For some time now I have been experimenting with Edison's observation that short sleep then wake up, the mind is better at figuring out things. So some hours ago I nodded off in sleep, after having read this article in Scientific American. Not
surprisingly in that sleep I figured out the proof solution.

But I am still groggy and so will wait for tomorrow to detail the proof.

Fortunately, this Benford Law comes smack in the middle of some of my present work in physics.

This will make a delightful short book and proof read.

AP

AP's 306th book of Science-- Proof of Benford's Law as discussed in Scientific American Dec2023, by Archimedes Plutonium

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December 2023, SCIENTIFIC AMERICAN page 82.

"This Unexpected Pattern of Numbers Is Everywhere"
"A curious mathematical phenomenon called Benford's law governs the numbers all around us." by Jack Murtagh

--- quoting SA chart graph ---
Distribution of Leading Digits in Many Real-World Data Sets

30.1% for 1


17.6% for 2


12.5% for 3


9.7% for 4


7.9% for 5


6.7% for 6


5.8% for 7


5.1% for 8


4.6% for 9


Alright the proof is rather simple and follows my recent work of Psi-squared in quantum mechanics backed up with the geometry of the Rectangle in Whirling Squares of Logarithmic Spiral. That diagram graph is a portion of the Log Spiral in the Fibonacci
Sequence.

While on this topic I should also prove that the trivial 1, and 144 are the only perfect squares and 1 and 8 in perfect cubes in that Fibonacci Sequence.

So, let us run a Sample Space of Probabilities.

1 --> 1
2 --> 4
3 --> 9
4 --> 16
5 --> 25
6 --> 36
7 --> 49
8 --> 64
9 --> 81
10 --> 100
11 --> 121
12 --> 144
13 --> 169
14 --> 196
15 --> 225
16 --> 256
17 --> 289
18 --> 324

And let me stop there for the moment to see who well that fits the data so f

Archimedes Plutonium<plutonium.archimedes@gmail.com>
1:12 AM (13 hours ago)
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Alright I am going to stop there and feel my table is more accurate than the graph given Jack Murtagh in his "Distribution of Leading Digits in Many Real-World Data Sets"

My graph is also logarithmic as a downward portion of the log spiral as seen in Rectangle of Whirling Squares of Fibonacci Sequence.

My results up to 109 and their perfect-squares is this.

There are 30 Ones from 1 to 109.
There are 14 Twos from 1 to 109.
There are 12 Threes from 1 to 109.
There are 12 Fours from 1 to 109.
There are 9 Fives from 1 to 109.
There are 9 Sixes from 1 to 109.
There are 8 Sevens from 1 to 109.
There are 8 Eights from 1 to 109.
There are 7 Nines from 1 to 109.

I suspect mine is the more accurate table and that Jack probably threw in skewering data to compose his. Mine comes directly from a pure mathematical mechanism-- square root, and who knows what data Jack threw into his data sets.

Now the main thrust of this book is not to dwell so much on the proof, but to gain insight into the Theory of Probability. We normally look upon Probability as a science of how many favorable outcomes divided by total possible outcomes. For example,
drawing the largest number from a box which has 10 numbers, is 1/10 or 10%. Drawing a number that starts with One digit in my table above is 30/109 = 27%, and drawing a number that starts with Two digit is 14/109 = 12%.

But the bigger question is how does squaring as in Psi-squared of physics is Probability itself? How is the square of a number become probability??? This sounds almost philosophical.

AP



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Archimedes Plutonium<plutonium.archimedes@gmail.com>
1:39 AM (13 hours ago)
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So, I do not yet have a proof of Benford's Law, until I find out what this new definition of Probability is.

The common definition of Probability is (number of favorable ways) divided by (total number of ways). And where certainty has a value of 1 or 100%, while probability is less than certain and so has a percentage less than 100% of occurring.

In Physics, probability entered that subject in a spectacular way in the 1930s with Schrodinger wave equation of quantum mechanics. Here a Psi function that is squared is a probability. Nothing like our commonsense probability of favorable ways/total
ways. Nothing like that. And it is viewed as saying the Probability of finding a magnetic monopole in a given region of a Atom.

So here in 1930s with Schrodinger the world is treated to a whole new theory and definition of Probability, involving simply the math mechanics of squaring.

