![{\displaystyle {\scriptstyle \mathbf {Incrementation} }\ {\xrightarrow {\text{ Repeat }}}\ {\scriptstyle \mathbf {Addition} }\ {\xrightarrow {\text{ Repeat }}}\ {\scriptstyle \mathbf {Multiplication} }\ {\xrightarrow {\text{ Repeat }}}\ {\scriptstyle \mathbf {Exponentiation} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cca7a51460fe49c82f83155e1205fce32228624)
![{\displaystyle {\scriptstyle \mathbf {Exponentiation} }\ {\xrightarrow {\text{ Repeat }}}\ {\scriptstyle \mathbf {Tetration} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af87b0d35cf2adb6dcaf750ac9d138e2235fb164)
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![{\displaystyle \scriptstyle \mathbf {0.} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/28c198874bf543d59d3cdb5179e402bcd547086b) |
![{\displaystyle \scriptstyle \mathbf {Incrementation} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1475b7c12a0ee08414869ce94afbdbb020c640c3) |
![{\displaystyle \scriptstyle \mathbf {{_{^{+}}}{_{^{+}}}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c11264f74d53ddc128e2aa1018e89f6a7ac3fba4) |
![{\displaystyle \scriptstyle \mathbf {Decrementation} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/86248a462920412605470a80f76c7c12371954fa) |
![{\displaystyle \scriptstyle \mathbf {{_{^{-}}}{_{^{-}}}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9d82c629b965792624b15bd48bff3787607ddfe) |
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![{\displaystyle \scriptstyle \mathbf {1.} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7868f9f47d82e692fe1b83547c16857fe366a12) |
![{\displaystyle \scriptstyle \mathbf {Addition} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee3add8f06fc6fcf49a34331edeb1df90a393a5) |
![{\displaystyle \scriptstyle \mathbf {+} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/308fbeb84eaec587c01c786d26e330d4b523e7cb) |
![{\displaystyle \scriptstyle \mathbf {Subtraction} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b05887f1e9abe8c5d35d16e4d8c3af553a85a757) |
![{\displaystyle \scriptstyle \mathbf {-} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/935ea079e3ee23a881a180fafa4aeae6a6266b29) |
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![{\displaystyle \scriptstyle \mathbf {2.} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f76ec628f7cc0c8cb893880c0f6a1d04302685c) |
![{\displaystyle \scriptstyle \mathbf {Multiplication} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/08f1c47177fd7af509dc9e5e1d500345360a9f00) |
![{\displaystyle \scriptstyle \mathbf {\times } }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d47cd0486e9ab3d1dad5732d1a6dd95df381f3e6) |
![{\displaystyle \scriptstyle \mathbf {Division} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/08c69dc7697da056c9055cefd0f5d78736eae4a6) |
![{\displaystyle \scriptstyle \mathbf {\div } }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4deffa0291f58854027eaabdbb5202d5a98ed65e) |
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![{\displaystyle \scriptstyle \mathbf {3.} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a3894d94f252cab50a5f09458804ff2c41ce0d8) |
![{\displaystyle \scriptstyle \mathbf {Exponentiation} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7213c3b13e2d0cb910e6be033d9ef0688e7ce57b) |
![{\displaystyle \scriptstyle {a^{n}\ =\ b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c015a15a2c958b96ae953a42102b8d87e176a242) |
![{\displaystyle {\begin{aligned}&\scriptstyle \mathbf {Radicals} \\&\scriptstyle \mathbf {Logarithms} \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f02ce33c939c9c5d2c8c0092782867a87f1d09a5) |
