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The geared fraction is an extension to the arithmetic fraction, consisting of multiple input spaces that define its numerical value. The common arithmetic fraction can be regarded as a 2-Gear fraction, as its value is purely defined by two numbers, the numerator at the top and the denominator at the bottom.

2-Gear fractions
The common properties of a two geared fraction are as follows for $$\forall\ a,b \in\mathbb{C}: $$

$$ =\frac{b}{b} =1$$
 * $$\frac{a}{b}=ab^{e^{i\pi}}=ab^{-1}
 * $$\frac{a}{a}
 * $$\left ( \frac{b}{a} \right )^{e^{i\pi}}=\frac{a}{b}$$

3-Gear fractions
Similarly, the properties of a three geared fraction are as follows for $$\forall\ a,b,c \in\mathbb{C}:$$

=ab^{e^{i\frac{2\pi}{3}}}c^{e^{i\frac{4\pi}{3}}} =ab^{-\frac{1}{2}+i\frac{\sqrt{3}}{2}}c^{-\frac{1}{2}-i\frac{\sqrt{3}}{2}}$$ =\frac{b}{b\mid b} =\frac{c}{c\mid c} =1$$ =\biggl(\frac{c}{a\mid b}\biggr)^{e^{i\frac{4\pi}{3}}} =\frac{a}{b\mid c}$$
 * $$\frac{a}{b\mid c}
 * $$\frac{a}{a\mid a}
 * $$\biggl(\frac{b}{c\mid a}\biggr)^{e^{i\frac{2\pi}{3}}}

Definitions
An n-geared fraction can be generalised:

\cdot a_2^{\Epsilon_N^{2}} \cdot a_3^{\Epsilon_N^{3}} \cdot a_4^{\Epsilon_N^{4}} \ ... \ a_N^{\Epsilon_N^{N}}$$ $$
 * The general form: $$a_1^{\Epsilon_N}
 * The product form: $$\prod_{k=1}^Na_k^{\Epsilon_N^{k}}$$
 * The exponential form: $$\exp\biggl(\sum_{k=1}^N\ln(a_k)\Epsilon_N^{k}\biggr)
 * Gamma notation: $$\Gamma (N)_{k=1}^N(a_k)$$with N terms within

Where $$\Epsilon_N=e^{i\frac{2\pi}{N}}$$and $$a_k\in\mathbb{C}$$and N is the number of input spaces ("cogs") the gear has.

Properties of geared fractions
The following properties hold for all geared fractions:

=\Gamma (N)_{k=1}^N(b_k)\times\Gamma (N)_{k=1}^N(a_k) =\Gamma (N)_{k=1}^N(a_kb_k)$$for $$\forall\ a_k,b_k \in\mathbb{C}$$ $$for $$\forall\ x \in\mathbb{C}$$
 * Equivalence of product form and gamma notation: $$\Pi_{k=1}^N(a_k^{\Epsilon_N^{k}})=\Gamma (N)_{k=1}^N(a_k)$$for $$\forall\ a_k \in\mathbb{C}$$and $$\Epsilon_N=e^{i\frac{2\pi}{N}}$$
 * Shorthand notation of a general n-geared fraction: $$\Gamma (N)$$for gear with N input spaces, $$\Gamma$$for any general gear
 * All gears belong to the set of complex numbers (all $$\Gamma \in\mathbb{C}$$)
 * Commutative multiplication: $$\Gamma (N)_{k=1}^N(a_k)\times\Gamma (N)_{k=1}^N(b_k)
 * Cancellation of identical terms: $$\Gamma (N)_{k=1}^N(x)=1
 * Expansive property upon multiplication: $$\Gamma (A)\times\Gamma (B)=\Gamma (\operatorname{lcm}(A, B))$$for $$\forall\ A,B \in\mathbb{N}$$

Example of expansive property:

A 3-Gear multiplied with a 2-Gear creates a 6-Gear

$$\frac{a}{b\mid c}\times \frac{d}{e}=\frac{1\mid ad \mid 1}{b \mid e \mid c}$$

=\Gamma (N)_{k=1}^N(a_k^x)$$for $$\forall\ x \in\mathbb{C}$$ =\Gamma (N)_{k=1}^N(a_{k-P})$$where $$\Epsilon_N=e^{i\frac{2\pi}{N}}$$
 * Compatibility with exponentiation: $$\Bigl(\Gamma (N)_{k=1}^N(a_k)\Bigr)^{x}
 * Rotational property upon exponentiation: $$\Bigl(\Gamma (N)_{k=1}^N(a_k)\Bigr)^{\Epsilon_N^{P}}

Definitions
A gearbox is defined as the natural logarithm of a geared fraction and can be defined as follows:

+a_2\Epsilon_N^{2} +a_3\Epsilon_N^{3} +\cdots +a_N\Epsilon_N^{N}$$ $$ \begin{array}{|c|c|c|} a_N \\ a_1 \\ a_2 \\ \cdots \\ a_{N-1} \end{array} $$with N terms throughout
 * The general form:$$a_1\Epsilon_N
 * The summation form: $$\sum_{k=1}^Na_k\Epsilon_N^{k}$$
 * The logarithmic form: $$\ln\Bigl(\Gamma (N)_{k=1}^N(e^{a_k})\Bigr)
 * Column notation: $$

Where $$\Epsilon_N=e^{i\frac{2\pi}{N}}$$and $$a_k\in\mathbb{C}$$and N is the number of input spaces the gearbox has.

Properties of gearboxes
The following properties hold for all gearboxes:

= \begin{array}{|c|c|c|} a_N \\ a_1 \\ a_2 \\ \cdots \\ a_{N-1} \end{array}$$for $$\forall\ a_k \in\mathbb{C}$$and $$\Epsilon_N=e^{i\frac{2\pi}{N}}$$ + \begin{array}{|c|c|c|} b_N \\ b_1 \\ b_2 \\ \cdots \\ b_{N-1} \end{array} = \begin{array}{|c|c|c|} b_N \\ b_1 \\ b_2 \\ \cdots \\ b_{N-1} \end{array} + \begin{array}{|c|c|c|} a_N \\ a_1 \\ a_2 \\ \cdots \\ a_{N-1} \end{array} = \begin{array}{|c|c|c|} a_N+b_N \\ a_1+b_1 \\ a_2+b_2 \\ \cdots \\ a_{N-1}+b_{N-1} \end{array}$$for $$\forall\ a_k,b_k \in\mathbb{C}$$ \begin{array}{|c|c|c|} x \\ x \\ x \\ \cdots \\ x \end{array} =0 $$for $$\forall\ x \in\mathbb{C}$$
 * Equivalence of summation form and column notation: $$\sum_{k=1}^Na_k\Epsilon_N^{k}
 * Shorthand notation of a general gearbox: $$\gamma (N)$$for gearbox with N input spaces, $$\gamma$$for any general gearbox
 * All gearboxes belong to the set of complex numbers (all $$\gamma \in\mathbb{C}$$)
 * Commutative addition: $$\begin{array}{|c|c|c|} a_N \\ a_1 \\ a_2 \\ \cdots \\ a_{N-1} \end{array}
 * Cancellation of identical terms: $$
 * Expansive property upon addition: $$\gamma (A)+\gamma (B)=\gamma (\operatorname{lcm}(A, B))$$for $$\forall\ A,B \in\mathbb{N}$$

Example of expansive property:

A 3-Gearbox multiplied with a 2-Gearbox creates a 6-Gearbox

$$ \begin{array}{|c|c|c|} a \\ b \\ c \end{array} + \begin{array}{|c|c|c|} d \\ e \end{array} = \begin{array}{|c|c|c|} a+d \\ 0 \\ b \\ e \\ c \\ 0 \end{array} $$