Squaring a number is geometrically a square area and in Fibonacci Sequence is a logarithmic spiral formed from the Rectangle of Whirling Squares. This is a new modern form of Probability theory, the likes of which surpass our old notions of probability
theory of favorable ways/total ways.

So in this book, I prove Denford's Law, but in so doing, I first need to find what the modern day definition of Probability is. And that is a tough tough job.

AP
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Archimedes Plutonium<plutonium.archimedes@gmail.com>
2:48 AM (11 hours ago)
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Troubles with Old Math's definition of Probability of Event as that of (number of favorable ways) divided by (total number of ways).

Which is a fine definition for cards or dice or gambling, but not useful in science.

We seldom know what is the total number of ways. We seldom know what is "favorable". So the Old Math definition gets us started in Probability theory but rather leaves us with a crude and primitive idea of probability. It is a anthropomorphic concept of (
number of favorable ways) divided by (total number of ways). And it is useless when 1930s rolled by with Schrodinger Wave Equation. You cannot fit (number of favorable ways) divided by (total number of ways) into the Magnetic Monopole wandering around
inside a Atom.

Not even the "chance of finding the monopole" near a proton makes sense with Old Math's probability.

So Chance and Probability need a make-over.

In New Physics, there is Superdeterminism and a total lack of free-will and how can one fit probability with superdeterminism?

There seems to be just one way. That Probability is a fancy term for Frequency of occurrence. The probability that the Sun rises tomorrow has a high probability, a high frequency.

And frequency involves time as denoted by 1/time = frequency. Frequency is in speed and thus in calculus. This makes sense that 1/time is a derivative of change in y divided by change in x is derivative.

So in this viewpoint we simply make all of Probability theory be that of Frequency, a part of the Calculus derivative.

Now we can start to home in on why squaring is probability.

AP
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Archimedes Plutonium<plutonium.archimedes@gmail.com>
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I will need to rename this new book as "The Modern New Definition of Probability, not the Old Math definition".

It just so happens that the Benford's Law is a good example of the Modern New definition of Probability.

One can say the Modern New Definition of Probability starts with Schrodinger Wave Equation circa 1930. When Schrodinger came out with his Wave Equation there were a few years of stumbling around with the equation with the Psi function. At first,
Schrodinger accepted and used the Psi function and then finally Max Born corrected him by telling him it was Psi-squared that was a probability of finding the Dirac magnetic monopole in a region around the Atom.

And from 1930 forward, we have a brand new definition of Probability itself, involving a square.

Of course Old Math definition of probability was borne from the 1700 and 1800s from dice rolling and card playing. This definition of (number of favorable ways) divided by (total number of ways) is woefully inadequate for a definition of Probability in
science especially physics. For example-- how silly is it to ask what is "favorable" in physics or biology, and as if we can measure favorable and/or total number of ways in biology or chemistry or physics.

What we can measure in all sciences is Frequency of occurrence. The frequency of weather raining. And this is where probability theory should have started-- not gaming of cards and dice, but rather in frequency of weather-- how frequent is rain? How
frequent are sunny days? The probability that tomorrow will rain is a frequency measure.

And here is where I am super fortunate to be doing Benford's Law while simultaneously working on that special math formula of Permeability x Permittivity is the speed of light squared, a math form of A x A^2 = A^3 = 10^18 or 10^-18.

You see the Permeability is Magnetic field in units of 10^-6 while Permittivity is Electric field in units of 10^-12.

You see Permeability squared is Permittivity.

Probability is a squaring phenomenon.

Probability is a frequency. The Old Math probability is too too narrow minded.

Frequency is part of speed just as distance / time where the 1/time is the frequency.

With frequency we write speed = velocity = distance X frequency.

So probability is part of the Calculus derivative of time being frequency.

Somewhat sad that Probability theory was not borne out of frequency, such as the frequency of weather rain, or the frequency of where we walk, or the frequency of where we drive, what roads and driveways we use and not use. The frequency of washing hands,
the frequency of sleeping, the frequency of eating potatoes, rather than Old Math's number of favorable ways/total number of ways.

With frequency, we can give measure to probability questions, which we could never measure "favorable" or "total ways".

This is why Schrodinger sputtered come 1930, and where Born figured out he was missing a square for the Psi function, because Probability theory was still in its primitive crude form. In fact, not until today, does AP revise all of Probability theory
with the new definition of Frequency.