![{\displaystyle {\begin{aligned}&\scriptstyle {a\ =\ {\sqrt[{n}]{b}}}\\&\scriptstyle {n\ =\ \log _{a}^{b}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc9aefffb8ebeb1873a67e038df44dc5f98eab18) |
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![{\displaystyle \scriptstyle \mathbf {4.} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/aedd8a7015cd6c336e68216c1614b0de9e77d504) |
![{\displaystyle \scriptstyle \mathbf {Tetration} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8b3d2736091ed67281d2810a745707555e982a8) |
![{\displaystyle \scriptstyle {^{n}a\ =\ b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/017091f4836a0f2016ca98c95833818bd5ca9747) |
![{\displaystyle {\begin{aligned}&\scriptstyle \mathbf {Super-Radicals} \\&\scriptstyle \mathbf {Super-Logarithms} \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3560164f82439c3d8d86414a66e2d5ae8a5ef2a) |
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![{\displaystyle a\ +\ b\ =\ b\ +\ a\ \qquad \quad \ =>\qquad \qquad \quad \ {\scriptstyle \mathbf {Subtraction} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81ff5860ff8f3fe7e4e3ccf54e534829f4283adf)
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![{\displaystyle a\ \ {_{^{\times }}}\ \ b\ =\ b\ \ {_{^{\times }}}\ \ a\ \qquad \quad \ =>\qquad \qquad \quad \ {\scriptstyle \mathbf {Division} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/062ef2448032087e0f4824cd2c766370c179ec46)
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![{\displaystyle a\ \uparrow \ {\color {white}.}b\ \neq \ b\ \uparrow \ a\ \qquad \quad \ =>\qquad \qquad \quad \ {\scriptstyle \mathbf {Radicals\ \neq \ Logarithms} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17d27b19c87f73821231f5433c7e24a18670ad4f)
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- Since incrementation, addition, and multiplication are symmetrical or commutative, they each have a
single inverse operation. But since exponentiation is asymmetrical or non-commutative, it possesses
two distinct inverse operations.
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![{\displaystyle \mathbb {X} ^{*}\ =\mathbb {X} _{*}\ =\mathbb {X} \smallsetminus {\Big \{}0{\Big \}}\qquad \qquad \quad ;\quad \mathbb {X} ^{^{+}}=\mathbb {X} _{_{+}}={\Big \{}\ x\in \mathbb {X} \ {\Big |}\ x\geqslant 0\ {\Big \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18fd7555dd70ec86fdc2cf58a03419488788f457)
![{\displaystyle \mathbb {X} ^{**}=\mathbb {X} _{**}=\mathbb {X} \smallsetminus {\Big \{}0,1{\Big \}}\qquad \qquad ;\quad \mathbb {X} ^{^{-}}=\mathbb {X} _{_{-}}={\Big \{}\ x\in \mathbb {X} \ {\Big |}\ x\leqslant 0\ {\Big \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2664fc319d8b75d53fb018e98c92f4bb67b8750a)
![{\displaystyle \mathbb {X} _{\mathbb {Y} }=\mathbb {Y} _{\mathbb {X} }=\mathbb {X} \ \cap \ \mathbb {Y} =\mathbb {Y} \ \cap \ \mathbb {X} \ \quad ;\quad {\overline {\mathbb {X} }}\ \ =\mathbb {X} \ \cup \ {\Big \{}\pm \infty {\Big \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/935542fda4d639e9481a4d7b182feec8f6858c87)
![{\displaystyle \mathbb {X} ^{\text{n}}=\underbrace {{\color {white}.}\mathbb {X} \times \mathbb {X} \times ...\times \mathbb {X} {\color {white}.}} _{\text{n times}}\ =\ {\Big \{}\ (x_{_{1}},x_{_{2}},...,x_{n})\ {\Big |}\ x_{_{\text{k}}}\in \mathbb {X} \ ;\ k\ \in \ {\overline {1...n}}\ {\Big \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e2abf71fef7a5836357de475394447d50b523a6)
