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% =========================================================
% Concept Boxes
% =========================================================
\newtcolorbox{proposedbox}{
  title=Proposed Concept,
  colback=blue!5!white,
  colframe=blue!75!black,
  colbacktitle=blue!15!white,
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  sharp corners,
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}

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  enhanced,
  sharp corners,
  breakable,
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  enhanced,
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}

\ifLuaTeX
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\author{}
\date{}

\begin{document}
\frontmatter

\renewcommand*\contentsname{Contents}
{
\hypersetup{linkcolor=}
\setcounter{tocdepth}{3}
\tableofcontents
}
\mainmatter
\hypertarget{introduction}{%
\chapter{Introduction}\label{introduction}}

This book builder template was currated by John
Haverlack.(\protect\hyperlink{ref-JohnHaverlackACEP}{{``John {Haverlack}
\textbar{} {ACEP}''} n.d.})

\begin{quote}
``\emph{If I have seen further it is by standing on the shoulders of
Giants.}''

-- Isaac Newton
\end{quote}

\hypertarget{conventions}{%
\section{Conventions}\label{conventions}}

In this book we'll use a few conventions.

\hypertarget{new-concepts}{%
\subsection{New Concepts}\label{new-concepts}}

As many of the topics discussed in this book are a mix of
\textbf{established} math and physics, \textbf{proposed} dualistic
interpretations of established ideas, and also \textbf{speculative}
ideas that I don't yet know how to address, I wanted a way to clearly
distinguish these concepts. I've come up with the following convention
to highlight these classes of concepts to indicate their level of
mainstream acceptance.

In this book, established concept may be highlighted in green, and
represent mainstream physics or math concepts.

\begin{establishedbox}
Established Concept Einsteins Relativistic Dynamics Equations
\[E^2 = (m_{0} \cdot c^2)^2 + (p \cdot c)^2 \]
\end{establishedbox}

New ideas proposed by the author which have not been peer reviewed,
verified or tested, and should be looked at with scrutiny.

\begin{proposedbox}
Proposed Concept With the speed of light, \(c = 1\):
\[E^2 = m_{0}^2 + p^2 \]
\end{proposedbox}

Speculative Idea, that the author wonders about, but does not know how
to demonstrate, or ideas that need further treatment to prove or
disprove.

\begin{speculativebox}
Speculative Concept With the speed of light, \(c = 1\):
\[E^2 = m_{0}^2 + p^2 \]
\end{speculativebox}

\hypertarget{citation-example}{%
\chapter{Citation Example}\label{citation-example}}

\begin{itemize}
\tightlist
\item
  Numerical Linear
  Algebra(\protect\hyperlink{ref-trefethenNumericalLinearAlgebra2022a}{Trefethen
  and Bau 2022})
\end{itemize}

\hypertarget{best-words-ever}{%
\chapter{Best Words Ever}\label{best-words-ever}}

Blah blah blah

\hypertarget{example-content}{%
\chapter{Example Content}\label{example-content}}

In \(R\nu\) the
\href{https://en.wikipedia.org/wiki/Planck_units\#Planck_length}{Planck
Length} is the universal unit for measurement of distance, and is
defined approximately to be:
\[\boxed{L_P=\sqrt{\hbar}=5.72928\times10^{-35}m=1 L}\] Where \(1\ L\),
is 1 Planck Length of distance.

\hypertarget{si-conversion-factors}{%
\subsection{SI Conversion Factors}\label{si-conversion-factors}}

The following conversion factors can be used to convert observable
quantities of measure from the \emph{SI} system of units to \(R\nu\) to
\textasciitilde6 significant digits.