AP, King of Science

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I wrote this last month, as I was deep in research over A x A^2 = Permeability x Permittivity = A^3. And we can see that A^2 is the squaring of A. So that implies magnetic field squared relates to electric field. And we can think of electric field as a
Frequency, a new modern revised Probability theory.

Archimedes Plutonium
Oct 23, 2023, 10:11:55 PM
to Plutonium Atom Universe,sci.physics,sci.math

I wrote:

So now, let me get going on this project and looking to Feynman Lectures on Physics.

In his Volume 3 of Lectures on Physics, 1965, page 21-6 he talks about "The meaning of the wave function"

--- quoting ---
When Schrodinger first discovered his equation he discovered the conservation law of Eq.(21.9) as a consequence of his equation. But he imagined incorrectly that P was the electric charge density of the electron and that J was the electric current

density, so he thought that the electrons interacted with the electromagnetic field through these charges and currents. When he solved his equations for the hydrogen atom and calculated Psi, he was not calculating the probability of anything --- there
were no amplitudes at that time -- the interpretation was completely different. The atomic nucleus was stationary but there were currents moving around; would radiate light. He soon found on doing a number of problems that it did not work out quite right.
It was at this point that Born made an essential contribution to our ideas regarding qauntum mechanics. It was Born who correctly (as far as we know) interpreted the Psi of the Schrodinger equation in terms of a probability amplitude-- that very
difficult idea that the square of the amplitude is not the charge density but is only the probability per unit volume of finding an electron there, and that when you do find the electron some place the entire charge is there. That whole idea is due to
Born.

--- end quoting ---

Now Feynman in his Volume 1, to his credit, does discuss Probability theory, to sort of lead a student to quantum mechanics probability. But in my opinion, his probability is far too abstract.

I want probability teaching so that High School students can breeze through it.

So I intend to write this on Psi^2 so that High School students follow it easily.

Going back to page 21-4

(21.9) delP/del(t) = Psi"(delPsi/del(t)) + Psi (delPsi"/del(t))

And the reason I focus upon that equation is because it is the Pythagorean theorem of A^2 + B^2 = C^2.

Quantum Mechanics is duality and the Wave is a duality of electricity to magnetism. The Circle formed from right-triangles in sine and cosine. Where one side is electricity, other is magnetism, or, current and magnetic monopole.

When we form the trigonometry half circle of sine or cosine, they are formed by the motion of right triangles where the Pythagorean right triangles A^2 + B^2 = C^2.

AP

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The Old Math definition of Probability no longer is tenable.

--- quoting Wikipedia on definition of probability ---
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates
impossibility of the event and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ('heads'
and 'tails') are both equally probable; the probability of 'heads' equals the probability of 'tails'; and since no other outcomes are possible, the probability of either 'heads' or 'tails' is 1/2 (which could also be written as 0.5 or 50%).

These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in areas of study such as statistics, mathematics, science, finance, gambling, artificial intelligen

Alright, so in Mathematics Calculus we define the derivative as dy/dx, the change in y divided by the change in x. And we often think of the y as distance and the x axis as time so as to see the derivative be distance divided by time as a speed. And a
second derivative of that speed is acceleration.

What I propose is that we change the view of dx to be a frequency. So we can multiply. We have dy x (1/dx) and call the 1/dx the frequency. And further still, we call this frequency as Probability.

The concept of Probability interchangeable with frequency as that portion of the derivative calculus of 1/dx.

Of course for mathematics, we still keep the definition of derivative as a division of dy/dx, for that is far more general than dy x (1/dx).

We logically retain the most general definition and see frequency as a secondary byproduct, just as speed is a secondary byproduct of dy/dx.

But the great advance in all of this is that we pull Probability theory out of the gambling and gaming arena it has been stuck in ever since its birth in 1700 and 1800s. We stop the definition of favorable outcomes divided by total outcomes as unworkable
for physics.

Frequency is everywhere in physics.

Favorable outcomes divided by total outcomes is almost nowhere in physics.

Physics needs a Probability definition that explains Schrodinger's Psi is really Psi-squared. And that comes from the fact that Permeability-squared is Permittivity. What is different about magnetic field compared to electric field is a factor of
squaring.

AP

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So AP is revising all of Probability theory of Old Math, starting with the definition of what Probability means.

In Old Math, their probability meant a division, a division of favorable outcome divided by total outcomes.