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Integers. Rationals. Radicals
[edit]
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![{\displaystyle \mathbb {N} +\mathbb {N} \ =\ \mathbb {N} \cdot \mathbb {N} \ =\ \mathbb {N} \quad ;\qquad \mathbb {N} -\mathbb {N} \ =\ \mathbb {Z} \qquad \quad ;\quad \qquad {\frac {\mathbb {Z} }{\mathbb {Z} }}\ =\ {\overline {\mathbb {Q} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e935f827952740624d617a28be261add6e41b25)
![{\displaystyle \mathbb {N} \ =\ {\Big (}\ \mathbb {N} ,\ +,\ \times \ {\Big )}\qquad ;\qquad {\overline {\mathbb {Q} }}\ =\ {\Big (}\ \mathbb {N} ,\ \pm ,\ \times ,\ \div \ {\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f4aa2e38840a0dde22ff17a14931eb9db5c3266)
![{\displaystyle \mathbb {Z} \ =\ {\Big (}\ \mathbb {N} ,\ \pm ,\ \times \ {\Big )}\qquad ;\qquad {\widehat {\overline {\mathbb {V} }}}\ =\ {\Big (}\ \mathbb {N} ,\ \pm ,\ \times ,\ \div ,\ {\sqrt[{^{n}}]{\ }}\ {\Big |}\ n\in \mathbb {N} ^{*}\ {\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3096228e3dab913d7aa2cb8365dc2162e8b2f43e)
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Algebraics & Transcendentals
[edit]
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![{\displaystyle {\widehat {\mathbb {A} }}_{n}=\ \left\{\ z\ \left|\ \exists \ P_{n}(z)=\sum _{k\ =\ 0}^{n}{a_{_{\text{k}}}z^{k}}=0\ ;\quad \ {\begin{aligned}&a_{n}\neq \ 0\\&a_{_{\text{k}}}\in \ \mathbb {Z} \end{aligned}}\quad ,\ k\ \in \ {\overline {0...n}}\ \right.\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d35dd2fb6482e50e414bcb4be1e40dd965c62b4)
![{\displaystyle {\widehat {\mathbb {A} }}_{1}=\ \mathbb {Q} \quad ;\qquad \ {\widehat {\mathbb {A} }}_{\infty }=\ \mathbb {C} \quad ;\qquad \ {\begin{aligned}&{\widehat {\mathbb {A} }}_{n}\subset \ {\widehat {\mathbb {A} }}_{n+1}\qquad \qquad ,\quad n\in \mathbb {N} ^{*}\\&\mathbb {A} _{n}=\ {\widehat {\mathbb {A} }}_{n}\ \smallsetminus \ {\widehat {\mathbb {A} }}_{n-1}\quad ,\quad n\in \mathbb {N} ^{**}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d721860745ff7af2ff22afe7fce61781de388a8)
![{\displaystyle \mathbb {A} _{1}=\ \mathbb {Q} \quad ;\qquad \ \mathbb {A} _{\infty }=\ \mathbb {T} \quad ;\qquad \ \mathbb {A} \ \ =\bigcup _{{\text{n}}\ \in \ \mathbb {N} ^{*}}\mathbb {A} _{n}\quad {\Bigg |}\ {\begin{aligned}\mathbb {V} \ =\ {\widehat {\mathbb {V} }}\smallsetminus \mathbb {Q} \\\mathbb {B} \ =\ \mathbb {A} \smallsetminus {\widehat {\mathbb {V} }}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/920c0c65b1e8cc9e9bbb3dc043f3f58900105f71)
![{\displaystyle \mathbb {V} _{n}=\ \mathbb {V} \ \ \cap \ \mathbb {A} _{n}\qquad \qquad \qquad \qquad \mathbb {V} \ \ =\bigcup _{{\text{n}}\ \in \ \mathbb {N} ^{*}}\mathbb {V} _{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f8a4719f0c099cf32a9576e360759545b024703)
![{\displaystyle \mathbb {B} _{n}=\ \mathbb {B} \ \ \cap \ \mathbb {A} _{n}\qquad \qquad \qquad \qquad {\color {white}.}\mathbb {B} \ \ =\bigcup _{{\text{n}}\ \in \ \mathbb {N} ^{*}}\mathbb {B} _{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb6de4d3452f8c5e59ede9d9ce98bb929a3d159a)
![{\displaystyle \mathbb {A} _{n}=\ \mathbb {V} _{n}\cup \ \mathbb {B} _{n}\quad ,\quad n\in \mathbb {N} ^{**}\quad ;\quad {\color {white}.}\mathbb {V} _{1}=\ \mathbb {B} _{1}=\ \mathbb {B} _{2}=\ \mathbb {B} _{3}=\ \mathbb {B} _{4}=\varnothing }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f876ae695c711a7ae0290d95e15d835c4009902b)
