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= p =
= p =

{| class="wikitable" style="margin: 1em auto 1em auto; background-color: #ffffff"
|+ Defining Constants
! scope="col" | Symbol
! scope="col" | SI value
! scope="col" | Bully
|-
| <math>c \,</math>
| <math>299 792 458 \ </math>
| <math> 1 </math>
|-
| <math>h \,</math>
| <math>\frac{4 \times 10^{-18}}{(25812.807) (483597.9)^2} \ </math>
| <math> 2\pi \,</math>
|-
| <math>\hbar=\frac{h}{2 \pi}</math>
| <math>\frac{2 \times 10^{-18}}{\pi (25812.807) (483597.9)^2} \ </math>
| <math> 1 \,</math>
|-
| <math>e \,</math>
| <math>\frac{2 \times 10^{-9}}{(25812.807) (483597.9)} \ </math>
| <math> 1 \,</math>
|-
| <math>K_J =\frac{2e}{h} \,</math>
| <math>483597.9 \times 10^9 \,</math>
| <math>\frac{1}{\pi} \,</math>
|-
| <math>R_K =\frac{h}{e^2} \,</math>
| <math>25812.807 \,</math>
| <math>2\pi \,</math>
|-
| <math>Z_0 = 2 \alpha R_K \,</math>
| <math>2 \alpha (25812.807) \,</math>
| <math>4 \pi \alpha \,</math>
|-
| <math> \varepsilon_0 = \frac{1}{Z_0 c} \,</math>
| <math>\frac{1}{2 \alpha (25812.807) (299792458)} \ </math>
| <math>\frac{1}{4 \pi \alpha} \,</math>
|-
| <math> \mu_0 = \frac{Z_0}{c} \,</math>
| <math>\frac{2 \alpha (25812.807)}{299792458} \ </math>
| <math>4 \pi \alpha \,</math>
|-
| <math>G \,</math>
| <math>- \,</math>
| <math>1 \,</math>
|}


The Bully system includes units of transformation which are defined by analogy with [https://en.wikipedia.org/wiki/Units_of_information units of information]. These include the nat (n), bit (b), trit(t), and dit or digit (d). For each type of transformation unit, one may convert from nats to bits, trits, or dits, by multiplication with the natural logarithm as shown below:

 b = n × log<sub>e</sub>(2)
 t = n × log<sub>e</sub>(3)
 d = n × log<sub>e</sub>(10)
 where log<sub>e</sub> is the natural logarithm.



The above definitions ensure normalization of Boltzmann's constant and Planck's constant when using Bully units:

 k<sub>B</sub> = 1.0 Tn (exact)
 <math display="inline">\hbar</math> = 1.0 An (exact)

= Planck units and the Bully Metric =
Since c and G are normalized in the Bully system, this ensures that Bully units should have a simple relationship with Planck's units. As illustrated below, multiplying each SI value from Table 1 by 566.66, results in the corresponding Bully value multiplied by 10<sup>-30</sup>:

 566.66 × t<sub>P</sub> = 566.66 × 5.391247(60)×10<sup>−44</sup> s
                        = 3055.004(34)×10<sup>−44</sup> s
                        = 30.55004(34)×10<sup>−30</sup> ps
                        = 1.00001(11)×10<sup>−30</sup> ta

 566.66 × l<sub>P</sub> = 566.66 × 1.616255(18)×10<sup>−35</sup> m
                        = 915.867(10)×10<sup>−35</sup> m
                        = 9.15867(10)×10<sup>−30</sup> mm
                        = 1.00001(11)×10<sup>−30</sup> la

 566.66 × m<sub>P</sub> = 566.66 × 2.176434(24)×10<sup>-8</sup> kg
                        = 1233.298(14)×10<sup>−8</sup> kg
                        = 12.33298(14)×10<sup>−30</sup> rg
                        = 1.00001(11)×10<sup>−30</sup> ma

SI prefixes will have the same meaning and conventions when used with apan variants as they have when used with standard SI units (see table in subsequent section for a list of SI prefixes). The "quecto" (symbol "q") metric prefix means 10<sup>-30</sup>. The relationship between Bully units and Planck's units can be summarized as:

 566.66 × t<sub>P</sub> = 1.00001(11) qta
 566.66 × l<sub>P</sub> = 1.00001(11) qla
 566.66 × m<sub>P</sub> = 1.00001(11) qma

Planck units are understood to represent the smallest meaningful size of each quantity. For example, the Planck length is the smallest possible length because looking at small objects requires energy.  If one were to build a microscope powerful enough to see objects of Planck length or smaller, the microscope would use so much energy that a black hole would form.  In terms of Bully units, the "quecto" of each unit is 566.66 times larger than the absolute minimum size for that unit.