In New Math, probability is a multiplication, not a division.

In New Math probability is the same as Frequency.

In New Math Calculus, the derivative is defined as dy/dx, the change in y divided by the change in x.

So all we do to have Probability theory is focus on the dy/dx and alter that to be a multiplication. We do that by dy*(1/dx). And thus we have 1/dx as a frequency. And we can call it the probability 1/dx.

This is needed, so very much needed because in the history of Physics comes 1930 with the Schrodinger Wave Equation talking about Psi-squared and being a probability of finding the Dirac magnetic monopole nearby a hydrogen atom. Totally inadequate
situation of probability defined as favorable outcome divided by total outcomes. And obviously, probability is a frequency.

The entire speed of light in the EM Spectrum is wavelength x frequency = speed of light. How silly and stupid it would be for wavelength x (favorable outcomes/total outcomes) as speed of light.

AP

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I thought it would be fun to check whether the list of physics constants from Wikipedia follows the Benford's law. It is a list of 57 constants.

c speed of light in vacuum 299792458 m⋅s−1 0 [1]
h
h Planck constant 6.62607015×10−34 J⋅Hz−1 0 [2]
ℏ
=
h
/
2
π
\hbar = h/2\pi reduced Planck constant 1.054571817...×10−34 J⋅s 0 [3]
μ
0
\mu_0 vacuum magnetic permeability 1.25663706212(19)×10−6 N⋅A−2 1.5×10−10 [4]
Z
0
=
μ
0
c
{\displaystyle Z_{0}=\mu _{0}c} characteristic impedance of vacuum 376.730313668(57) Ω 1.5×10−10 [5]
ε
0
=
1
/
μ
0
c
2
{\displaystyle \varepsilon _{0}=1/\mu _{0}c^{2}} vacuum electric permittivity 8.8541878128(13)×10−12 F⋅m−1 1.5×10−10 [6]
k
,
k
B
{\displaystyle k,k_{\text{B}}} Boltzmann constant 1.380649×10−23 J⋅K−1 0 [7]
G
G Newtonian constant of gravitation 6.67430(15)×10−11 m3⋅kg−1⋅s−2 2.2×10−5 [8]
k
e
=
1
/
4
π
ε
0
{\displaystyle k_{\text{e}}=1/4\pi \varepsilon _{0}} Coulomb constant 8.9875517923(14)×109 N⋅m2⋅C−2 1.5×10−10 [9]
Λ
\Lambda cosmological constant 1.089(29)×10−52 m−2 0.027 [10]
σ
=
π
2
k
B
4
/
60
ℏ
3
c
2
{\displaystyle \sigma =\pi ^{2}k_{\text{B}}^{4}/60\hbar ^{3}c^{2}} Stefan–Boltzmann constant 5.670374419...×10−8 W⋅m−2⋅K−4 0 [11]
c
1
=
2
π
h
c
2
{\displaystyle c_{1}=2\pi hc^{2}} first radiation constant 3.741771852...×10−16 W⋅m2 0 [12]
c
1L
=
2
h
c
2
/
s
r
{\displaystyle c_{\text{1L}}=2hc^{2}/\mathrm {sr} } first radiation constant for spectral radiance 1.191042972...×10−16 W⋅m2⋅sr−1 0 [13]
c
2
=
h
c
/
k
B
{\displaystyle c_{2}=hc/k_{\text{B}}} second radiation constant 1.438776877...×10−2 m⋅K 0 [14]
b
b [c] Wien wavelength displacement law constant 2.897771955...×10−3 m⋅K 0 [15]
b
′
b' [d] Wien frequency displacement law constant 5.878925757...×1010 Hz⋅K−1 0 [16]
b
entropy
{\displaystyle b_{\text{entropy}}} Wien entropy displacement law constant 3.002916077...×10−3 m⋅K 0 [17]
e
e elementary charge 1.602176634×10−19 C 0 [18]
G
0
=
2
e
2
/
h
{\displaystyle G_{0}=2e^{2}/h} conductance quantum 7.748091729...×10−5 S 0 [19]
G
0
−
1
=
h
/
2
e
2
{\displaystyle G_{0}^{-1}=h/2e^{2}} inverse conductance quantum 12906.40372... Ω 0 [20]
R
K
=
h
/
e
2
{\displaystyle R_{\text{K}}=h/e^{2}} von Klitzing constant 25812.80745... Ω 0 [21]
K
J
=
2
e
/
h
{\displaystyle K_{\text{J}}=2e/h} Josephson constant 483597.8484...×109 Hz⋅V−1 0 [22]
Φ
0
=
h
/
2
e
{\displaystyle \Phi _{0}=h/2e} magnetic flux quantum 2.067833848...×10−15 Wb 0 [23]
α
=
e
2
/
4
π
ε
0
ℏ
c
{\displaystyle \alpha =e^{2}/4\pi \varepsilon _{0}\hbar c} fine-structure constant 7.2973525693(11)×10−3 1.5×10−10 [24]
α
−
1
{\displaystyle \alpha ^{-1}} inverse fine-structure constant 137.035999084(21) 1.5×10−10 [25]
