User:Etewilak/sandbox

Sunday, 8 August 2241

ראשון ל׳ באב ו׳״א

May a total solar eclipse be visible from Jerusalem on the date mentioned above.

See https://eclipse.gsfc.nasa.gov/SEsearch/SEsearchmap.php?Ecl=22410808 for the map of the solar eclipse of 8 August 2241.

Eastern Arabic numerals

	0	1	2	3	4	5	6	7	8	9
Persian	۰	۱	۲	۳	۴	۵	۶	۷	۸	۹
Urdu	۰	۱	۲	۳	۴	۵	۶	۷	۸	۹
Sindhi	۰	۱	۲	۳	۴	۵	۶	۷	۸	۹

Hebrew calendar

I’ve been longing for the arrival of a 384-day Hebrew year

The Hebrew year 5782 will be the first 384-day Hebrew year since 5755.

Formulas

The real root of the cubic equation $x^{3}+2x^{2}+10x-20=0$ is given as follows:

$x={\dfrac {{\sqrt[{3}]{352+6{\sqrt {3930}}}}+{\sqrt[{3}]{352-6{\sqrt {3930}}}}-2}{6}}\approx 1.368808108$

It can also be expressed in terms of hyperbolic sine and its inverse:

$x={\dfrac {-2+2{\sqrt {26}}\sinh {\dfrac {\sinh ^{-1}{\dfrac {88{\sqrt {26}}}{169}}}{3}}}{3}}$

Plastic number

The plastic number, denoted by $\rho$, is the real root of the cubic equation $x^{3}-x-1=0$.

$\rho ={\dfrac {{\sqrt[{3}]{108+12{\sqrt {69}}}}+{\sqrt[{3}]{108-12{\sqrt {69}}}}}{6}}\approx 1.324717957$

It can also be expressed in terms of hyperbolic cosine and its inverse:

$\rho ={\dfrac {2{\sqrt {3}}\cosh {\dfrac {\cosh ^{-1}{\dfrac {3{\sqrt {3}}}{2}}}{3}}}{3}}$

Its algebraic conjuagates are $A\pm Bi$, where

$A={\frac {-{\sqrt[{3}]{108+12{\sqrt {69}}}}-{\sqrt[{3}]{108-12{\sqrt {69}}}}}{12}}\approx -0.662358979$

$B={\frac {{\sqrt[{3}]{12{\sqrt {3}}+4{\sqrt {23}}}}-{\sqrt[{3}]{12{\sqrt {3}}-4{\sqrt {23}}}}}{4}}\approx 0.562279512$

Each complex conjugate has an absolute value of

${\frac {\sqrt {6{\sqrt[{3}]{100-12{\sqrt {69}}}}+6{\sqrt[{3}]{100+12{\sqrt {69}}}}-12}}{6}}\approx 0.868836962$

Supergolden ratio

The supergolden ratio, denoted by $\psi$, is the real root of the cubic equation $x^{3}-x^{2}-1=0$.

$\psi ={\frac {2+{\sqrt[{3}]{116+12{\sqrt {93}}}}+{\sqrt[{3}]{116-12{\sqrt {93}}}}}{6}}\approx 1.465571232$

It can also be expressed in terms of hyperbolic cosine and its inverse:

$\psi ={\dfrac {1+2\cosh {\dfrac {\cosh ^{-1}{\dfrac {29}{2}}}{3}}}{3}}$

Its algebraic conjugates are $A\pm Bi$, where

$A={\frac {4-{\sqrt[{3}]{116+12{\sqrt {93}}}}-{\sqrt[{3}]{116-12{\sqrt {93}}}}}{12}}\approx -0.232785616$

$B={\frac {{\sqrt[{3}]{348{\sqrt {3}}+108{\sqrt {31}}}}-{\sqrt[{3}]{348{\sqrt {3}}-108{\sqrt {31}}}}}{12}}\approx 0.792551993$

Each complex conjugate has an absolute value of

${\frac {\sqrt {6{\sqrt[{3}]{108+12{\sqrt {93}}}}+6{\sqrt[{3}]{108-12{\sqrt {93}}}}}}{6}}\approx 0.826031358$

Tribonacci constant

The tribonacci constant is the real root of the cubic equation $x^{3}-x^{2}-x-1=0$.

$x={\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}\approx 1.839286755$

It can also be expressed in terms of hyperbolic cosine and its inverse:

$x={\dfrac {1+4\cosh {\dfrac {\cosh ^{-1}{\dfrac {19}{8}}}{3}}}{3}}$

Its algebraic conjugates are $A\pm Bi$, where

$A={\frac {2-{\sqrt[{3}]{19+3{\sqrt {33}}}}-{\sqrt[{3}]{19-3{\sqrt {33}}}}}{6}}\approx -0.419643378$

$B={\frac {{\sqrt[{3}]{57{\sqrt {3}}+27{\sqrt {11}}}}-{\sqrt[{3}]{57{\sqrt {3}}-27{\sqrt {11}}}}}{6}}\approx 0.606290729$

Each complex conjugate has an absolute value of

${\frac {\sqrt {3{\sqrt[{3}]{17+3{\sqrt {33}}}}+3{\sqrt[{3}]{17-3{\sqrt {33}}}}-3}}{3}}\approx 0.737352706$