisely upon the decimal placement.

square root of 0.01 = 0.1

square root of 0.1 = 0.316227766

square root of 1 = 1

square root of 10 = 3.16227766

square root of 100 = 10

square root of 1000 = 31.6227766

square root of 10000 = 100

square root of 100000 = 316.227766

...infinitely so.

With that well-established pattern, one realizes that the square root of fractal one [1] varies according to the decimal placement. The scientific notation of the terms based on 0.01, 0.1, 1.0, 10, 100, 1000...n would have their decimal places listed by the generally accepted notation: 1.0 x 10n . [Further reading: The Square Root of the Speed of Light in Vacuum and Planck Units,]

In this manner, one realizes that the general practice of scientific notation is in fact the management of fractal numbers. For some reason, as I have been employing the fractal expressions in the Earth/matriX essays, without the specification of the decimal placement, this has upset some physicists. They have challenged me on this point as though the fractal numbers that I am using were contradictory and unlike the scientific notation. Were I to place all of the decimal places on all of the discussions of fractal numerical values in my analyses, the amount of space required to post those expanded texts would explode the confines of my web-site, not to speak of cluttering up the economics in abstracted reasoning about the significant terms that I have been attempting to achieve in my writing.

Various physicists have asked me to place the scientific notation indicating the decimal places in the analyses. I find this not only unnecessary but counter to the reasoning. In fact, because over the decades scientists have employed the scientific notation of fractal numerical values together with the indication of decimal placement, I consider that a possible reason why they have not detected faulty reasoning in the computations of the symbolic formulae of the fundamental physical constants that contain root expressions.

Fractal numbers written as in the Earth/matriX essays with implied regard for the decimal placement of terms derive the same fractal numerical values for computational procedures of multiplication and division. No matter where the decimal placements of the fractal terms 1.0 and 3.16227766 lie, when these terms are either multiplied or divided in whichever order of first and second terms, the answers shall always be the same [fractal 3.16227766] irrespective of the decimal placement.

Now, when these two values are added and subtracted then the decimal placement must be maintained; that is basic math, with no need to offer examples.

One could write the fractal expressions in various manners, without regard for the decimal placement: 0.316227766; 3.16227766, etc. For the Earth/matriX essays, I chose to use the decimal placement after the first significant digit, as in 1.0, for all fractal numbers expressed in my essays simply because that placement reflects similarity to the scientific notation. As I mentioned, in fractal terms, it matters not in reasoning the numerical values whether one multiplies or divides 1.0 times 3.1622776 or, 0.1 times 0.316227766 in theoretically abstracted terms. Once one has done that theoretical exercise, as I do in each essay, then one can place the decimal where it is required for the case under study.

Ultimately, scientists do not, in theoretically abstracted mental exercises look at the shorthand form of scientific notation of say the mass of a proton [1.672621637] and the mass of a neutron [1.674927211], each with a decimal notation of x 10-27 and actually visualize the cited number of zeros before their cited significant numerical values. Scientists merely compute the significant numbers and then take care of the number of zeros after. That was the reason scientific notation was invented to begin with as a shorthand expression for very long numbers.

To ask scientists to consider the fractal numbers, without reference for the analytical moment to the decimal places does not appear to me as too much to ask. I have yet to understand why certain physicists have requested of me that I follow the general practice of stipulating the number of decimal places, when the purpose of the analysis is to consider the interrelationships of the fractal numerical expressions.

That is all well and good. Scientific notation is shorthand form of expressing fractal numerical values in order to avoid early on lengthy expressions in the computations. The fractal numbers used in the Earth/matriX essays follow upon a similar purpose of economy in mentally abstracted reasoning. One must know how the fractal numerical values function; proper decimal placement comes after, just as it comes after in the scientific notation system, after each significant term.

Now, why is all of this important? It is not some kind of struggle between two systems of numerical notation. I could easily have employed the scientific notation in all of my essays. But, the reason that did not occur was because my studies began with considering the fractal expressions in the ancient reckoning systems of time. The ancient Maya Long Count has some very long numbers in it, such as the Alautun [23,040,000,000 days]; in scientific notation 2.304 x 1010; in fractal expression, simply 2.304]. In my studies of the ancient reckoning system and in my studies of physics and chemistry today, I have found it effective in abstracted reasoning to use the concept of fractal numbers, knowing that in the final analysis one would place the proper number of zeros to a specific term in a given context.

With regard to the fundamental physical constants, the importance of distinguishing the use of symbolic formulae and the scientific notation of terms cannot be emphasized enough. And, this is the point that I have been reviewing in my recent writing about the CODATA fundamental physical constants and the deficiency of using the symbolic formulae to express how certain fundamental physical constants are derived.

