User:GX, May 1971/Math/Arithmetic

Basic Operations

${\scriptstyle \mathbf {Incrementation} }\ {\xrightarrow {\text{ Repeat }}}\ {\scriptstyle \mathbf {Addition} }\ {\xrightarrow {\text{ Repeat }}}\ {\scriptstyle \mathbf {Multiplication} }\ {\xrightarrow {\text{ Repeat }}}\ {\scriptstyle \mathbf {Exponentiation} }$

${\scriptstyle \mathbf {Exponentiation} }\ {\xrightarrow {\text{ Repeat }}}\ {\scriptstyle \mathbf {Tetration} }$

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$\scriptstyle \mathbf {0.}$

$\scriptstyle \mathbf {Incrementation}$

$\scriptstyle \mathbf {{_{^{+}}}{_{^{+}}}}$

$\scriptstyle \mathbf {Decrementation}$

$\scriptstyle \mathbf {{_{^{-}}}{_{^{-}}}}$

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$\scriptstyle \mathbf {1.}$

$\scriptstyle \mathbf {Addition}$

$\scriptstyle \mathbf {+}$

$\scriptstyle \mathbf {Subtraction}$

$\scriptstyle \mathbf {-}$

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$\scriptstyle \mathbf {2.}$

$\scriptstyle \mathbf {Multiplication}$

$\scriptstyle \mathbf {\times }$

$\scriptstyle \mathbf {Division}$

$\scriptstyle \mathbf {\div }$

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$\scriptstyle \mathbf {3.}$

$\scriptstyle \mathbf {Exponentiation}$

$\scriptstyle {a^{n}\ =\ b}$

${\begin{aligned}&\scriptstyle \mathbf {Radicals} \\&\scriptstyle \mathbf {Logarithms} \end{aligned}}$

${\begin{aligned}&\scriptstyle {a\ =\ {\sqrt[{n}]{b}}}\\&\scriptstyle {n\ =\ \log _{a}^{b}}\end{aligned}}$

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$\scriptstyle \mathbf {4.}$

$\scriptstyle \mathbf {Tetration}$

$\scriptstyle {^{n}a\ =\ b}$

${\begin{aligned}&\scriptstyle \mathbf {Super-Radicals} \\&\scriptstyle \mathbf {Super-Logarithms} \end{aligned}}$

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$a\ +\ b\ =\ b\ +\ a\ \qquad \quad \ =>\qquad \qquad \quad \ {\scriptstyle \mathbf {Subtraction} }$

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$a\ \ {_{^{\times }}}\ \ b\ =\ b\ \ {_{^{\times }}}\ \ a\ \qquad \quad \ =>\qquad \qquad \quad \ {\scriptstyle \mathbf {Division} }$

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$a\ \uparrow \ {\color {white}.}b\ \neq \ b\ \uparrow \ a\ \qquad \quad \ =>\qquad \qquad \quad \ {\scriptstyle \mathbf {Radicals\ \neq \ Logarithms} }$

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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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Basic Sets

Basic Notations

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$\mathbb {X} ^{*}\ =\mathbb {X} _{*}\ =\mathbb {X} \smallsetminus {\Big \{}0{\Big \}}\qquad \qquad \quad ;\quad \mathbb {X} ^{^{+}}=\mathbb {X} _{_{+}}={\Big \{}\ x\in \mathbb {X} \ {\Big |}\ x\geqslant 0\ {\Big \}}$

$\mathbb {X} ^{**}=\mathbb {X} _{**}=\mathbb {X} \smallsetminus {\Big \{}0,1{\Big \}}\qquad \qquad ;\quad \mathbb {X} ^{^{-}}=\mathbb {X} _{_{-}}={\Big \{}\ x\in \mathbb {X} \ {\Big |}\ x\leqslant 0\ {\Big \}}$

$\mathbb {X} _{\mathbb {Y} }=\mathbb {Y} _{\mathbb {X} }=\mathbb {X} \ \cap \ \mathbb {Y} =\mathbb {Y} \ \cap \ \mathbb {X} \ \quad ;\quad {\overline {\mathbb {X} }}\ \ =\mathbb {X} \ \cup \ {\Big \{}\pm \infty {\Big \}}$