=\begin{array}{|c|c|c|} xa_{N} \\ xa_{1} \\ xa_{2} \\ \cdots \\ xa_{N-1} \end{array}$$for $$\forall\ x \in\mathbb{C}$$ =\begin{array}{|c|c|c|} a_{N-P} \\ a_{1-P} \\ a_{2-P} \\ \cdots \\ a_{N-1-P} \end{array}$$where $$\Epsilon_N=e^{i\frac{2\pi}{N}}$$
 * Compatibility with multiplication: $$x\times\begin{array}{|c|c|c|} a_N \\ a_1 \\ a_2 \\ \cdots \\ a_{N-1} \end{array}
 * Rotational property upon multiplication: $$\Epsilon_N^{P}\times\begin{array}{|c|c|c|} a_N \\ a_1 \\ a_2 \\ \cdots \\ a_{N-1} \end{array}

Gearboxes as a vector space
Gearboxes satisfy all eight axioms in order to be regarded as a subset of a vector space, however the basis units of all gearboxes are not linearly independent, as it is possible to linearly combine multiple basis units to represent a completely different one.

For example:

\begin{array}{|c|c|c|} 1 \\ 1 \\ 0 \\ 1 \end{array} = \begin{array}{|c|c|c|} 0 \\ 0 \\ -1 \\ 0 \end{array} $$for a four term gearbox

\begin{array}{|c|c|c|} 0 \\ 1 \\ 0  \end{array} = \begin{array}{|c|c|c|} -1 \\ 0 \\ -1 \end{array} $$for a three term gearbox

For this reason, gearboxes cannot be fully utilised as proper vectors.

Gearbox multiplication
Gearbox multiplication is defined as follows:

$$\begin{array}{|c|c|c|} a_N \\ a_1 \\ a_2 \\ \cdots \\ a_{N-1} \end{array} \times \begin{array}{|c|c|c|} b_N \\ b_1 \\ b_2 \\ \cdots \\ b_{N-1} \end{array} = a_N \begin{array}{|c|c|c|} b_N \\ b_1 \\ b_2 \\ \cdots \\ b_{N-1} \end{array} +a_1 \begin{array}{|c|c|c|} b_{N-1} \\ b_N \\ b_1 \\ \cdots \\ b_{N-2} \end{array} +a_2 \begin{array}{|c|c|c|} b_{N-2} \\ b_{N-1} \\ b_N \\ \cdots \\ b_{N-3} \end{array} +\cdots+a_{N-1} \begin{array}{|c|c|c|} b_{1} \\ b_{2} \\ b_3 \\ \cdots \\ b_{N} \end{array}$$for $$\forall\ a_k,b_k \in\mathbb{C}$$

Gearboxes are simultaneously commutative, associative, and distributive under the multiplicative operation.

Cyclic derivatives
Functions with cyclic derivatives are equal to their own Nth order derivative, where N $$ \in\mathbb{N}$$.

The general n-cycle function is defined as follows:

=a_1e^{x\Epsilon_N} +a_2e^{x\Epsilon^2_N} +a_3e^{x\Epsilon^3_N} +\cdots+ a_Ne^{x\Epsilon^N_N}$$for $$\forall\ a_k \in\mathbb{C}$$and $$\Epsilon_N=e^{i\frac{2\pi}{N}}$$
 * Direct definition: $$y=C_N(x)={\operatorname{d}^N\!y\over\operatorname{d}\!x^N}$$
 * General definition: $$C_N(x)=\sum_{k=1}^Na_ke^{x\Epsilon^k_N}
 * General definition using gamma notation: $$C_N(x)=\sum_{k=1}^Na_k\Bigl(\Gamma (N)_{i=k}^k(e^x)\Bigr)$$

The function is real only when all $$a_k$$terms are identical and real. It can be shown that there are n unique and real general n-cycle functions.

For example:


 * The two unique and real general 2-cycle functions are: $$C_2(x)=2\cosh x$$and $$C_2(x)=2\sinh x$$following from the general definition
 * The three unique and real general 3-cycle functions are: $$C_3(x)=2e^{-\frac{1}{2}x}\cos(\frac{\sqrt{3}}{2}x)+e^x$$and both of its derivatives following from the general definition

Selector functions
Selector functions choose an encoded value to be the output for a given integer input.