One thing is the reality of spacetime/motion, and its forms of matter-energy, and a completely different aspect is the symbolic representation of that reality. It has been extremely difficult for me to detect the apparent corrections required in order to understand the nature of the fundamental physical constants. One looks upon the work of others with great respect, especially when an entire community of academics firmly believes that the data that they work with is exact and precise, as in the case of the CODATA fundamental physical constants.

I have been consulting and referencing the CODATA fundamental physical constants for years. And, as I embarked upon a self-imposed task of trying to understand how Max Planck derived the constants that now carry his name, I was extremely surprised to detect errors and omission in the CODATA listing of numerical values. I have been point out the deficiency of using square roots in the Planck constants of mass, temperature, length and time. I have pointed out in other essays the same deficiency that appears in other fundamental physical constants. But, at present, I am concerned with the Planck constants.

I consider that since scientists have employed the scientific notation, they have not perceived the initial fallacy embedded in the symbolic formulae as presented by Max Planck and as repeated for the past 110 years or so. In my previous essays, I have drawn attention to the fact that when equations contain square roots, the number of possible answers that may be derived thereof are two. When cube roots are employed, the number of possible corresponding answers is three; to the fourth root there exist four possible answers; to the fifth root, there exist five possible answers; and so on, infinitely so.

The CODATA present the symbolic formula for each of the cited Planck constants as though that were the only numerical value that could possibly be derived from each particular constant. Given the fact that the Planck constants of mass, temperature, length and time are based on square roots, there exist two possible answers for each corresponding symbolic equation; not one. The CODATA lists a single answer for each Planck constant. According to the symbolic formula of each Planck constant that contains a square root expression in it, there exist two possible answers as shall be demonstrated below.

Without wanting to go into extreme detail for each symbolic formula, I present the relationship of the terms whereby first the speed of light in vacuum is multiplied by itself a number of times [1, 3, 5], and then is subject to deriving the square roots thereof. The important point is to lay bare the deficiency in the computational procedure, and not explain the actual numerical values of the constants themselves. I seek only to demonstrate the contradiction and ambiguity in employing powers and roots in the symbolic formulae presented on the CODATA list of recommended values for the fundamental physical constants. There is no attempt here to present the actual values of the constants themselves, rather only the internal mechanism leading up to those values.

These internal computational procedures in fact contradict the concept of a constant based on a symbolic formula using root expressions. This is all that is required to be demonstrated for now in the analysis. I have presented alternate results for the equations and they may be considered as well ---

[]. I do not consider it necessary to repeat those findings here. The analytical purpose of this essay is to demonstrate the internal computational ambiguity and deficiency of the symbolic formulae, and not the resolution thereof for each constant.

In order to illustrate the deficiency, I shall once again point out the fractal difference in deriving square roots from the velocity of the speed of light in a vacuum. The significance of the decimal placement becomes apparent when treating the speed of light, whether it is expressed in meters [centimeters] or kilometers. The CODATA scientific notation for the speed of light in vacuum in meters is given as 299792458 m s-1. The numerical value of the speed of light in vacuum expressed in kilometers is 299792.458.

The square roots for these two numerical values for the speed of light in vacuum are as follows:

17314.51582 for meters

547.5330657 for kilometers

One would thus expect that in the symbolic formulae of the cited Planck constants, the numerical value of 17314.51582 for meters is employed. Thus, without consideration for the moment of the fact that c, the speed of light in vacuum is given as c [for Planck mass], c3 [for Planck length], and c5 [for Planck temperature and time] in the symbolic formulae of the cited Planck constants, consider the fact that these terms are placed within a square root expression. The current notation of the symbolic formulae in the CODATA is expressed as of the 0.5 power, which essentially means the square root expression, as Max Planck employed the sign of square roots in his original formulae.

All numerical expressions have two square root values depending upon the decimal placement, with the exception of unit 1; any numerical term above or below 1 has two square root value derivations.

The square root of the speed of light in vacuum has as its square root values those of fractal 1.731451582 and 5.475330657 [again, taking into consideration everything that I have mentioned regarding the decimal placement].

For this illustration, I have chosen the Planck constant length as it is expressed in meters in the CODATA. The original symbolic formula for Planck length is as follows:

lp = √ħG/c3

Planck Length is given today as 1.616252(81) x 10- 35 meters

In the CODATA 2006, the symbolic formula is written in the following manner:

lp = ħ/mpc = (ħG/c3 )1/2

The two square roots of fractal c3 are: 5.190761256 and 1.641462836.

In the Planck formulae for temperature and time c5 is employed.