$\mathbb {X} ^{\text{n}}=\underbrace {{\color {white}.}\mathbb {X} \times \mathbb {X} \times ...\times \mathbb {X} {\color {white}.}} _{\text{n times}}\ =\ {\Big \{}\ (x_{_{1}},x_{_{2}},...,x_{n})\ {\Big |}\ x_{_{\text{k}}}\in \mathbb {X} \ ;\ k\ \in \ {\overline {1...n}}\ {\Big \}}$

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Integers. Rationals. Radicals

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$\mathbb {N} +\mathbb {N} \ =\ \mathbb {N} \cdot \mathbb {N} \ =\ \mathbb {N} \quad ;\qquad \mathbb {N} -\mathbb {N} \ =\ \mathbb {Z} \qquad \quad ;\quad \qquad {\frac {\mathbb {Z} }{\mathbb {Z} }}\ =\ {\overline {\mathbb {Q} }}$

$\mathbb {N} \ =\ {\Big (}\ \mathbb {N} ,\ +,\ \times \ {\Big )}\qquad ;\qquad {\overline {\mathbb {Q} }}\ =\ {\Big (}\ \mathbb {N} ,\ \pm ,\ \times ,\ \div \ {\Big )}$

$\mathbb {Z} \ =\ {\Big (}\ \mathbb {N} ,\ \pm ,\ \times \ {\Big )}\qquad ;\qquad {\widehat {\overline {\mathbb {V} }}}\ =\ {\Big (}\ \mathbb {N} ,\ \pm ,\ \times ,\ \div ,\ {\sqrt[{^{n}}]{\ }}\ {\Big |}\ n\in \mathbb {N} ^{*}\ {\Big )}$

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Algebraics & Transcendentals

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${\widehat {\mathbb {A} }}_{n}=\ \left\{\ z\ \left|\ \exists \ P_{n}(z)=\sum _{k\ =\ 0}^{n}{a_{_{\text{k}}}z^{k}}=0\ ;\quad \ {\begin{aligned}&a_{n}\neq \ 0\\&a_{_{\text{k}}}\in \ \mathbb {Z} \end{aligned}}\quad ,\ k\ \in \ {\overline {0...n}}\ \right.\right\}$

${\widehat {\mathbb {A} }}_{1}=\ \mathbb {Q} \quad ;\qquad \ {\widehat {\mathbb {A} }}_{\infty }=\ \mathbb {C} \quad ;\qquad \ {\begin{aligned}&{\widehat {\mathbb {A} }}_{n}\subset \ {\widehat {\mathbb {A} }}_{n+1}\qquad \qquad ,\quad n\in \mathbb {N} ^{*}\\&\mathbb {A} _{n}=\ {\widehat {\mathbb {A} }}_{n}\ \smallsetminus \ {\widehat {\mathbb {A} }}_{n-1}\quad ,\quad n\in \mathbb {N} ^{**}\end{aligned}}$

$\mathbb {A} _{1}=\ \mathbb {Q} \quad ;\qquad \ \mathbb {A} _{\infty }=\ \mathbb {T} \quad ;\qquad \ \mathbb {A} \ \ =\bigcup _{{\text{n}}\ \in \ \mathbb {N} ^{*}}\mathbb {A} _{n}\quad {\Bigg |}\ {\begin{aligned}\mathbb {V} \ =\ {\widehat {\mathbb {V} }}\smallsetminus \mathbb {Q} \\\mathbb {B} \ =\ \mathbb {A} \smallsetminus {\widehat {\mathbb {V} }}\end{aligned}}$

$\mathbb {V} _{n}=\ \mathbb {V} \ \ \cap \ \mathbb {A} _{n}\qquad \qquad \qquad \qquad \mathbb {V} \ \ =\bigcup _{{\text{n}}\ \in \ \mathbb {N} ^{*}}\mathbb {V} _{n}$

$\mathbb {B} _{n}=\ \mathbb {B} \ \ \cap \ \mathbb {A} _{n}\qquad \qquad \qquad \qquad {\color {white}.}\mathbb {B} \ \ =\bigcup _{{\text{n}}\ \in \ \mathbb {N} ^{*}}\mathbb {B} _{n}$