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![{\displaystyle \mathbb {R} \rightarrow {\begin{cases}\mathbb {A} _{1}=\mathbb {Q} \rightarrow {\begin{cases}\mathbb {Z} \rightarrow {\begin{cases}\mathbb {Z} _{_{+}}=\mathbb {N} \rightarrow \mathbb {P} \\\mathbb {Z} _{_{-}}\end{cases}}\\\mathbb {F} \end{cases}}\\\\\ \qquad \ \mathbb {I} \rightarrow {\begin{cases}\mathbb {I} _{\mathbb {T} }\ =\ \mathbb {R} _{\mathbb {T} }\ =\ \mathbb {T} _{\mathbb {R} }\\\\\mathbb {I} _{\mathbb {A} }\rightarrow {\begin{cases}\mathbb {I} _{\mathbb {V} }\ =\ \mathbb {R} _{\mathbb {V} }\ =\ \mathbb {V} _{\mathbb {R} }\\\mathbb {I} _{\mathbb {B} }\ =\ \mathbb {R} _{\mathbb {B} }\ =\ \mathbb {B} _{\mathbb {R} }\end{cases}}\end{cases}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d885d26addea708d95b1a471ae7c8c5a0fef3f61)
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![{\displaystyle \mathbb {C} \ =\ \mathbb {R} ^{2}\ =\ \mathbb {R} \times \mathbb {R} \ =\ \mathbb {R} +i\cdot \mathbb {R} \ =\ \mathbb {A} \ \cup \ \mathbb {T} \ =\ \mathbb {Q} \ \cup \ \mathbb {V} \ \cup \ \mathbb {B} \ \cup \ \mathbb {T} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a192d1c6f2c35b567d7f75c365e5061dfb96c213)
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![{\displaystyle \mathbb {C} \rightarrow {\begin{cases}\mathbb {A} \rightarrow {\begin{cases}\mathbb {Q} \qquad \mathbb {A} _{1}\\\mathbb {V} \qquad {\sqrt {\ }}\quad \ {\color {white}.}{\text{radicals}}\\\mathbb {B} \qquad {\text{Bring radicals}}\end{cases}}\\\\\mathbb {A} _{\infty }=\ \mathbb {T} \end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c5f1bd3992c4344a0f1c225a1007c3e376ca044)
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The Basic Circle Constant
[edit]
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![{\displaystyle e\ =\ 2.\ 72^{^{-}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4df7037b0bf4bd401c44e3f08c50658d6b03732)
- Describes a circle or a spiral, by transforming a translation alongside a vertical or inclined straight
line from the complex plane into a rotation around the point of origin :
![](//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/300px-Euler%27s_formula.svg.png)
![{\displaystyle e^{\mathcal {Z}}\ =\ e^{x\ +\ iy}\ =\ e^{x}\cdot e^{iy}\ =\ e^{x}\cdot (\cos {y}\ +\ i\cdot \sin {y})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12339aa8762682aaf14b85e1db5c0a3f42229e3d) |
- Transcendental, and therefore irrational, since the exponential function generates a transcendental
polynomial :
![{\displaystyle e^{x}\ =\sum _{n=0}^{\infty }\ {\frac {x^{n}}{n!}}\qquad ,\qquad {\frac {1}{n!}}\in \mathbb {Q} \qquad ,\qquad \deg(e^{x})=\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f72474b7dd851464cd90e38a62b3d34d4fcc790)
- whereas algebraics are characterized by polynomial equations of finite degree over the rationals.
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The Basic Factorial Constant
[edit]
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![{\displaystyle \pi \ =\ \Gamma ^{2}\left({\tfrac {1}{2}}\right)\ =\ 4\cdot {\Big (}{\tfrac {1}{2}}!{\Big )}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aeae48481332cff3d79f633c9935c2f26f8b619c)
- Transcendental, and therefore irrational, since each n-sided polygon is described by a polynomial of
the n-th degree, generated by the product of all first degree polynomials corresponding to the linear
equation of each one of the polygon's n sides. But the circle, on the other hand, has an infinite number
of such sides, each the size of a point, characterized by the linear equation of the tangent in that point.