The two square roots of fractal c5 are: 1.556151076 and 4.920981783.

I have explained these computational possibilities in various essays regarding the variations in results for symbolic formulae that contain root expressions. However, today I receive challenges from physicists who continue to tell me that this particular problem of roots, as I have identified it in the Planck constants, is a trivial question in as much as one is able to simply convert meters and kilometers after having derived the numerical value for the Planck length. That is not the point that I have been making throughout the Planck essays posted on the Earth/matriX website.

Let me make another attempt at illustrating the ambiguity and deficiency that I perceive in this regard. The point that I have been making is that it is impossible to employ root expressions in the symbolic formulae and expect to produce a single constant numerical value as the CODATA propose in those physical constants that contain square root expressions.

Without repeating the possible answers for each symbolic formulae of each Planck constant as shown in other essays, the following is what I perceive to be amiss in the computational procedure suggested by the CODATA symbolic formulae. Consider the following commentary within the boundaries of the symbolic equation for Planck length.

The conversion from meters to kilometers is straight forward within the terms of the metric system.

a) In order to derive meters from the symbolic equation the following must occur: Employing the symbolic equation for Planck length in meters one would expect the c3 expression to have a fractal root value of 5.190761256, given that this root value corresponds to the expression of the speed of light in vacuum cubed in meters.

c3 in meters is 2.69440024225 the square root of which is 5.129076125612

And, any computation that is derived thereof, in order to convert from meters to kilometers, one would merely divide by the numerical 1000, since there are one thousand meters in a kilometer. This is pretty basic stuff.

5.129076125612 divided by 1000 equals 5190761256 kilometers

b) But, because of the symbolic equation given for Planck length in that it contains a square root expression this is not what is happens. In order to derive kilometers from the symbolic equation one would have to employ c3 in kilometers in the symbolic equation itself from the start. Simply making the conversion after obtaining the previous result would make sense, if there were only a single possible answer to be derived from the equation.

In order to derive the symbolic equation in kilometers, then one should employ the expression c-cubed in kilometers from the start, without having to make the conversion after obtaining the result in meters.

c3 in kilometers is 299792.458 cubed is 2.69440024216 the square root of which is 164146283.6

Further, any computation that is derived thereof in kilometers from the original equation could then be converted from kilometers to meters. In order to obtain meters then, one would simply multiply the kilometers figure by the number 1000, since there are one thousand meters in a kilometer. This is pretty basic stuff.

However, given the fact that the speed of light in vacuum produce one numerical value for its expression in meters [fractal 1.731451582] and another numerical value for its expression in kilometers [fractal 5.475330657], the results of the equation are two distinct numerical values in the final analysis. The symbolic formula for Planck length produces necessarily two distinct constant values; not just the one cited in the CODATA list. [Ultimately, the two possible fractal numerical values for Planck length are then: 1.616252 fractal for meters and, 5.11104 fractal for kilometers.]

Regarding the internal mechanism of computation, each derivative term as of the square roots would have to be either multiplied or divided by the square root of ten, depending on the case, in order to obtain the equivalency for meters/kilometers according to the previous computations.

164146283.6 times the square root of ten = 51907612.37

51907612.37 divided by the square root of ten = 16414628.3

and so on...

In other words, the procedure for converting meters into kilometers as in example a) violates the meaning of the symbolic expressions. In order to achieve kilometers as a constant as of the symbolic expression one would have to employ the square root of the speed of light in vacuum in kilometers from the outset in the symbolic formula.

Obviously one may convert the result obtained in meters into kilometers as shown. But, the fact that the symbolic formula expression is said to represent the nature of the constant for length, then one should carry out the computation in kilometers from the beginning of the equation's terms.

The problem is not in the results obtained, as meters converted to kilometers and/or kilometers converted to meters. The problem arises that in both procedures the final results are contradictory. In other words, if one computes the symbolic formula in meters and then makes the conversion to kilometers, those two results do not match if one computes the symbolic formula in kilometers and then converts to meters.

a) c3 in meters is 2.69440024225 the square root of which is 5.129076125612 meters

5.129076125612 meters divided by 1000 equals 5190761256 kilometers


b) c3 in kilometers is 299792.458 cubed is 2.69440024216 the square

root of which is 164146283.6

164146283.6 kilometers times 1000 equals 1.641462836 x 1011 meters

The conversion of the two possible answers to the same equation is achieved through the square root of ten [3.16227766] due to its relationship to the nature of square roots.