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3256}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.0930}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.5814}}@{}}
\toprule
\begin{minipage}[b]{\linewidth}\raggedright
Conversion Factor
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Symbol
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Value
\end{minipage} \\
\midrule
\endhead
meters to Planck Length & \(\chi_P\) &
\(1.74542\times10^{34} \frac{L}{m}\) \\
seconds to Planck Length & \(\tau_p\) &
\(5.23264\times10^{42} \frac{L}{s}\) \\
mass to Planck Length & \(G_P\) & \(1.62871\times10^8 \frac{L}{kg}\) \\
energy to Planck Length & \(E_P\) & \(1.81219\times10^9 \frac{L}{J}\) \\
momentum to Planck Length & \(P_P\) &
\(5.43280\times10^{-1} \frac{L\cdot s}{kg \cdot m}\) \\
temperature to Planck Length & \(k_P\) &
\(2.501998\times10^{-14} \frac{L}{K}\) \\
charge to Planck Length & \(C_P\) &
\(1.89007\times10^{18} \frac{L}{C}\) \\
\bottomrule
\end{longtable}

\hypertarget{physical-constants}{%
\subsection{Physical Constants}\label{physical-constants}}

Applying conversion factors from the table above, we can convert SI
values to Reduced Natural Units. For example, performing this analysis
on the the speed of light yields a unit-less number with a value of 1:

\(c = 299792458 \frac{m}{s} = 299792458 \frac{m}{s} \cdot 1.74542\times10^{34} \frac{L}{m} \cdot \frac{1}{5.23264\times10^{42} \frac{L}{s}} = 1.00000\)

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2273}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.0909}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.4545}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2273}}@{}}
\toprule
\begin{minipage}[b]{\linewidth}\raggedright
Quantity
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Symbol
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
SI
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
\(\nu\)
\end{minipage} \\
\midrule
\endhead
Speed of Light & \(c\) & \(299792458 \frac{m}{s}\) & 1 \\
Reduced Gravitational Constant & \(G_0\) &
\(8.38659\times10^{-10} \frac{m^3}{kg \cdot s^2}\) & 1 \\
Boltzmann's Constant & \(k\) & \(k=1.380649\times10^-23 \frac{J}{K}\) &
1 \\
Permittivity of Free Space & \(\epsilon_o\) &
\(8.854187817620\times10^{-12} \frac{C^{2}s^2}{kg \cdot m^3}\) & 1 \\
Permeability of Free Space & \(\mu_o\) &
\(\huge{\frac{1}{\epsilon_{o} \cdot c^{2}}}\) & 1 \\
Reduced Planck's Constant & \(\hbar\) &
\(1.054571726\times10^-34 \frac{kg \cdot m^2}{s}\) & \(1 L^2\) \\
Mass of the Electron & \(m_e\) & \(9.10938\times10^{-31} kg\) &
\(1.48366\times10^{-22} L\) \\
Charge of the Electron & \(e^-\) & \(-1.60218\times10^{-19} C\) &
\(-3.02822\times10^{-1} L\) \\
Unit Cycle & \(\Theta\) & \(2\pi = 6.28318...\ Radians\) &
\(1 \tau = 6.28318...\ Radians\) \\
\bottomrule
\end{longtable}

\hypertarget{fine-structure-constant}{%
\section{Fine Structure Constant}\label{fine-structure-constant}}

As a consistency check, we compute the
\emph{\href{https://en.wikipedia.org/wiki/Fine-structure_constant}{Fine
Structure Constant}} using Reduced Natural Units which is a unit less
ratio that should be independent of our system of units.

\(\huge{\alpha=\frac{e^2}{4\pi\epsilon_o\hbar c}=\frac{e^2}{2\tau}=0.00729735≈\frac{1}{137}}\)

\hypertarget{dimensional-analysis}{%
\subsubsection{Dimensional Analysis}\label{dimensional-analysis}}

The reader should be familiar with high school physics and chemistry
\href{https://en.wikipedia.org/wiki/Dimensional_analysis}{dimensional
analysis}.

\begin{itemize}
\tightlist
\item
  \(1\ meter\ (m) = 100\ centimeters\ (cm)\)
\item
  \(1\ kilometer\ (km) = 1000\ meters\ (m)\)
\item
  \(1\ mile = 5280\ feet\ (ft\ or\ ')\)
\item
  \(1\ foot\ (ft\ or\ ') = 12\ inches\ (in\ or\ ")\)
\item
  \(1\ inch\ (") = 2.54\ centimeters\ (cm)\)
\end{itemize}