m
e
{\displaystyle m_{\text{e}}} electron mass 9.1093837015(28)×10−31 kg 3.0×10−10 [26]
m
p
{\displaystyle m_{\text{p}}} proton mass 1.67262192369(51)×10−27 kg 3.1×10−10 [27]
m
n
{\displaystyle m_{\text{n}}} neutron mass 1.67492749804(95)×10−27 kg 5.7×10−10 [28]
m
μ
{\displaystyle m_{\mu }} muon mass 1.883531627(42)×10−28 kg 2.2×10−8 [29]
m
τ
{\displaystyle m_{\tau }} tau mass 3.16754(21)×10−27 kg 6.8×10−5 [30]
m
t
{\displaystyle m_{\text{t}}} top quark mass 3.0784(53)×10−25 kg 1.7×10−3 [31]
m
p
/
m
e
{\displaystyle m_{\text{p}}/m_{\text{e}}} proton-to-electron mass ratio 1836.15267343(11) 6.0×10−11 [32]
m
W
/
m
Z
{\displaystyle m_{\text{W}}/m_{\text{Z}}} W-to-Z mass ratio 0.88153(17) 1.9×10−4 [33]
sin
2
⁡
θ
W
=
1
−
(
m
W
/
m
Z
)
2
{\displaystyle \sin ^{2}\theta _{\text{W}}=1-(m_{\text{W}}/m_{\text{Z}})^{2}} weak mixing angle 0.22290(30) 1.3×10−3 [34]
g
e
{\displaystyle g_{\text{e}}} electron g-factor −2.00231930436256(35) 1.7×10−13 [35]
g
μ
g_{{\mu }} muon g-factor −2.0023318418(13) 6.3×10−10 [36]
g
p
{\displaystyle g_{\text{p}}} proton g-factor 5.5856946893(16) 2.9×10−10 [37]
h
/
2
m
e
{\displaystyle h/2m_{\text{e}}} quantum of circulation 3.6369475516(11)×10−4 m2⋅s−1 3.0×10−10 [38]
μ
B
=
e
ℏ
/
2
m
e
{\displaystyle \mu _{\text{B}}=e\hbar /2m_{\text{e}}} Bohr magneton 9.2740100783(28)×10−24 J⋅T−1 3.0×10−10 [39]
μ
N
=
e
ℏ
/
2
m
p
{\displaystyle \mu _{\text{N}}=e\hbar /2m_{\text{p}}} nuclear magneton 5.0507837461(15)×10−27 J⋅T−1 3.1×10−10 [40]
r
e
=
e
2
k
e
/
m
e
c
2
{\displaystyle r_{\text{e}}=e^{2}k_{\text{e}}/m_{\text{e}}c^{2}} classical electron radius 2.8179403262(13)×10−15 m 4.5×10−10 [41]
σ
e
=
(
8
π
/
3
)
r
e
2
{\displaystyle \sigma _{\text{e}}=(8\pi /3)r_{\text{e}}^{2}} Thomson cross section 6.6524587321(60)×10−29 m2 9.1×10−10 [42]
a
0
=
ℏ
2
/
k
e
m
e
e
2
=
r
e
/
α
2
{\displaystyle a_{0}=\hbar ^{2}/k_{\text{e}}m_{\text{e}}e^{2}=r_{\text{e}}/\alpha ^{2}} Bohr radius 5.29177210903(80)×10−11 m 1.5×10−10 [43]
E
h
=
α
2
c
2
m
e
{\displaystyle E_{\text{h}}=\alpha ^{2}c^{2}m_{\text{e}}} Hartree energy 4.3597447222071(85)×10−18 J 1.9×10−12 [44]
R
y
=
h
c
R
∞
=
E
h
/
2
{\displaystyle \mathrm {Ry} =hcR_{\infty }=E_{\text{h}}/2} Rydberg unit of energy 2.1798723611035(42)×10−18 J 1.9×10−12 [45]
R
∞
=
α
2
m
e
c
/
2
h
{\displaystyle R_{\infty }=\alpha ^{2}m_{\text{e}}c/2h} Rydberg constant 10973731.568160(21) m−1 1.9×10−12 [46]
G
F
/
(
ℏ
c
)
3
{\displaystyle G_{\text{F}}/(\hbar c)^{3}} Fermi coupling constant 1.1663787(6)×10−5 GeV−2 5.1×10−7 [47]
N
A
,
L
{\displaystyle N_{\text{A}},L} Avogadro constant 6.02214076×1023 mol−1 0 [48]
R
=
N
A
k
B
{\displaystyle R=N_{\text{A}}k_{\text{B}}} molar gas constant 8.31446261815324 J⋅mol−1⋅K−1 0 [49]
F
=
N
A
e
{\displaystyle F=N_{\text{A}}e} Faraday constant 96485.3321233100184 C⋅mol−1 0 [50]
N
A
h
{\displaystyle N_{\text{A}}h} molar Planck constant 3.9903127128934314×10−10 J⋅s⋅mol−1 0 [51]
m
(
12
C
)
{\displaystyle m({}^{12}{\text{C}})} atomic mass of carbon-12 1.99264687992(60)×10−26 kg 3.0×10−10 [52]
M
(
12
C
)
=
N
A
m
(
12
C
)
{\displaystyle M({}^{12}{\text{C}})=N_{\text{A}}m({}^{12}{\text{C}})} molar mass of carbon-12 11.9999999958(36)×10−3 kg⋅mol−1 3.0×10−10 [53]
m
u
=
m
(
12
C
)
/
12
{\displaystyle m_{\text{u}}=m({}^{12}{\text{C}})/12} atomic mass constant 1.66053906660(50)×10−27 kg 3.0×10−10 [54]
M
u
=
M
(
12
C
)
/
12
{\displaystyle M_{\text{u}}=M({}^{12}{\text{C}})/12} molar mass constant 0.99999999965(30)×10−3 kg⋅mol−1 3.0×10−10 [55]
V
m
(
Si
)
{\displaystyle V_{\text{m}}({\text{Si}})} molar volume of silicon 1.205883199(60)×10−5 m3⋅mol−1 4.9×10−8 [56]
Δ
ν
Cs
{\displaystyle \Delta \nu _{\text{Cs}}} hyperfine transition frequency of 133Cs 9192631770 Hz 0 [57]