The Planck mass may seem unexpectedly large for a minimum mass value, but keep in mind that in this case the unit is for gravitational mass.  There obviously are well defined and detectable masses that are smaller than the Planck mass (for example the electron and proton masses), but the Planck mass represents the boundary between gravitational mass and quantum mass.  If an object has a mass larger than the Planck mass then gravitational effects will dominate.  If the mass is smaller than the Planck mass then quantum mechanical effects will dominate.

= The Bully Constants =
There are a surprising number of physical constants that can be approximated using various algebraic combinations of the following four numbers [[The Bully Mnemonic|(click here to learn more)]]:

 1.033
 30.55
 2
 0.00004

== Earth's radius (r) ==
An approximate relationship of the speed of light to the [https://en.wikipedia.org/wiki/Earth_radius Earth's radius (r)]:

<math display="block"> r_{earth} \approx \frac{c \times 30.55\,ms}{\sqrt{2 \times 1.033}} = 6371 \, km</math>

== Earth's Schwarzschild radius (R) ==
An approximate relationship of the speed of light to the Earth's Schwarzschild radius (R):

<math display="block"> R_{earth} = \frac{2 \times GM_{earth}}{c^{2}} \approx \frac{ c \times 30.55\,ps}{1.033} \approx {8.866 \, mm} </math>

== Earth's standard gravitational parameter (μ = MG) ==
An approximate relationship of the speed of light to the [https://en.wikipedia.org/wiki/Standard_gravitational_parameter Earth's standard gravitational parameter (μ = MG)]:

<math display="block"> {\mu}_{earth} = GM_{earth} \approx \frac{ c^{3} \times 30.55\,ps}{2 \times 1.033} \approx {398,400,000,000,000\, \frac{m^{3}}{s^2}} </math>

== Earth's gravity (g) ==
A typical [https://en.wikipedia.org/wiki/Gravity_of_Earth gravitational acceleration] on earth's surface can be approximated as:

<math display="block">g_{earth} = \frac{{\mu}_{earth}}{{r_{earth}}^{2}} \approx \frac{c}{30.55 Ms} \approx {9.813 \, \frac{m}{s^{2}}} </math>

= Definition of the apan =

The above approximations for earth's physical parameters can be further simplified by introducing new measurement units.  The '''apan''' will be defined with two variants. The '''time apan''' (symbol '''ta''') is by definition exactly 30.55 picoseconds.  The '''length apan''' (or light apan) (symbol '''la''') is by definition the distance light travels in vacuum in exactly 30.55 picoseconds.

SI prefixes will have the same meaning and conventions when used with the apan as the have when used with standard SI units (see table in subsequent section).  For example:
 One million ta = 1,000,000 ta = 1 mega time apan = 1 Mta = 30.55 μs
 One million la = 1,000,000 la = 1 mega light apan = 1 Mla = 9.1586595919 km

Approximations for earth's physical parameters can be written in terms of the apan as follows:

== Sidereal year ==
The number of seconds in the Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] can be approximated as:

<math display="block">{1 \, Sidereal \, Year} = {1.033 \, Eta} = {31,558,150 \, s} </math>

== Tropical year ==
The number of seconds in the Earth's [https://en.wikipedia.org/wiki/Tropical_year tropical year] can be approximated as:

<math display="block"> {1 \, Tropical \, Year} = (1.033 - 0.00004) \, Eta - 2 \,s = 31,556,926 \, s </math>

== Earth's radius (r) ==
An apan based approximation of [https://en.wikipedia.org/wiki/Earth_radius Earth's radius (r)]:

<math display="block"> r_{earth} \approx \frac{ 1 \, Gla}{\sqrt{2 \times 1.033}} = 6371 \, km </math>

== Earth's Schwarzschild radius (R) ==
An apan based approximation of the Earth's Schwarzschild radius (R):

<math display="block"> R_{earth} \approx \frac{la}{1.033} \approx {8.866 \, mm} </math>

== Earth's standard gravitational parameter (μ = MG) ==
An apan based approximation of the [https://en.wikipedia.org/wiki/Standard_gravitational_parameter Earth's standard gravitational parameter (μ = MG)]:

<math display="block"> {\mu}_{earth} \approx \frac{ 1 \, \frac{la^{3}}{ta^{2}} }{2 \times 1.033} \approx {398,400,000,000,000\, \frac{m^{3}}{s^2}} </math>

== Earth's gravity (g) ==
A typical [https://en.wikipedia.org/wiki/Gravity_of_Earth gravitational acceleration] on earth's surface can be approximated as:

<math display="block">g_{earth} = \frac{{\mu}_{earth}}{{r_{earth}}^{2}} \approx 1 \frac{Ela}{Eta^{2}} \approx {9.813 \, \frac{m}{s^{2}}} </math>