How many kilometers are in 1 mile?
\(1\ mile = 1\ mile \times \frac{5280\ ft}{mile} \times \frac{12\ in}{ft} \times \frac{2.54\ cm}{in}\times \frac{1\ m}{100 cm} \times \frac{1\ km}{1000 m} = \frac{5280 \times 12 \times 2.54}{100 \times 1000}\ km = \frac{160934.40}{100000}\ km = 1.6\ km\)
Note that each unit in the denominator cancels with one if the numerator
until we are left with only km.

\hypertarget{newtons-law-of-gravity}{%
\section{Newton's Law of Gravity}\label{newtons-law-of-gravity}}

The force of gravity (\(F_g\)) between 2 masses, \(m1\) and \(m2\)
separated by distance \(r\) is given by
\href{https://en.wikipedia.org/wiki/Newton\%27s_law_of_universal_gravitation}{Newton's
Law of Gravity}:

\(F_{g} = G \frac{m_{1} m_{2}}{r^{2}}\)

Where \(G\), is the
\href{http://en.wikipedia.org/wiki/Gravitational_Constant}{Gravitational
Constant}.

\(G = 6.67430 \times 10^{-11}\ N\frac{m^2}{kg^2}\)

The strength of gravitational force follow the inverse square law
distributing gravitational flux over the surface area of a sphere
(\(4\pi r^2\)).

\hypertarget{inverse-square-law}{%
\subsubsection{Inverse Square Law}\label{inverse-square-law}}

Any source of a signal strength (\(S_0\)) that radiates isotropically in
3-dimensional space will distribute that signal strength (\(S_0\)) over
the surface area of a sphere (\(SA = 4 \pi r\)) of radius (\(r\)). Such
that the intensity (\(I\)) at distance (\(r\)) is:

\[I(r) = \frac{S_0}{4 \pi r^{2}}=\frac{S_0}{2 \tau r^{2}}\]
\includegraphics{lib/img/Inverse_square_law.svg.png} \#\#\#\# \(R\nu\)
Reduced Gravitational Constant In this version of Newton's Law of
Gravity we introduce a new constant \(G_0\), the reduced gravitational
constant to accommodate for the factor of \(4\pi = 2\tau\) which is has
been integrated in the SI version of the gravitational constant.

\(F_g =G \frac{m_{1} m_{2}}{r^{2}}= G_0 \frac{m_{1} m_{2}}{4 \pi r^{2}}=G_0 \frac{m_{1} m_{2}}{2 \tau r^{2}}\)

Where:

\(G = \frac{G_{0}}{2\tau} = 6.67384 \times 10^{-11} \frac{N \cdot m^2}{kg^2}\)

Analyzing the units:
\[\frac{N \cdot m^2}{kg^2} = \left( \frac{\left( kg \cdot \frac{m}{s^2} \right) \cdot m^2}{kg^2} \right)=\frac{m^3}{s^2 kg}\]
Converting seconds to meters with the SI speed of light as a conversion
factor:
\[\frac{m^3}{s^2 kg}\cdot\frac{1}{c^2}=\frac{m^3}{s^2 kg}\cdot\frac{s^2}{m^2}=\frac{m}{kg}\]

Thus where space and time are measured in units of meters, the reduced
gravitational constant, is:

\[\boxed{G_0=\frac{2\tau G}{c^2}=\frac{2\tau \cdot 6.67384 \times 10^{-11}}{299792458^2} \frac{m}{kg} = 9.33135 \times 10^-27 \frac{m}{kg}}\]

\begin{quote}
Observation This implies that not only can space an time be measure in
units of meters, but so can mass.
\end{quote}

\hypertarget{relativistic-energy-momentum-relation}{%
\subsection{Relativistic Energy Momentum
Relation}\label{relativistic-energy-momentum-relation}}

Einsteins
\href{https://en.wikipedia.org/wiki/Energy\%E2\%80\%93momentum_relation}{Relativistic
Energy Momentum} relationship shows a Pythagorean relation between the
total energy (\(E\)), rest mass (\(m_0\)) and momentum (\(p\)) of a
system.