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Rational Approximations
[edit]
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![{\displaystyle {\dot {\mathbf {I} }}\quad \ \simeq \quad \mathbf {Q} _{_{\tfrac {1}{10}}}\qquad \qquad \qquad \qquad \quad \ \simeq \qquad \quad \mathbf {Q} _{_{\tfrac {1}{7}}}\ \qquad \quad \simeq \quad \ \mathbf {Q} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/54b03ddff4aa08c3ada15a45b41cb0d8ac66d77b)
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![{\displaystyle e^{\pi }\ \ =23.\ 14^{^{+}}\qquad \qquad \qquad \qquad \quad \simeq \quad 23{\tfrac {1}{7}}\ =\ {\tfrac {162}{7}}\qquad \qquad \qquad \qquad \mathbb {T} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/05ae7b9c737e73dfb698d4ad27a4469cf73c5451)
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![{\displaystyle \pi ^{2}\ \ =\ 9.\ 87^{^{-}}\ \qquad \qquad \qquad \qquad \quad \simeq \quad \ 9{\tfrac {6}{7}}\ =\ \ {\tfrac {69}{7}}\qquad \qquad \qquad \qquad \quad \mathbb {T} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/07d297879588b61cd5c2c72113d99994c3b49ac9)
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![{\displaystyle \pi \quad =\ 3.\ 14\ 16^{^{-}}\qquad \qquad \qquad \qquad \simeq \quad \ 3{\tfrac {1}{7}}\ =\ \ {\tfrac {22}{7}}\qquad \simeq \quad {\tfrac {3{\color {white}.}55}{11{\color {white}.}3}}\qquad \quad \ \mathbb {T} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5bf30eb1e6d7933d7d6782ad54e042f126e1bef)
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![{\displaystyle e\quad =\ 2.\ 7\ 1828\ 1828\ 45\ 90\ 45^{^{+}}\ \quad \simeq \quad \ 2{\tfrac {5}{7}}\ =\ \ {\tfrac {19}{7}}\qquad \qquad \qquad \qquad \quad \ \mathbb {T} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1be47a70901fcb56d6e8fcb369868e04dd58ac)
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![{\displaystyle {\sqrt {3}}\ =1.\ 73\ 20\ 50\ 80^{^{+}}\ {\color {white}.}\qquad \qquad \quad \simeq \quad \ 1{\tfrac {5}{7}}\ =\ \ {\tfrac {12}{7}}\qquad \simeq \quad {\tfrac {97}{8\times 7}}\qquad \quad \mathbb {A} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/546a45f466770f3c614934d360fb2623f2409185)
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![{\displaystyle {\sqrt {2}}\ =1.\ 41\ 42\ 135^{^{+}}\ \qquad \qquad \qquad \simeq \quad \ 1{\tfrac {3}{7}}\ =\ \ {\tfrac {10}{7}}\qquad \simeq \quad \ {\tfrac {99}{70}}\qquad \quad \ \mathbb {A} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df75d9e54d1587c43ed9cb296118066bcd238347)
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![{\displaystyle {\frac {e}{\pi }}\ \ =\ 0.\ 865^{^{+}}\qquad \qquad \qquad \qquad \quad \simeq \qquad \qquad \quad {\tfrac {6}{7}}\qquad \qquad \qquad \qquad \quad \ \mathbb {T} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0877081928ad177530fe0b9b60a73a055e10b35a)
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![{\displaystyle \gamma \ \ =\ 0.\ 577^{^{+}}\qquad \qquad \qquad \qquad \quad \simeq \qquad \qquad \quad {\color {white}.}{\tfrac {4}{7}}\qquad \simeq \quad {\tfrac {8\times 7}{97}}\qquad \quad \ \mathbb {T} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f33132716379f23f3affba7adff2ab6e7005698)