$\mathbb {A} _{n}=\ \mathbb {V} _{n}\cup \ \mathbb {B} _{n}\quad ,\quad n\in \mathbb {N} ^{**}\quad ;\quad {\color {white}.}\mathbb {V} _{1}=\ \mathbb {B} _{1}=\ \mathbb {B} _{2}=\ \mathbb {B} _{3}=\ \mathbb {B} _{4}=\varnothing$

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Reals & Complex

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$\mathbb {R} \rightarrow {\begin{cases}\mathbb {A} _{1}=\mathbb {Q} \rightarrow {\begin{cases}\mathbb {Z} \rightarrow {\begin{cases}\mathbb {Z} _{_{+}}=\mathbb {N} \rightarrow \mathbb {P} \\\mathbb {Z} _{_{-}}\end{cases}}\\\mathbb {F} \end{cases}}\\\\\ \qquad \ \mathbb {I} \rightarrow {\begin{cases}\mathbb {I} _{\mathbb {T} }\ =\ \mathbb {R} _{\mathbb {T} }\ =\ \mathbb {T} _{\mathbb {R} }\\\\\mathbb {I} _{\mathbb {A} }\rightarrow {\begin{cases}\mathbb {I} _{\mathbb {V} }\ =\ \mathbb {R} _{\mathbb {V} }\ =\ \mathbb {V} _{\mathbb {R} }\\\mathbb {I} _{\mathbb {B} }\ =\ \mathbb {R} _{\mathbb {B} }\ =\ \mathbb {B} _{\mathbb {R} }\end{cases}}\end{cases}}\end{cases}}$

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$\mathbb {C} \ =\ \mathbb {R} ^{2}\ =\ \mathbb {R} \times \mathbb {R} \ =\ \mathbb {R} +i\cdot \mathbb {R} \ =\ \mathbb {A} \ \cup \ \mathbb {T} \ =\ \mathbb {Q} \ \cup \ \mathbb {V} \ \cup \ \mathbb {B} \ \cup \ \mathbb {T}$

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$\mathbb {C} \rightarrow {\begin{cases}\mathbb {A} \rightarrow {\begin{cases}\mathbb {Q} \qquad \mathbb {A} _{1}\\\mathbb {V} \qquad {\sqrt {\ }}\quad \ {\color {white}.}{\text{radicals}}\\\mathbb {B} \qquad {\text{Bring radicals}}\end{cases}}\\\\\mathbb {A} _{\infty }=\ \mathbb {T} \end{cases}}$

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Basic Irrationals

Basic Constants

The Basic Circle Constant

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$e\ =\ 2.\ 72^{^{-}}$

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 :

$e^{\mathcal {Z}}\ =\ e^{x\ +\ iy}\ =\ e^{x}\cdot e^{iy}\ =\ e^{x}\cdot (\cos {y}\ +\ i\cdot \sin {y})$

Transcendental, and therefore irrational, since the exponential function generates a transcendental
polynomial :

$e^{x}\ =\sum _{n=0}^{\infty }\ {\frac {x^{n}}{n!}}\qquad ,\qquad {\frac {1}{n!}}\in \mathbb {Q} \qquad ,\qquad \deg(e^{x})=\infty$

whereas algebraics are characterized by polynomial equations of finite degree over the rationals.

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The Basic Factorial Constant

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$\pi \ =\ \Gamma ^{2}\left({\tfrac {1}{2}}\right)\ =\ 4\cdot {\Big (}{\tfrac {1}{2}}!{\Big )}^{2}$

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

Synoptic Table

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${\dot {\mathbf {I} }}\quad \ \simeq \quad \mathbf {Q} _{_{\tfrac {1}{10}}}\qquad \qquad \qquad \qquad \quad \ \simeq \qquad \quad \mathbf {Q} _{_{\tfrac {1}{7}}}\ \qquad \quad \simeq \quad \ \mathbf {Q}$

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$e^{\pi }\ \ =23.\ 14^{^{+}}\qquad \qquad \qquad \qquad \quad \simeq \quad 23{\tfrac {1}{7}}\ =\ {\tfrac {162}{7}}\qquad \qquad \qquad \qquad \mathbb {T}$

