1. Number Systems
Background: Number Systems is a post to explore number systems in general and for use in the
physical and computational sciences.
Post 8
Natural Events in Fibonacci Number Space
Dimensionless Derivatives
Posts 1 – 7 have established:
1 𝐷 = (1 +
𝛾∞
𝑓
𝑇𝐷
)
−1
(1 +
𝛾 𝐷
𝑓
𝑇𝐷
)
+1
For natural events, this definition should correlate to the Bernoulli base of natural logarithms:
∫
1
𝑥
𝑑𝑥
𝑒
1
= 1 where lim
𝑛→∞
(1 +
1
𝑛
)
𝑛
= 𝑒
A mathematical description of nature should not be accurate unless the number system complies
with both natural conditions of the number one shown above.
Natural examples:
1
𝑐3
2 =
1
35
2 𝑥 10−16
meter-2 sec+2
h = 6.6260700 E-34 = 6.6260700 x (1∞ − 𝑅 𝐸
𝑓{3}
) x 10-34
meter+2 kg+1 sec-1
𝒘𝒉𝒆𝒓𝒆 𝒂 𝒈 = 𝒈
when g = gEarthSurface <g units: acceleration+1 second+2>
𝐸
𝐸 𝐵
= 𝑚𝑉𝐵
Then the dimensionless ratio for energy equals mass x volume of space.
For F(n) = 4 where D = 5:
15 = (1 +
𝛾∞
𝑓
𝑇5→13
)
−1
(1 +
𝛾5
𝑓
𝑇5→13
)
+1

2. ∫
1
𝑥
𝑒3
1
𝑑𝑥 = 1 𝑤ℎ𝑒𝑟𝑒 lim
𝑛→∞
(1 +
1
𝑛
)
𝑛
= 𝑒3 = 𝑒
ℎ = ℎ3 = 𝑏3 𝐸 𝐵 𝑥 𝑘𝑎𝑝𝑝𝑎 𝑤ℎ𝑒𝑟𝑒 𝐸 = (𝑚𝑎 𝑔)𝑥𝑏3
To be rigorous, for the observed constant h:
𝑎 𝑔 = 𝑔 𝑟𝐸𝑎𝑟𝑡ℎ
is not indicated in general number space F(n).
Fibonacci number space derives:
𝒂 𝒈 = 𝒈 𝒓
Where r is a radius from a center of mass in space. From mathematical rigor, r should represent
any radius from any center of mass.
To be rigorous, the numerical value of h should be the value h = h(r) while physical results at
spatial location r should be dimensionless.
Define
𝑥2 = 𝑚𝑥1 + 𝑏
𝑑𝑥2
𝑑𝑥1
= 𝑚
𝑑𝑥 𝐷+1𝐷
𝑑𝑥 𝐷
= 𝑚 𝐷
𝑚 𝐷 =
𝑑ℎ 𝐷+1𝐷
𝑑ℎ 𝐷
𝑚 𝐷 =
(1 − 𝑅 𝐸
𝑓{𝐷}
)
𝑒 𝐷
𝐷
𝐷+1𝐷
ℎ 𝐷+1𝐷 = 𝑚 𝐷ℎ 𝐷 + 𝑏
ℎ 𝐷 = 𝑚 𝐷− 1𝐷ℎ 𝐷− 1𝐷 + 𝑏
ℎ5 = 𝑚3ℎ3 + 𝑏3
Post 9 is intended to further clarify nomenclature through natural examples of Fibonacci Number
Space beginning from the value F(n) = D = 3.