\(E^{2} = (m_0 \cdot c^2)^2 + (p \cdot c)^2\)

Where space and time are both measure in units of meters, c=1.

\(E^2=(m_0)^2+(p)^2\)

From this we can see that Energy, Momentum and Mass have equivalent
units.

\begin{quote}
\emph{While we do not really know what energy, mass and momentum are we
know that they are fundamentally ``made'' out of the same stuff because
they have the same units.}
\end{quote}

\hypertarget{objects-of-mass-at-rest}{%
\paragraph{Objects of mass at rest}\label{objects-of-mass-at-rest}}

For an object at rest with no momentum (\(p = 0\)) we see Einstein's
famous equations:

\(E = m_{0} \cdot c^2\)

Or, with \(c=1\), this is much simpler to understand. Energy = Mass

\(E = m_0\) \#\#\#\#\# Zero mass objects moving at the speed of light
And for objects with no mass, like photos, (\(m_{0}= 0\)):

\(E=pc\)

Or, with \(c=1\), this is much simpler to understand. Energy = Momentum

\(E=p\)

\hypertarget{plancks-constant}{%
\section{Planck's Constant}\label{plancks-constant}}

The \href{https://en.wikipedia.org/wiki/Planck_constant}{Reduced Planck
constant} , ħ, represents a conversion factor for relating the
frequency, \(\omega\) (in \(2\pi\) radians per second), of a photon to
the energy of that photon. This can easily be seen from the simple but
profound relationship:

\(E=\hbar\omega\)

Where:

\(\hbar=1.054571726 \times 10^{−34} J \cdot s\)

and

\(J \cdot s = {kg}\cdot\frac{m^2}{s}\)

\begin{quote}
Reduced Planck's Constant \(\hbar = \frac{h}{2\pi} = \frac{h}{\tau}\)
\end{quote}

Simplifying our units by converting time and mass to units of meters:
\[\boxed{\hbar=1.054571726 \times 10^{−34} {kg}\cdot\frac{m^2}{s}\cdot\frac{G_0}{c}=3.282462\times10^{-69}m^2}\]

Which suggest that the Plank constant can be interpreted as an areas for
which the square root of is suspiciously close to the Plank length:

\[\boxed{\sqrt{\hbar}=\sqrt{3.282462\times10^{-69}m^2}=5.72928\times10^{-35}m}\]

\hypertarget{planck-area}{%
\subsubsection{Planck Area}\label{planck-area}}

The
\href{https://en.wikipedia.org/wiki/Planck_units\#Derived_units}{Planck
Area} is the square of the
\href{https://en.wikipedia.org/wiki/Planck_units\#Planck_length}{Planck
Length}.

\(l_{P}= \sqrt{\frac{\hbar G}{c^3}}\)

and \(l_{P}^{2}= \frac{\hbar G}{c^3}\)

In \(R\nu\) units both \(c\) and \(G_o\) are 1.

\(l_{P} = \sqrt{\hbar}\)

and \(l_P^{2}=\hbar\) \#\# Bekenstein's Bound After having recently read
\emph{Three Roads to Quantum Gravity} by Lee Smolin, I now suspect the
meaning of this areas is related to the
\href{https://en.wikipedia.org/wiki/Bekenstein_bound}{Bekensteins Law}
as applied to a surface areas surrounding a mass. Where the
\href{https://en.wikipedia.org/wiki/Entropy_(statistical_thermodynamics)}{thermodynamic
entropy}, \emph{S}, is proportional to the the enclosed surface area,
\(A\).