164146283.6 times square root ten = 51907612.37

51907612.37 divided by square root of ten = 16414628.3

In this manner, it is proven that the possible numerical values for the Planck length constant are two, not one. And so obtains for each symbolic formula of the recommended fundamental constants that contain a square root expression. There is no single answer for those constants, but rather two possible answers, which as shown are distinct each time given the nature of square roots of all numbers except unit 1.

It should be obvious now, that if the symbolic equation listed for Planck length were correct, then the distinct numerical would show equivalency in both computational options. But, the fact that two different numerical values exist as of the square root of the speed of light in vacuum, then the results are distinct and in non-equivalency. And, further, it is now obvious that those results are translated one from/to the other by way of the square root of ten as shown above, confirming the non-correspondence of a single numerical value for Planck length.

Were the symbolic formula correct, then both results should show the same numerical value for the product in either meters or kilometers. And, as illustrated, if one works out the symbolic equation in meters and converts that result to kilometers, one obtains a different answer from the case when one works out the same equation with the same terms in kilometers and converts that result into meters. Both results should be the same were the symbolic formula correct. Hence, since the symbolic formula contains a square root expression, the possible results of the computation are two, and two different results are obtained as shown.

To summarize, if the CODATA symbolic equation is worked out in meters and converted to kilometers one result obtains. If the same CODATA symbolic equation is worked out in kilometers and converted to meters a different result obtains. The numerical value of kilometers in both equations is different; and, the numerical value of meters in both equations is different. The numerical value in meters in both equations should be the same; and the numerical value in kilometers in both expressions should be the same.

Remember, in this essay I have not worked out the final result of these two equations, but simply illustrated the different internal numerical values that derive within the equation for the speed of light first taken to the third power and then deriving its two square root values. Obviously, if one works out the symbolic equation completely, two different numerical values for the Planck length would obtain posing a contradiction of terms and results. The two possible fractal numerical values for Planck length are fractal 1.616252 for meters and fractal 5.11104 for kilometers. This obtains according to the CODATA formula, not according to spacetime/motion events of matter-energy.

The two possible expressions for Planck length would be normalized and made equivalent by multiplying them by the square root of ten as shown. 1.616252 times the square root of ten equals 5.11104. Again the two different numerical values are derived as of using meters and/or kilometers in the symbolic formula and not as of simply converting meters to kilometers or vice versa.

The question then becomes one of which is the correct numerical value for the Planck length and may it be derived in any system of units of measurement inasmuch as it supposedly represents a dimensionless number for all time and all space. That discussion requires yet another lengthy essay from a totally distinct perspective, one that does not involve the use of symbolic formulae based on square roots.

March 20, 2010 Copyrighted by Charles William Johnson. Reproduction prohibited. All rights reserved.

Earth/matriX Editions

P.O. Box 231126, New Orleans, Louisiana 70183-1126, USA

Earth/matriX, Science Today

Science in Ancient Artwork


- Natural Units and Atomic Units

- The Planck Constants Based on the Fundamental Physical Constants

- Relationships Regarding Select Fundamental Physical and Chemical Constants

- Proposal to Include Planck Implied Values in the CODATA Listing

- The Compton Wavelength Represents the Fractal Reciprocal of the Planck Implied Length

The Earth/matriX Tables
of Planck Constants Based on the Fundamental Physical Constants

Our Price $17.95
Paperback In Stock
Our Price $19.95
Paperback In Stock
Our Price $18.95
Paperback In Stock
The physics paradigm today is based mainly upon the concept of c-square, the squaring of the speed of light in a vacuum. Numerous fundamental physical and chemical constants provided in the physics literature [CODATA] reflect numerical values based upon powers of c, the speed of light in vacuo. The speed of light in a vacuum is determined to represent the upper limit of movement of mass|energy by physicists.
The upper speed limit for a light photon is 299792458 meters/second. The square of that number produces a numerical value that does not exist in any form of matter-energy. The c-square actually represents a number that corresponds to a near massless event: a light photon. The author goes beyond a critique of Albert Einstein’s famous formula based upon this unreal number. The rejection of Einstein’s formula is explored through basic math, the summation of powers in the equation’s terms.
A common procedure followed in deriving many of the CODATA recommendations is to divide certain fundamental physical constants by the value of the elementary charge, e, 1.602176487. With regard to the Planck constants and units of measurement, the case is argued that Max Planck may have simply reversed engineered this procedure in order to derive his natural units.


[Presentation] [Books]   [Reviews]
[All Essays]

Charles William Johnson, e-mail:

Science in Ancient Artwork Series ISSN-1526-3312
©1995 - 2012 Copyrighted by Charles William Johnson. All rights reserved. Reproduction prohibited.
The text and all images of this website are copyright protected.

webmaster: jorge luna.