The general uncoded (inflexible) selector function is as follows:

$$S_N(n)=\frac{A}{N} \biggl(\sum_{k=1}^N \Epsilon_N^{nk} {\operatorname{d}^k(C_N(x))\!\over\operatorname{d}\!x^k} \Biggr|_{x=B} \biggr)$$for $$\forall\ A,B \in\mathbb{C}$$, $$\forall\ n \in\mathbb{N}$$ and $$\Epsilon_N=e^{i\frac{2\pi}{N}}$$

The inflexible selector function provides a basis on which the flexible selector can be derived.

The general coded (flexible) selector function is as follows:

$$S_N(n)=\frac{1}{N}\sum_{k=1}^N\Epsilon^{kn}_N\sum_{i=1}^Na_i\Epsilon^{ki}_N$$for $$\forall\ a_i \in\mathbb{C}$$, $$\forall\ n \in\mathbb{N}$$ and $$\Epsilon_N=e^{i\frac{2\pi}{N}}$$

The the outputs of the function are the chosen values for $$a_i$$.

$$ $$ $$ $$ $$ $$
 * $$S_N(0)=a_N
 * $$S_N(1)=a_1
 * $$S_N(2)=a_2
 * $$S_N(3)=a_3
 * $$\cdots
 * $$S_N(N-1)=a_{N-1}

$$S_N(N)=a_N $$

The first two relevant selector functions:

S_2(n)=\frac{1}{2}\Biggl( (a_2+a_1)+\Epsilon_2^n

\begin{array}{|c|c|c|} a_2 \\ a_1 \end{array}\Biggr) $$ S_3(n)=\frac{1}{3}\Biggl( (a_3+a_2+a_1) +\Epsilon_3^n \begin{array}{|c|c|c|} a_3 \\ a_1 \\ a_2 \end{array} +\Epsilon_3^{2n} \begin{array}{|c|c|c|} a_3 \\ a_2 \\ a_1 \end{array}\Biggr) $$

Derivation of polynomial solving formulas from selector functions
The selector functions can be used to derive both the quadratic and the cubic formula to find the roots of polynomials.

Quadratic formula derivation:

S_2(n)= \frac{-A}{2} + \frac{\Epsilon_2^n}{2} \begin{array}{|c|c|c|} b \\ a \end{array} $$ S_2(n)= \frac{-A}{2} + \frac{\Epsilon_2^n}{2} \sqrt{\begin{array}{|c|c|c|} a^2+b^2 \\ 2ab \end{array}}
 * Equation to solve is $$x^2+Ax+B=0$$
 * The roots of the polynomial are $$a_1=a$$and $$a_2=b$$
 * From the properties of the coefficients of the polynomial, $$-A=(a+b)$$and $$B=ab$$
 * Therefore: $$
 * Squaring the gearbox: $$

$$ S_2(n)= \frac{-A}{2} + \frac{\Epsilon_2^n}{2} \sqrt{ a^2+2ab+b^2-4ab }
 * Manipulation of expression within the root: $$

$$ S_2(n)= \frac{-A}{2} + \frac{\Epsilon_2^n}{2} \sqrt{ (a+b)^2-4B }
 * Factorise expressions: $$

$$ S_2(n)= \frac{-A}{2} + \Epsilon_2^n \frac{\sqrt{ A^2-4B }}{2}
 * Obtain the desired form: $$

$$

The cubic formula can be obtained by manipulating the gearboxes in a similar manner and defining the roots accordingly.

There is no proof of the quartic formula being obtainable using this method.