\(S=\frac{1}{4}\cdot\frac{A}{G\hbar}\)

\(S=\frac{k c^{3} A}{4 G \hbar}\)

\(S \le \frac{2\pi k R E}{\hbar c} = \frac{\tau R k E}{\hbar c}\)

From our new values for \(G_0\)and \(\hbar\) we can likely rewrite this:

\(S=\frac{\pi\cdot A}{\hbar G_0}\)

With the limiting case being at the Plank scale.

\(S=\frac{\pi\cdot \sqrt{\hbar}}{\hbar G_0}\)

\hypertarget{planck-length}{%
\section{Planck Length}\label{planck-length}}

https://en.wikipedia.org/wiki/Planck\_length

The concept of the Planck Length comes from exploring the limits of
Quantum Mechanics and General Relativity. The limits of General
Relativity can be seen a the event horizon of a black hole, described by
the Schwarzschild Radius. And the limits of Quantum Mechanics can be
found in the Compton Wavelength for a given quanta.

The
\href{https://simple.wikipedia.org/wiki/Schwarzschild_radius}{Schwarzschild
Radius} is defined as the distance at which light cannot escape from the
gravitational field of a mass (m):

Classic Derivation.

\(r_S=\frac{2G m}{c^2}\)

The reduced
\href{https://en.wikipedia.org/wiki/Compton_wavelength}{Compton
Wavelength} represents a lower limit on the wavelength for quanta that
can interact with a quantum particle with mass (m):

\(\lambda_C=\frac{h}{m c}\)

\(\bar{\lambda_C}=\frac{2\pi\hbar}{m c}=\frac{\tau\hbar}{m c}\)

And set the Schwarzschild Radius equal to the Compton Wavelength:
\(r_S=\lambda_C\)

\(\frac{2Gm}{c^{2}}=\frac{h}{m c}\)

\(m^{2}= \frac{hc}{2G}\)

\(m = \sqrt{\frac{hc}{2G}}\)

\(l_P=\frac{2G\sqrt{\frac{hc}{2G}}}{c^2}\)
\(l_P=\frac{2G\sqrt{\frac{hc}{2G}}}{c^{2}}= \sqrt{\frac{2Gh}{c^2}}\)

With reduced Compton Wavelength \(r_S=\bar{\lambda_C}\)

\(\frac{2Gm}{c^{2}}=\frac{\tau\hbar}{m c}\)

\(m^2=\frac{\tau\ \hbar\ c}{2G}\)

\(m = \sqrt{\frac{\tau\ \hbar\ c}{2G}}\)

\(l_P=\frac{2G\sqrt{\frac{\tau\ \hbar\ c}{2G}}}{c^2}\)

\(l_P=\frac{2G\sqrt{\frac{\tau\ \hbar\ c}{2G}}}{c^{2}}=\sqrt{\frac{4\ \tau\ G\ \hbar}{c^3}}\)

If we reduce the units in these equation to those of mass and time
measured in meters.

\(l_P=\sqrt{4\tau\hbar G_{o}}\)

and

\(\lambda_C=\frac{\hbar}{m}\)

\(m=R_s=\lambda_C=\frac{\hbar}{m}\)

This is known as the Planck Mass, \(M_P\). \(M_P=m=\sqrt{\hbar}\)

Solving the Compton Wavelength for distance we find the classic Plank
Length:

\(\lambda_C=\frac{\hbar}{\sqrt{\hbar}}=\frac{\sqrt{\hbar}}{\sqrt{\hbar}}\cdot\frac{\hbar}{\sqrt{\hbar}}=\sqrt{\hbar}=L_P\)

Which is in precise agreement with the value we found in above. Thus the
Plank Length is:

\(L_P=\sqrt{\hbar}=5.72928\times10^{-35}m\)

When we measure distance, time, and mass in units of distance, c=1, and
the Plank Time, \(T_P\), is equal to Plank Length, \(L_P\), which is
equal to the Plank Mass, \(M_P\):

\[\boxed{L_P=T_P=M_P}\]