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$\pi ^{2}\ \ =\ 9.\ 87^{^{-}}\ \qquad \qquad \qquad \qquad \quad \simeq \quad \ 9{\tfrac {6}{7}}\ =\ \ {\tfrac {69}{7}}\qquad \qquad \qquad \qquad \quad \mathbb {T}$

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$\pi \quad =\ 3.\ 14\ 16^{^{-}}\qquad \qquad \qquad \qquad \simeq \quad \ 3{\tfrac {1}{7}}\ =\ \ {\tfrac {22}{7}}\qquad \simeq \quad {\tfrac {3{\color {white}.}55}{11{\color {white}.}3}}\qquad \quad \ \mathbb {T}$

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$e\quad =\ 2.\ 7\ 1828\ 1828\ 45\ 90\ 45^{^{+}}\ \quad \simeq \quad \ 2{\tfrac {5}{7}}\ =\ \ {\tfrac {19}{7}}\qquad \qquad \qquad \qquad \quad \ \mathbb {T}$

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${\sqrt {3}}\ =1.\ 73\ 20\ 50\ 80^{^{+}}\ {\color {white}.}\qquad \qquad \quad \simeq \quad \ 1{\tfrac {5}{7}}\ =\ \ {\tfrac {12}{7}}\qquad \simeq \quad {\tfrac {97}{8\times 7}}\qquad \quad \mathbb {A} _{2}$

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${\sqrt {2}}\ =1.\ 41\ 42\ 135^{^{+}}\ \qquad \qquad \qquad \simeq \quad \ 1{\tfrac {3}{7}}\ =\ \ {\tfrac {10}{7}}\qquad \simeq \quad \ {\tfrac {99}{70}}\qquad \quad \ \mathbb {A} _{2}$

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${\frac {e}{\pi }}\ \ =\ 0.\ 865^{^{+}}\qquad \qquad \qquad \qquad \quad \simeq \qquad \qquad \quad {\tfrac {6}{7}}\qquad \qquad \qquad \qquad \quad \ \mathbb {T}$

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$\gamma \ \ =\ 0.\ 577^{^{+}}\qquad \qquad \qquad \qquad \quad \simeq \qquad \qquad \quad {\color {white}.}{\tfrac {4}{7}}\qquad \simeq \quad {\tfrac {8\times 7}{97}}\qquad \quad \ \mathbb {T}$

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$\Omega _{1}=\ 0.\ 567^{^{+}}\qquad \qquad \qquad \qquad \quad \simeq \qquad \qquad \quad {\color {white}.}{\tfrac {4}{7}}\qquad \qquad \qquad \qquad \quad \ \mathbb {T}$

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As Multiples of 1/7

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$e^{\pi }\ \ =\ 23{\frac {1}{7}}^{^{-}}=\ {\frac {162}{7}}^{^{-}}\qquad \qquad \qquad \qquad \qquad \qquad \ \pi ^{2}\ =\ 9{\frac {6}{7}}^{^{+}}=\ {\frac {69}{7}}^{^{+}}$

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$\pi \quad =\ \ 3{\frac {1}{7}}^{^{-}}\ =\ \ {\frac {22}{7}}^{^{-}}\qquad \qquad \qquad \qquad \qquad \qquad \quad e\ =\ 2{\frac {5}{7}}^{^{+}}=\ {\frac {19}{7}}^{^{+}}$

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${\sqrt {3}}\ =\ \ 1{\frac {5}{7}}^{^{+}}\ =\ \ {\frac {12}{7}}^{^{+}}\qquad \qquad \qquad \qquad \qquad \qquad {\sqrt {2}}\ =\ 1{\frac {3}{7}}^{^{-}}=\ {\frac {10}{7}}^{^{-}}$

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${\frac {e}{\pi }}\quad =\quad {\frac {6}{7}}^{^{+}}\qquad \qquad \qquad \ \gamma \ =\ {\frac {4}{7}}^{^{+}}\qquad \qquad \qquad \ \Omega _{1}\ =\ \ {\frac {4}{7}}^{^{-}}$

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Clarifications

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${\sqrt {2}}\ =\ {\sqrt {2\cdot {\frac {49}{49}}}}\ =\ \ {\sqrt {\frac {98}{49}}}\ \ =\ {\sqrt {\frac {100^{^{-}}}{49\ }}}\ =\ {\frac {10}{7}}^{^{-}}\ \qquad \qquad \qquad \qquad \qquad \ \mathbb {A} _{2}$