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![{\displaystyle \Omega _{1}=\ 0.\ 567^{^{+}}\qquad \qquad \qquad \qquad \quad \simeq \qquad \qquad \quad {\color {white}.}{\tfrac {4}{7}}\qquad \qquad \qquad \qquad \quad \ \mathbb {T} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7e8bc8c79cab92b8874687fe65d3fe5ba23f95)
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As Multiples of 1/7
[edit]
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![{\displaystyle e^{\pi }\ \ =\ 23{\frac {1}{7}}^{^{-}}=\ {\frac {162}{7}}^{^{-}}\qquad \qquad \qquad \qquad \qquad \qquad \ \pi ^{2}\ =\ 9{\frac {6}{7}}^{^{+}}=\ {\frac {69}{7}}^{^{+}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4afcf112bc3ef9f7f4d159fef2a11a0fa3caa975)
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![{\displaystyle \pi \quad =\ \ 3{\frac {1}{7}}^{^{-}}\ =\ \ {\frac {22}{7}}^{^{-}}\qquad \qquad \qquad \qquad \qquad \qquad \quad e\ =\ 2{\frac {5}{7}}^{^{+}}=\ {\frac {19}{7}}^{^{+}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75992b0fab204e2d0aead22a76f9788bb7f3da44)
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![{\displaystyle {\sqrt {3}}\ =\ \ 1{\frac {5}{7}}^{^{+}}\ =\ \ {\frac {12}{7}}^{^{+}}\qquad \qquad \qquad \qquad \qquad \qquad {\sqrt {2}}\ =\ 1{\frac {3}{7}}^{^{-}}=\ {\frac {10}{7}}^{^{-}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a246d2bd7de6e9a46131effc4a530c00f75c88a1)
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![{\displaystyle {\frac {e}{\pi }}\quad =\quad {\frac {6}{7}}^{^{+}}\qquad \qquad \qquad \ \gamma \ =\ {\frac {4}{7}}^{^{+}}\qquad \qquad \qquad \ \Omega _{1}\ =\ \ {\frac {4}{7}}^{^{-}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23403a13733725bd065a5d16561f4b4f3aeb9b11)
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![{\displaystyle {\sqrt {2}}\ =\ {\sqrt {2\cdot {\frac {49}{49}}}}\ =\ \ {\sqrt {\frac {98}{49}}}\ \ =\ {\sqrt {\frac {100^{^{-}}}{49\ }}}\ =\ {\frac {10}{7}}^{^{-}}\ \qquad \qquad \qquad \qquad \qquad \ \mathbb {A} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6212a0418e1075d43371f49e16c794b0bb77bf1e)
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![{\displaystyle {\sqrt {3}}\ =\ {\sqrt {3\cdot {\frac {49}{49}}}}\ =\ {\sqrt {\frac {147}{49}}}\ =\ {\sqrt {\frac {144^{^{+}}}{49\ }}}\ =\ {\frac {12}{7}}^{^{+}}\ \qquad \qquad \qquad \qquad \qquad \ \mathbb {A} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f7961f456f43fd00d5b9b2fb58d32127c32debb)
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![{\displaystyle {\sqrt {2}}\ =\ {\sqrt {2\cdot {\frac {25}{25}}}}\ =\ \ {\sqrt {\frac {50}{25}}}\ \ =\ {\sqrt {\frac {49^{^{+}}}{25\ \ }}}\ \ =\ {\frac {7}{5}}^{^{+}}\ \qquad \qquad \qquad \qquad \qquad \quad \mathbb {A} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b73f17d4331e243895ce701b5e59b214cde4bc49)
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![{\displaystyle {\sqrt {3}}\ =\ {\sqrt {3\cdot {\frac {16}{16}}}}\ =\ {\sqrt {\frac {48}{16}}}\ \ =\ \ {\sqrt {\frac {49^{^{-}}}{16\ \ }}}\ \ =\ {\frac {7}{4}}^{^{-}}\ \qquad \qquad \qquad \qquad \qquad \quad \mathbb {A} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a37112b21a5bb35ee3ec765c083a89b73c57e370)