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3000}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3000}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.4000}}@{}}
\toprule
\begin{minipage}[b]{\linewidth}\raggedright
Conversion Factor
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Symbol
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Value
\end{minipage} \\
\midrule
\endhead
meters to Planck Length & \(\chi_P\) &
\(1.74542\times10^{34} \frac{L}{m}\) \\
seconds to Planck Length & \(\tau_p\) &
\(5.23264\times10^{42} \frac{L}{s}\) \\
mass to Planck Length & \(G_P\) & \(1.62871\times10^8 \frac{L}{kg}\) \\
energy to Planck Length & \(E_P\) & \(1.81219\times10^9 \frac{L}{J}\) \\
momentum to Planck Length & \(P_P\) &
\(5.43280\times10^{-1} \frac{L\cdot s}{kg \cdot m}\) \\
temperature to Planck Length & \(k_P\) &
\(2.501998\times10^{-14} \frac{L}{K}\) \\
charge to Planck Length & \(C_P\) &
\(1.89007\times10^{18} \frac{L}{C}\) \\
\bottomrule
\end{longtable}

Applying conversion factors from the table above, we can convert SI
values to Reduced Natural Units. \(c=\frac{1}{\sqrt{\epsilon_o \mu_o}}\)

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2308}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2308}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2308}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.3077}}@{}}
\toprule
\begin{minipage}[b]{\linewidth}\raggedright
Quantity
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Symbol
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
SI
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
\(\nu\)
\end{minipage} \\
\midrule
\endhead
Speed of Light & \(c\) & \(299792458 \frac{m}{s}\) & 1 \\
Gravitational Constant & \(G_0\) &
\(8.38659\times10^{-10} \frac{m^3}{kg \cdot s^2}\) & 1 \\
Boltzmann's Constant & \(k\) & \(k=1.380649\times10^-23 \frac{J}{K}\) &
1 \\
Permittivity of Free Space & \(\epsilon_o\) &
\(8.854187817620\times10^{-12} \frac{C^{2}s^2}{kg \cdot m^3}\) & 1 \\
Permeability of Free Space & \(\mu_o\) &
\(\huge{\frac{1}{\epsilon_{o} \cdot c^{2}}}\) & 1 \\
Planck's Constant & \(\hbar\) &
\(1.054571726\times10^-34 \frac{kg \cdot m^2}{s}\) & \(1 L^2\) \\
Mass of the Electron & \(m_e\) & \(9.10938\times10^{-31} kg\) &
\(1.48366\times10^{-22} L\) \\
Charge of the Electron & \(e^-\) & \(-1.60218\times10^{-19} C\) &
\(-3.02822\times10^{-1} L\) \\
\bottomrule
\end{longtable}

\hypertarget{fine-structure-constant-1}{%
\section{Fine Structure Constant}\label{fine-structure-constant-1}}

https://en.wikipedia.org/wiki/Fine-structure\_constant As a consistency
check, we compute the \emph{Fine Structure Constant} using Reduced
Natural Units which is a unit less ratio that should be independent of
our system of units.

\(\huge{\alpha=\frac{e^2}{4\pi\epsilon_o\hbar c}=\frac{e^2}{4\pi}=0.00729735≈\frac{1}{137}}\)

This check confirms that our system of Reduced Natural Units has
internally consistent values for \(c\), \(\epsilon_o\), \(\hbar\) and
\(e-\). And also \(G_o\) which was used to computer prior values is also
consistent.

\hypertarget{sage-code}{%
\section{Sage Code}\label{sage-code}}

Unit Analysis computations have been performed with
\href{https://www.sagemath.org/}{Sage Math}.