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On Thursday, November 23, 2023 at 2:36:22 AM UTC-6, Archimedes Plutonium wrote:

I thought it would be fun to check whether the list of physics constants from Wikipedia follows the Benford's law. It is a list of 57 constants.

So I have 57 constants of physics in that list of Wikipedia. And now tally the leading digit.

1 is 2
2 is 6
3 is 1
4 is 1
5 is 3
6 is 8
7 is 1
8 is 6
9 is 8
10 is 1
11 is 5
12 is 3
13 is 1
14 is 1
15 is 2
16 is 5
17 is 3
18 is 1
19 is 7
20 is 1
21 is 2
22 is 4
23 is 2
24 is 7
25 is 1
26 is 9
27 is 1
28 is 1
29 is 1
30 is 3
31 is 3
32 is 1
33 is 8
34 is 2
35 is 2
36 is 2
37 is 5
38 is 3
39 is 9
40 is 5
41 is 2
42 is 6
43 is 5
44 is 4
45 is 2
46 is 1
47 is 1
48 is 6
49 is 8
50 is 9
51 is 3
52 is 1
53 is 1
54 is 1
55 is 9
56 is 1
57 is 9

The tally is 19 Ones, 9 Twos, 7 Threes, 2 Fours, 5 Fives, 4 Sixes, 2 Sevens, 4 Eights, 5 Nines.

I would say it starts out fine with Ones and Twos and Threes, but then, thereafter is ruined with Fours.

I would say the table of Physics Constants hints of a Benford's law.

AP

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