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![{\displaystyle {\sqrt {2}}\ \in \ \left({\frac {7}{5}}\ ,\ {\frac {10}{7}}\right)\qquad =>\qquad {\sqrt {2}}\ \simeq \ {\frac {{\tfrac {7}{5}}+{\tfrac {10}{7}}}{2}}\ =\ \ {\frac {99}{70}}\qquad \qquad \qquad \quad \mathbb {A} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b12ab466c6d66e9727645e0044f526c946295ea)
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![{\displaystyle {\sqrt {3}}\ \in \ \left({\frac {12}{7}}\ ,\ {\frac {7}{4}}\right)\qquad =>\qquad {\sqrt {3}}\ \simeq \ {\frac {{\tfrac {7}{4}}+{\tfrac {12}{7}}}{2}}\ =\ {\frac {97}{8\cdot 7}}\ \qquad \qquad \qquad \ \mathbb {A} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73d3f30f2cfc9d028e457dff4d2272cf6ce77beb)
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![{\displaystyle {\sqrt {2}}\ =\ 1.41^{^{+}}\ =\ {\frac {1.40+1.42^{^{+}}}{2}}\ =\ {\frac {1{\tfrac {2}{5}}\ +\ 1{\tfrac {3}{7}}^{^{-}}}{2}}\ =\ \ {\frac {99}{70}}^{^{-}}\ \qquad \qquad \qquad \ \mathbb {A} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3791b2925cbc9a776d4c9ca8f3247354bca5588)
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![{\displaystyle {\sqrt {3}}\ =\ 1.73^{^{+}}\ =\ {\frac {1.71^{^{+}}+1.75}{2}}\ =\ {\frac {1{\tfrac {5}{7}}^{^{-}}+\ 1{\tfrac {3}{4}}}{2}}\ \ =\ {\frac {\ 97^{^{-}}}{8\cdot 7}}\ \qquad \qquad \qquad \ \mathbb {A} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29a9a79365c8589f3bd46d81e114399b2c320059)
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![{\displaystyle {\frac {1}{\sqrt {2}}}\ =\ 0.\ 707^{^{+}}\ =\ 0.\ (70)^{^{+}}\ =\ {\frac {70}{99}}^{^{+}}\ =>\ {\sqrt {2}}\ =\ \ {\frac {99}{70}}^{^{-}}\ \qquad \qquad \qquad \ \mathbb {A} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c596b37a7c2f4f6031ee314b0cd296ce67e21c)
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![{\displaystyle {\color {white}.}\ \gamma \ =\ {\frac {\ \ 1^{^{-}}}{\sqrt {3}}}\ =\ {\frac {8\cdot 7}{97}}^{^{-}}\qquad \qquad \qquad \mathbb {T} =\mathbb {A} _{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2956ca93c253c569f19ff4fdb71dd97f968621d5)
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![{\displaystyle \pi ^{2}\ \simeq \ {\Bigg (}3{\frac {1}{7}}{\Bigg )}^{2}=\ 3^{2}+2\cdot 3\cdot {\tfrac {1}{7}}+{\Big (}{\tfrac {1}{7}}{\Big )}^{2}=\ 9{\frac {6}{7}}^{^{+}}\ =\ \ {\frac {69}{7}}^{^{+}}\ \qquad \qquad \quad \mathbb {T} =\mathbb {A} _{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83333a1da4e1edd7dbd0616c90d657822ad63d57)
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![{\displaystyle {\frac {e}{\pi }}\ \simeq \ {\frac {19}{7}}{\Bigg /}{\frac {22}{7}}\ =\ {\frac {19}{22}}\ \simeq \ {\frac {19-1}{22-1}}\ =\ {\frac {18}{21}}\ =\ {\frac {6\cdot 3}{7\cdot 3}}\ =\ {\frac {6}{7}}\qquad \qquad \quad \mathbb {T} =\mathbb {A} _{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a31497393e26029faf7177b8f59866f885b3992)
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Algebraic Approximations