\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# Define constance}
\ExtensionTok{one}\NormalTok{ = 1.n}\ErrorTok{(}\VariableTok{digits}\OperatorTok{=}\NormalTok{6}\KeywordTok{)}
\ExtensionTok{pi}\NormalTok{ = pi.n}\ErrorTok{(}\VariableTok{digits}\OperatorTok{=}\NormalTok{6}\KeywordTok{)}
\ExtensionTok{tau}\NormalTok{ = 2 }\PreprocessorTok{*}\NormalTok{ pi}
\ExtensionTok{t}\NormalTok{ = tau}

\CommentTok{\# Define the units}
\ExtensionTok{meters}\NormalTok{ = var}\ErrorTok{(}\StringTok{\textquotesingle{}m\textquotesingle{}}\KeywordTok{)}
\ExtensionTok{m}\NormalTok{ = one}\PreprocessorTok{*}\NormalTok{meters}

\ExtensionTok{seconds}\NormalTok{ = var}\ErrorTok{(}\StringTok{\textquotesingle{}s\textquotesingle{}}\KeywordTok{)}
\ExtensionTok{s}\NormalTok{ = seconds}

\ExtensionTok{kilograms}\NormalTok{ = var}\ErrorTok{(}\StringTok{\textquotesingle{}kg\textquotesingle{}}\KeywordTok{)}
\ExtensionTok{kg}\NormalTok{ = kilograms}

\ExtensionTok{newtons}\NormalTok{ = kg }\PreprocessorTok{*}\NormalTok{ m / s\^{}2}
\ExtensionTok{N}\NormalTok{ = newtons}

\ExtensionTok{joules}\NormalTok{ = N }\PreprocessorTok{*}\NormalTok{ m}
\ExtensionTok{J}\NormalTok{ = joules}

\ExtensionTok{print}\ErrorTok{(}\StringTok{"pi    ="}\ExtensionTok{,}\NormalTok{ pi}\KeywordTok{)}
\ExtensionTok{print}\ErrorTok{(}\StringTok{"tau   ="}\ExtensionTok{,}\NormalTok{ t}\KeywordTok{)}

\CommentTok{\# Speed of light in meters/second}
\ExtensionTok{speed\_of\_light}\NormalTok{ = 299792458 }\PreprocessorTok{*}\NormalTok{ meters/seconds}
\ExtensionTok{sol}\NormalTok{ = speed\_of\_light}
\ExtensionTok{c}\NormalTok{ = sol}
\ExtensionTok{print}\ErrorTok{(}\StringTok{"si c  ="}\ExtensionTok{,}\NormalTok{ c}\KeywordTok{)}

\ExtensionTok{rnu\_c}\NormalTok{ = c / c}
\ExtensionTok{print}\ErrorTok{(}\StringTok{"R\textbackslash{}u03BD c  ="}\ExtensionTok{,}\NormalTok{ rnu\_c}\KeywordTok{)}

\CommentTok{\# Gravitational Constant}
\ExtensionTok{gravitational\_constant}\NormalTok{ = 6.67384e{-}11 }\PreprocessorTok{*}\NormalTok{ N}\PreprocessorTok{*(}\NormalTok{m\^{}2/kg\^{}2}\PreprocessorTok{)}
\ExtensionTok{G}\NormalTok{ = gravitational\_constant}
\ExtensionTok{print}\ErrorTok{(}\StringTok{"si G  ="}\ExtensionTok{,}\NormalTok{ G}\KeywordTok{)}

\ExtensionTok{rnu\_G}\NormalTok{ = 4}\PreprocessorTok{*}\NormalTok{pi}\PreprocessorTok{*}\NormalTok{G/c\^{}2}
\ExtensionTok{Go}\NormalTok{ = rnu\_G}
\ExtensionTok{print}\ErrorTok{(}\StringTok{"R\textbackslash{}u03BD Go ="}\ExtensionTok{,}\NormalTok{ Go}\KeywordTok{)}


\CommentTok{\# Planck\textquotesingle{}s Constant}
\ExtensionTok{reduced\_plancks\_constant}\NormalTok{ = 1.054571726e{-}34 }\PreprocessorTok{*}\NormalTok{ J}\PreprocessorTok{*}\NormalTok{s}
\ExtensionTok{h\_bar}\NormalTok{ = reduced\_plancks\_constant}
\ExtensionTok{print}\ErrorTok{(}\StringTok{"si \textbackslash{}u210F  ="}\ExtensionTok{,}\NormalTok{ h\_bar}\KeywordTok{)}