Extra Sandbox Stuff
Quadratic formula from selector form:

S_2(n)=\frac{1}{2}\Biggl( \begin{array}{|c|c|c|} a+b \end{array} +\Epsilon_2^n \begin{array}{|c|c|c|} a \\ b  \end{array}\Biggr) $$
 * Polynomial is: $$x^2+Ax+B=0$$
 * Roots of polynomial are: $$a,b$$
 * $$-A=\Sigma a$$
 * $$B=\Pi a$$
 * $$A^2-2B=\Sigma a^2$$

\begin{array}{lcr} S_2(n)=\frac{1}{2} \Sigma a \\ +\frac{1}{2}\Epsilon_2^n \sqrt{\Sigma a^2-2\Pi a} \end{array} $$ S_2(n)= \frac{-A}{2} + \Epsilon_2^n \frac{\sqrt{ A^2-4B }}{2}

$$

Cubic formula from selector form:

S_3(n)=\frac{1}{3}\Biggl( \begin{array}{|c|c|c|} a+b+c \end{array} +\Epsilon_3^n \begin{array}{|c|c|c|} a \\ b \\ c  \end{array} +\Epsilon_3^{2n} \begin{array}{|c|c|c|} a \\ c \\ b  \end{array} \Biggr) $$ \begin{array}{lcl} S_3(n)=\frac{1}{3}\Sigma a \\ +\frac{1}{3}\Epsilon_3^n \sqrt[3]{\Sigma a^3+6\Pi a-\frac{3}{2}\Sigma a^2 b+\frac{3\sqrt{3}}{2} \sqrt{-\Sigma a^4b^2-2\Pi a\Sigma a^2b+2\Sigma a^3b^3+2\Pi a\Sigma a^3+6\Pi a^2}} \\ +\frac{1}{3}\Epsilon_3^{2n} \sqrt[3]{\Sigma a^3+6\Pi a-\frac{3}{2}\Sigma a^2 b-\frac{3\sqrt{3}}{2} \sqrt{-\Sigma a^4b^2-2\Pi a\Sigma a^2b+2\Sigma a^3b^3+2\Pi a\Sigma a^3+6\Pi a^2}} \end{array} $$ \begin{array}{lcl} S_3(n)=-\frac{1}{3}A
 * Polynomial is: $$x^3+Ax^2+Bx+C=0$$
 * Roots of polynomial are: $$a,b,c$$
 * $$-A=\Sigma a$$
 * $$B=\Sigma ab$$
 * $$-C=\Pi a$$

\\ +\frac{1}{3}\Epsilon_3^n \sqrt[3]{\frac{9}{2}AB-\frac{27}{2}C-A^3+\frac{3\sqrt{3}}{2} \sqrt{27C^2+4B^3+4A^3C-A^2B^2-18ABC}} \\ +\frac{1}{3}\Epsilon_3^{2n} \sqrt[3]{\frac{9}{2}AB-\frac{27}{2}C-A^3-\frac{3\sqrt{3}}{2} \sqrt{27C^2+4B^3+4A^3C-A^2B^2-18ABC}} \end{array} $$

Quartic formula from selector form:

B^3+\frac{27}{2}A^2D+\frac{27}{2}C^2-36BD-\frac{9}{2}ABC=X_1 $$ 4A^2B^3D-16B^4D+27A^4D^2-144A^2BD^2-128B^2D^2-256D^3-18A^3BCD +80AB^2CD+192ACD^2-A^2B^2C^2+4B^3C^2+6A^2C^2D-144BC^2D+4A^3C^3 -18ABC^3+27C^4=X_2 $$ \frac{1}{4}\Epsilon_2^n \sqrt{A^2-\frac{7}{3}B+ \frac{2}{3} \sqrt[3]{ X_1 +\frac{3\sqrt{3}}{2} \sqrt{ X_2 } } + \frac{2}{3} \sqrt[3]{ X_1 -\frac{3\sqrt{3}}{2} \sqrt{ X_2 } } } $$
 * Polynomial is: $$x^4+Ax^3+Bx^2+Cx+D=0$$
 * Roots of polynomial are: $$a,b,c,d$$
 * $$-A=\Sigma a$$
 * $$B=\Sigma ab$$
 * $$-C=\Sigma abc$$
 * $$D=\Pi a$$
 * E2 part $$