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![{\displaystyle \pi \quad \simeq \quad {\sqrt {10}}\ \quad \simeq \quad \ {\sqrt[{3}]{31}}\ \quad \simeq \quad \ {\sqrt {2}}\ +\ {\sqrt {3}}\ \quad \simeq \quad \ 3\ +\ {\frac {\sqrt {2}}{10}}\ +\ {\frac {\sqrt {3}}{10^{^{4}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a94ece76f40cdfe5917a17e3b8a9750dbf5208f6)
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![{\displaystyle e\ =\ \varphi \ +\ 1.1^{^{+}}\ \simeq \ \left({\frac {1}{\sqrt {2}}}\ +\ {\frac {1}{\sqrt {3}}}\right)^{4}\ \simeq \ 1+{\sqrt {3}}\ \qquad \quad ;\quad \qquad \ \gamma \ \simeq \ {\frac {1}{\sqrt {3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8010230b27c592d0b5ae30b7a5b9f47faa1bf93)
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Transcendental Approximations
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![{\displaystyle e^{\pi }\ \simeq \ 20\ +\ \pi \ -\ {\tfrac {1}{1111}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5689cc44e70977eec646a5c1b530c8f7496a83b9)
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![{\displaystyle \pi \ \simeq \ {\sqrt[{9}]{10\cdot e^{8}}}\ \simeq \ e^{^{^{\displaystyle }{\sqrt[{3}]{\tfrac {3}{2}}}}}\ \simeq \ {\frac {5\ +\ {\sqrt[{4}]{e}}}{2\quad }}\ \simeq \ {\frac {2\,e^{^{\gamma }}\ +\ 9}{\quad 4}}\ \simeq \ {\frac {2\,e}{\sqrt {3}}}\ \simeq \ 2\,e\,\gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c1b88510d6a7abf2cafe91331edd7cdc3c063e)
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Other Identities & Approximations
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![{\displaystyle \varphi \ ={\frac {1+{\sqrt {5}}}{2}}\qquad =>\qquad \varphi ^{2}=\varphi \ +1\ \qquad \ =>\ \quad \ {\color {white}.}{\begin{aligned}{\tfrac {1}{\varphi }}\,\ +\,\ {\tfrac {1}{\varphi \ +\ 1}}\ =\ 1\\\\{\tfrac {1}{\varphi }}\,\ +\,\ {\tfrac {1}{\ \varphi ^{{\color {white}.}2^{\color {white}1}}}}\ \ =\ 1\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c94e56cc61f6d34fb7813e27e5f1f89f02994759)
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![{\displaystyle {\begin{aligned}e=2.\ 718^{^{+}}\\\varphi =1.\ 618^{^{+}}\end{aligned}}\qquad =>\qquad {\begin{aligned}&e\ \simeq \ \varphi \ +1.1\\\\&e^{\ }\simeq \ \varphi ^{2}+{\tfrac {1}{10}}\end{aligned}}\qquad =>\qquad {\begin{aligned}&{\tfrac {1}{\mathbf {e} }}\,\ +\,\ {\tfrac {1}{\varphi }}\,\ +{\tfrac {1}{71}}\ \simeq \ 1\\\\&{\tfrac {1}{\mathbf {e} }}\,\ +\,\ {\tfrac {1}{\sqrt {\mathbf {e} }}}+{\tfrac {1}{39}}\ \simeq \ 1\\\\&{\tfrac {1}{\sqrt {\mathbf {e} }}}+{\tfrac {1}{\varphi ^{{\color {white}.}2^{\color {white}1}}}}+{\tfrac {1}{87}}\ \simeq \ 1\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/004a445ef7662647ea8064fc8c3270bb1e36580a)
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![{\displaystyle {\begin{aligned}{\frac {1}{\sqrt {2}}}\ =\ 0.\ 707^{^{+}}\ =\ {\frac {700\ +\ 7^{^{+}}}{10^{3}}}\\\\\ln 2\ =\ 0.\ 693^{^{+}}\ =\ {\frac {700\ -\ 7^{^{-}}}{10^{3}}}\end{aligned}}\qquad \qquad \qquad =>\qquad \ \ln 2\ +\ {\frac {1}{\sqrt {2}}}\ =\ {\frac {7}{5}}^{^{+}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a1edd9d785b1c20a1ee1aa5daae3859c722780c)
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