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${\sqrt {3}}\ =\ {\sqrt {3\cdot {\frac {49}{49}}}}\ =\ {\sqrt {\frac {147}{49}}}\ =\ {\sqrt {\frac {144^{^{+}}}{49\ }}}\ =\ {\frac {12}{7}}^{^{+}}\ \qquad \qquad \qquad \qquad \qquad \ \mathbb {A} _{2}$

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${\sqrt {2}}\ =\ {\sqrt {2\cdot {\frac {25}{25}}}}\ =\ \ {\sqrt {\frac {50}{25}}}\ \ =\ {\sqrt {\frac {49^{^{+}}}{25\ \ }}}\ \ =\ {\frac {7}{5}}^{^{+}}\ \qquad \qquad \qquad \qquad \qquad \quad \mathbb {A} _{2}$

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${\sqrt {3}}\ =\ {\sqrt {3\cdot {\frac {16}{16}}}}\ =\ {\sqrt {\frac {48}{16}}}\ \ =\ \ {\sqrt {\frac {49^{^{-}}}{16\ \ }}}\ \ =\ {\frac {7}{4}}^{^{-}}\ \qquad \qquad \qquad \qquad \qquad \quad \mathbb {A} _{2}$

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${\sqrt {2}}\ \in \ \left({\frac {7}{5}}\ ,\ {\frac {10}{7}}\right)\qquad =>\qquad {\sqrt {2}}\ \simeq \ {\frac {{\tfrac {7}{5}}+{\tfrac {10}{7}}}{2}}\ =\ \ {\frac {99}{70}}\qquad \qquad \qquad \quad \mathbb {A} _{2}$

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${\sqrt {3}}\ \in \ \left({\frac {12}{7}}\ ,\ {\frac {7}{4}}\right)\qquad =>\qquad {\sqrt {3}}\ \simeq \ {\frac {{\tfrac {7}{4}}+{\tfrac {12}{7}}}{2}}\ =\ {\frac {97}{8\cdot 7}}\ \qquad \qquad \qquad \ \mathbb {A} _{2}$

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${\sqrt {2}}\ =\ 1.41^{^{+}}\ =\ {\frac {1.40+1.42^{^{+}}}{2}}\ =\ {\frac {1{\tfrac {2}{5}}\ +\ 1{\tfrac {3}{7}}^{^{-}}}{2}}\ =\ \ {\frac {99}{70}}^{^{-}}\ \qquad \qquad \qquad \ \mathbb {A} _{2}$

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${\sqrt {3}}\ =\ 1.73^{^{+}}\ =\ {\frac {1.71^{^{+}}+1.75}{2}}\ =\ {\frac {1{\tfrac {5}{7}}^{^{-}}+\ 1{\tfrac {3}{4}}}{2}}\ \ =\ {\frac {\ 97^{^{-}}}{8\cdot 7}}\ \qquad \qquad \qquad \ \mathbb {A} _{2}$

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${\frac {1}{\sqrt {2}}}\ =\ 0.\ 707^{^{+}}\ =\ 0.\ (70)^{^{+}}\ =\ {\frac {70}{99}}^{^{+}}\ =>\ {\sqrt {2}}\ =\ \ {\frac {99}{70}}^{^{-}}\ \qquad \qquad \qquad \ \mathbb {A} _{2}$

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${\color {white}.}\ \gamma \ =\ {\frac {\ \ 1^{^{-}}}{\sqrt {3}}}\ =\ {\frac {8\cdot 7}{97}}^{^{-}}\qquad \qquad \qquad \mathbb {T} =\mathbb {A} _{\infty }$

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$\pi ^{2}\ \simeq \ {\Bigg (}3{\frac {1}{7}}{\Bigg )}^{2}=\ 3^{2}+2\cdot 3\cdot {\tfrac {1}{7}}+{\Big (}{\tfrac {1}{7}}{\Big )}^{2}=\ 9{\frac {6}{7}}^{^{+}}\ =\ \ {\frac {69}{7}}^{^{+}}\ \qquad \qquad \quad \mathbb {T} =\mathbb {A} _{\infty }$

