![Number Systems
Background: Numbers Systems is a post to explore number systems in general and for use in the
physical and computational sciences.
Post 2
The Number One for Fibonacci n
F(n)
Post 1 has established:
1 =
[
1 +
2
5
3
1
5
2
8
3
13
5
21
8 . . . ^
1
𝛾
]
−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.
Then to be rigorous, we need to write:
1 = 1∞
For the infinite limit defining 1:
1 = 1∞ = 𝜙 + 𝛾
The Fibonacci definition of infinite ratios:
ϕ = lim
𝑛→ ∞
𝐹 𝑛−1
𝐹 𝑛
𝛾 = lim
𝑛→ ∞
𝐹 𝑛−2
𝐹 𝑛
𝐹(𝑛) = 𝐹(𝑛 − 2) + 𝐹(𝑛 − 1)
with seed values 0 and 1
𝐹 = {0, 1: 1, 2, 3, 5, 8, 13, 21, 34, 55, … }](https://image.slidesharecdn.com/9f46e55a-a524-4f91-97c0-3605543650de-160228031938/85/Post_Number-Systems_2-1-320.jpg)
![To continue required mathematical rigor for natural events, we would need to define values of one that
apply to individual Fibonacci integers n in addition to the universal value of one = 1∞.
For integers defining F(n) as above, to define 1n, we need to interpolate from the infinite definition of one.
To be rigorous, we need to state the following nomenclature and mathematics:
1 = 1∞
1 𝑛 = 𝜙 𝑛 + 𝛾𝑛
Define
𝐷 = 𝐹(𝑛)
𝜙 𝐷 =
𝐷
𝐷 + 𝛥1𝐷
𝛾 𝐷 =
𝐷 − 𝛥1𝐷
𝐷 + 𝛥1𝐷
1 𝐷 = 𝜙 𝐷 + 𝛾 𝐷
𝑓{𝐷} = γ-1
{D}
γ-1
{D} =
3
1
^
5
2
^
8
3
^
(𝐷+𝛥1𝐷)
(𝐷−𝛥1𝐷)
𝑅 𝐸 = [(𝛾−1)^(𝛾−1)]−1
𝑇1→𝐷 = (1 + 𝑅 𝐸
2
1⁄ 3
2⁄ 5
3⁄ 8
5⁄ 13
8⁄ 21
13⁄ 34
21⁄ …
(𝐷+𝛥1𝐷)
𝐷⁄
)
Then
1 𝐷 = (1 +
𝛾(∞)
𝑓{𝐷}
𝑇𝐷→(𝐷+𝛥2𝐷)
)
−1
(1 +
𝛾(𝐷+𝛥2𝐷)
𝑓{𝐷}
𝑇𝐷→(𝐷+𝛥2𝐷)
)
+1
A more concise form:
1 𝐷 = (1 +
𝛾∞
𝑓
𝑇𝐷
)
−1
(1 +
𝛾 𝐷
𝑓
𝑇𝐷
)
+1
This is difficult nomenclature. Post 3 is intended to clarify nomenclature through examples of the
Fibonacci definition of one.](https://image.slidesharecdn.com/9f46e55a-a524-4f91-97c0-3605543650de-160228031938/85/Post_Number-Systems_2-2-320.jpg)
Post_Number Systems_2
- 1. Number Systems Background: Numbers Systems is a post to explore number systems in general and for use in the physical and computational sciences. Post 2 The Number One for Fibonacci n F(n) Post 1 has established: 1 = [ 1 + 2 5 3 1 5 2 8 3 13 5 21 8 . . . ^ 1 𝛾 ] −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. Then to be rigorous, we need to write: 1 = 1∞ For the infinite limit defining 1: 1 = 1∞ = 𝜙 + 𝛾 The Fibonacci definition of infinite ratios: ϕ = lim 𝑛→ ∞ 𝐹 𝑛−1 𝐹 𝑛 𝛾 = lim 𝑛→ ∞ 𝐹 𝑛−2 𝐹 𝑛 𝐹(𝑛) = 𝐹(𝑛 − 2) + 𝐹(𝑛 − 1) with seed values 0 and 1 𝐹 = {0, 1: 1, 2, 3, 5, 8, 13, 21, 34, 55, … }
- 2. To continue required mathematical rigor for natural events, we would need to define values of one that apply to individual Fibonacci integers n in addition to the universal value of one = 1∞. For integers defining F(n) as above, to define 1n, we need to interpolate from the infinite definition of one. To be rigorous, we need to state the following nomenclature and mathematics: 1 = 1∞ 1 𝑛 = 𝜙 𝑛 + 𝛾𝑛 Define 𝐷 = 𝐹(𝑛) 𝜙 𝐷 = 𝐷 𝐷 + 𝛥1𝐷 𝛾 𝐷 = 𝐷 − 𝛥1𝐷 𝐷 + 𝛥1𝐷 1 𝐷 = 𝜙 𝐷 + 𝛾 𝐷 𝑓{𝐷} = γ-1 {D} γ-1 {D} = 3 1 ^ 5 2 ^ 8 3 ^ (𝐷+𝛥1𝐷) (𝐷−𝛥1𝐷) 𝑅 𝐸 = [(𝛾−1)^(𝛾−1)]−1 𝑇1→𝐷 = (1 + 𝑅 𝐸 2 1⁄ 3 2⁄ 5 3⁄ 8 5⁄ 13 8⁄ 21 13⁄ 34 21⁄ … (𝐷+𝛥1𝐷) 𝐷⁄ ) Then 1 𝐷 = (1 + 𝛾(∞) 𝑓{𝐷} 𝑇𝐷→(𝐷+𝛥2𝐷) ) −1 (1 + 𝛾(𝐷+𝛥2𝐷) 𝑓{𝐷} 𝑇𝐷→(𝐷+𝛥2𝐷) ) +1 A more concise form: 1 𝐷 = (1 + 𝛾∞ 𝑓 𝑇𝐷 ) −1 (1 + 𝛾 𝐷 𝑓 𝑇𝐷 ) +1 This is difficult nomenclature. Post 3 is intended to clarify nomenclature through examples of the Fibonacci definition of one.