User:Jacobolus/sandbox

Division by zero sources:

• Boyer, Carl Benjamin (1943). "An Early Reference to Division by Zero". American Mathematical Monthly. 50 (8): 487–491. JSTOR 2304187.
• Romig, H. G. (1924). "II. Early History of Division by Zero". Questions & Discussions. American Mathematical Monthly. 31 (8): 387–389. JSTOR 2298825.

${\begin{aligned}\left({\sqrt[{\oplus }]{\varepsilon }}\right)^{2}={\frac {1\ominus \delta \ominus \phi }{1\ominus \delta \oplus \phi }}\\[10mu]\left({\sqrt[{\oplus }]{\varepsilon }}\right)^{2}=1\ominus {\frac {S(\phi )}{S(1\ominus \delta )}}\\[10mu]C_{1/2}\left(\varepsilon \right)={\frac {S(\phi )}{C(\delta )}}\end{aligned}}$

Or as angle measures:

$\cos {\tfrac {1}{2}}\varepsilon ={\frac {\sin \phi }{\cos \delta }}$

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$x=a_{0}+\!{{} \atop {{\big |}\!}}\!{\frac {\ 1\ }{\,a_{1}}}\!{{\!{\big |}} \atop {}}+\!{{} \atop {{\big |}\!}}\!{\frac {\ 1\ }{\,a_{2}}}\!{{\!{\big |}} \atop {}}+\!{{} \atop {{\big |}\!}}\!{\frac {\ 1\ }{\,a_{3}}}\!{{\!{\big |}} \atop {}}+\!{{} \atop {{\big |}\!}}\!{\frac {\ 1\ }{\,a_{4}}}\!{{\!{\big |}} \atop {}},$

$x=a_{0}+{1 \over a_{1}+}\,{1 \over a_{2}+}\,{1 \over a_{3}+}\,{1 \over a_{4}}$

$x=a_{0}+{1 \over a_{1}}{{} \atop +}{1 \over a_{2}}{{} \atop +}{1 \over a_{3}}{{} \atop +}{1 \over a_{4}}.$