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${\frac {e}{\pi }}\ \simeq \ {\frac {19}{7}}{\Bigg /}{\frac {22}{7}}\ =\ {\frac {19}{22}}\ \simeq \ {\frac {19-1}{22-1}}\ =\ {\frac {18}{21}}\ =\ {\frac {6\cdot 3}{7\cdot 3}}\ =\ {\frac {6}{7}}\qquad \qquad \quad \mathbb {T} =\mathbb {A} _{\infty }$

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Algebraic Approximations

$\pi \quad \simeq \quad {\sqrt {10}}\ \quad \simeq \quad \ {\sqrt[{3}]{31}}\ \quad \simeq \quad \ {\sqrt {2}}\ +\ {\sqrt {3}}\ \quad \simeq \quad \ 3\ +\ {\frac {\sqrt {2}}{10}}\ +\ {\frac {\sqrt {3}}{10^{^{4}}}}$

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$e\ =\ \varphi \ +\ 1.1^{^{+}}\ \simeq \ \left({\frac {1}{\sqrt {2}}}\ +\ {\frac {1}{\sqrt {3}}}\right)^{4}\ \simeq \ 1+{\sqrt {3}}\ \qquad \quad ;\quad \qquad \ \gamma \ \simeq \ {\frac {1}{\sqrt {3}}}$

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Transcendental Approximations

$e^{\pi }\ \simeq \ 20\ +\ \pi \ -\ {\tfrac {1}{1111}}$

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$\pi \ \simeq \ {\sqrt[{9}]{10\cdot e^{8}}}\ \simeq \ e^{^{^{\displaystyle }{\sqrt[{3}]{\tfrac {3}{2}}}}}\ \simeq \ {\frac {5\ +\ {\sqrt[{4}]{e}}}{2\quad }}\ \simeq \ {\frac {2\,e^{^{\gamma }}\ +\ 9}{\quad 4}}\ \simeq \ {\frac {2\,e}{\sqrt {3}}}\ \simeq \ 2\,e\,\gamma$

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Other Identities & Approximations

$\varphi \ ={\frac {1+{\sqrt {5}}}{2}}\qquad =>\qquad \varphi ^{2}=\varphi \ +1\ \qquad \ =>\ \quad \ {\color {white}.}{\begin{aligned}{\tfrac {1}{\varphi }}\,\ +\,\ {\tfrac {1}{\varphi \ +\ 1}}\ =\ 1\\\\{\tfrac {1}{\varphi }}\,\ +\,\ {\tfrac {1}{\ \varphi ^{{\color {white}.}2^{\color {white}1}}}}\ \ =\ 1\end{aligned}}$

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${\begin{aligned}e=2.\ 718^{^{+}}\\\varphi =1.\ 618^{^{+}}\end{aligned}}\qquad =>\qquad {\begin{aligned}&e\ \simeq \ \varphi \ +1.1\\\\&e^{\ }\simeq \ \varphi ^{2}+{\tfrac {1}{10}}\end{aligned}}\qquad =>\qquad {\begin{aligned}&{\tfrac {1}{\mathbf {e} }}\,\ +\,\ {\tfrac {1}{\varphi }}\,\ +{\tfrac {1}{71}}\ \simeq \ 1\\\\&{\tfrac {1}{\mathbf {e} }}\,\ +\,\ {\tfrac {1}{\sqrt {\mathbf {e} }}}+{\tfrac {1}{39}}\ \simeq \ 1\\\\&{\tfrac {1}{\sqrt {\mathbf {e} }}}+{\tfrac {1}{\varphi ^{{\color {white}.}2^{\color {white}1}}}}+{\tfrac {1}{87}}\ \simeq \ 1\end{aligned}}$

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${\begin{aligned}{\frac {1}{\sqrt {2}}}\ =\ 0.\ 707^{^{+}}\ =\ {\frac {700\ +\ 7^{^{+}}}{10^{3}}}\\\\\ln 2\ =\ 0.\ 693^{^{+}}\ =\ {\frac {700\ -\ 7^{^{-}}}{10^{3}}}\end{aligned}}\qquad \qquad \qquad =>\qquad \ \ln 2\ +\ {\frac {1}{\sqrt {2}}}\ =\ {\frac {7}{5}}^{^{+}}$

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