\ExtensionTok{rnu\_h\_bar}\NormalTok{ = h\_bar }\PreprocessorTok{*}\NormalTok{ Go / c}
\ExtensionTok{print}\ErrorTok{(}\StringTok{"R\textbackslash{}u03BD "}\ExtensionTok{u}\StringTok{"\textbackslash{}u210F  ="}\ExtensionTok{,}\NormalTok{ rnu\_h\_bar}\KeywordTok{)}

\CommentTok{\# Planck Length}
\ExtensionTok{rnu\_h\_bar\_str}\NormalTok{ = str}\ErrorTok{(}\ExtensionTok{rnu\_h\_bar}\KeywordTok{)} 
\ExtensionTok{numerical\_part\_str}\NormalTok{ = rnu\_h\_bar\_str.split}\ErrorTok{(}\StringTok{\textquotesingle{}*\textquotesingle{}}\KeywordTok{)}\ExtensionTok{[0]}  
\ExtensionTok{numerical\_part\_str}\NormalTok{ = numerical\_part\_str.strip}\ErrorTok{(}\StringTok{\textquotesingle{}()\textquotesingle{}}\KeywordTok{)}
\ExtensionTok{numerical\_part}\NormalTok{ = float}\ErrorTok{(}\ExtensionTok{numerical\_part\_str}\KeywordTok{)}
\ExtensionTok{rnu\_sqrt\_h\_bar}\NormalTok{ = numerical\_part\^{}}\ErrorTok{(}\ExtensionTok{1/2}\KeywordTok{)}
\CommentTok{\# \^{} Sage cannot process sqrt on units... Lame.}
\ExtensionTok{lP}\NormalTok{ = rnu\_sqrt\_h\_bar }\PreprocessorTok{*}\NormalTok{ m}
\ExtensionTok{print}\ErrorTok{(}\StringTok{"R\textbackslash{}u03BD \textbackslash{}u221A\textbackslash{}u210F ="}\ExtensionTok{,}\NormalTok{ lP}\KeywordTok{)}
\end{Highlighting}
\end{Shaded}

\hypertarget{output}{%
\subsubsection{Output}\label{output}}

\begin{verbatim}
pi    = 3.14159
tau   = 6.28319
si c  = 299792458*m/s
Rν c  = 1
si G  = (6.67384e-11)*m^3/(kg*s^2)
Rν Go = (9.33135e-27)*m/kg
si ℏ  = (1.05457e-34)*kg*m^2/s
Rν ℏ  = (3.28246e-69)*m^2
Rν √ℏ = (5.72928e-35)*m
si lP = (1.61620e-35)*sqrt(m^2)
Rν lP = (2.77455e-47)*sqrt(m^3/kg)
\end{verbatim}

\hypertarget{terminology}{%
\chapter{Terminology}\label{terminology}}

\hypertarget{citations}{%
\chapter*{Citations}\label{citations}}
\addcontentsline{toc}{chapter}{Citations}

\hypertarget{refs}{}
\begin{CSLReferences}{1}{0}
\leavevmode\vadjust pre{\hypertarget{ref-JohnHaverlackACEP}{}}%
{``John {Haverlack} \textbar{} {ACEP}.''} n.d.
https://www.uaf.edu/acep/about/our-team/john-haverlack.php. Accessed
September 30, 2025.

\leavevmode\vadjust pre{\hypertarget{ref-trefethenNumericalLinearAlgebra2022a}{}}%
Trefethen, Lloyd N., and III David Bau. 2022. \emph{Numerical {Linear
Algebra}: {Twenty-Fifth Anniversary Edition}}. Philadelphia, PA:
{SIAM-Society for Industrial and Applied Mathematics}.

\end{CSLReferences}

\backmatter
\end{document}
