Eastern Arabic numerals
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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
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.
The real root of the cubic equation
x
3
+
2
x
2
+
10
x
−
20
=
0
{\displaystyle x^{3}+2x^{2}+10x-20=0}
x
=
352
+
6
3930
3
+
352
−
6
3930
3
−
2
6
≈
1.368808108
{\displaystyle 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
=
−
2
+
2
26
sinh
sinh
−
1
88
26
169
3
3
{\displaystyle x={\dfrac {-2+2{\sqrt {26}}\sinh {\dfrac {\sinh ^{-1}{\dfrac {88{\sqrt {26}}}{169}}}{3}}}{3}}}
The plastic number, denoted by
ρ
{\displaystyle \rho }
x
3
−
x
−
1
=
0
{\displaystyle x^{3}-x-1=0}
ρ
=
108
+
12
69
3
+
108
−
12
69
3
6
≈
1.324717957
{\displaystyle \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:
ρ
=
2
3
cosh
cosh
−
1
3
3
2
3
3
{\displaystyle \rho ={\dfrac {2{\sqrt {3}}\cosh {\dfrac {\cosh ^{-1}{\dfrac {3{\sqrt {3}}}{2}}}{3}}}{3}}}
Its algebraic conjuagates are
A
±
B
i
{\displaystyle A\pm Bi}
A
=
−
108
+
12
69
3
−
108
−
12
69
3
12
≈
−
0.662358979
{\displaystyle A={\frac {-{\sqrt[{3}]{108+12{\sqrt {69}}}}-{\sqrt[{3}]{108-12{\sqrt {69}}}}}{12}}\approx -0.662358979}
B
=
12
3
+
4
23
3
−
12
3
−
4
23
3
4
≈
0.562279512
{\displaystyle 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
6
100
−
12
69
3
+
6
100
+
12
69
3
−
12
6
≈
0.868836962
{\displaystyle {\frac {\sqrt {6{\sqrt[{3}]{100-12{\sqrt {69}}}}+6{\sqrt[{3}]{100+12{\sqrt {69}}}}-12}}{6}}\approx 0.868836962}
The supergolden ratio, denoted by
ψ
{\displaystyle \psi }
x
3
−
x
2
−
1
=
0
{\displaystyle x^{3}-x^{2}-1=0}
ψ
=
2
+
116
+
12
93
3
+
116
−
12
93
3
6
≈
1.465571232
{\displaystyle \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:
ψ
=
1
+
2
cosh
cosh
−
1
29
2
3
3
{\displaystyle \psi ={\dfrac {1+2\cosh {\dfrac {\cosh ^{-1}{\dfrac {29}{2}}}{3}}}{3}}}
Its algebraic conjugates are
A
±
B
i
{\displaystyle A\pm Bi}
A
=
4
−
116
+
12
93
3
−
116
−
12
93
3
12
≈
−
0.232785616
{\displaystyle A={\frac {4-{\sqrt[{3}]{116+12{\sqrt {93}}}}-{\sqrt[{3}]{116-12{\sqrt {93}}}}}{12}}\approx -0.232785616}
B
=
348
3
+
108
31
3
−
348
3
−
108
31
3
12
≈
0.792551993
{\displaystyle 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
6
108
+
12
93
3
+
6
108
−
12
93
3
6
≈
0.826031358
{\displaystyle {\frac {\sqrt {6{\sqrt[{3}]{108+12{\sqrt {93}}}}+6{\sqrt[{3}]{108-12{\sqrt {93}}}}}}{6}}\approx 0.826031358}
The tribonacci constant is the real root of the cubic equation
x
3
−
x
2
−
x
−
1
=
0
{\displaystyle x^{3}-x^{2}-x-1=0}
x
=
1
+
19
+
3
33
3
+
19
−
3
33
3
3
≈
1.839286755
{\displaystyle 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
=
1
+
4
cosh
cosh
−
1
19
8
3
3
{\displaystyle x={\dfrac {1+4\cosh {\dfrac {\cosh ^{-1}{\dfrac {19}{8}}}{3}}}{3}}}
Its algebraic conjugates are
A
±
B
i
{\displaystyle A\pm Bi}
A
=
2
−
19
+
3
33
3
−
19
−
3
33
3
6
≈
−
0.419643378
{\displaystyle A={\frac {2-{\sqrt[{3}]{19+3{\sqrt {33}}}}-{\sqrt[{3}]{19-3{\sqrt {33}}}}}{6}}\approx -0.419643378}
B
=
57
3
+
27
11
3
−
57
3
−
27
11
3
6
≈
0.606290729
{\displaystyle 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
3
17
+
3
33
3
+
3
17
−
3
33
3
−
3
3
≈
0.737352706
{\displaystyle {\frac {\sqrt {3{\sqrt[{3}]{17+3{\sqrt {33}}}}+3{\sqrt[{3}]{17-3{\sqrt {33}}}}-3}}{3}}\approx 0.737352706}
![{\displaystyle x={\dfrac {{\sqrt[{3}]{352+6{\sqrt {3930}}}}+{\sqrt[{3}]{352-6{\sqrt {3930}}}}-2}{6}}\approx 1.368808108}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f01554d160a902b23d1529310f6c4d3d7b392840)
![{\displaystyle \rho ={\dfrac {{\sqrt[{3}]{108+12{\sqrt {69}}}}+{\sqrt[{3}]{108-12{\sqrt {69}}}}}{6}}\approx 1.324717957}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07e74301b18192493a8fd8f131a4b657de2eabbf)
![{\displaystyle A={\frac {-{\sqrt[{3}]{108+12{\sqrt {69}}}}-{\sqrt[{3}]{108-12{\sqrt {69}}}}}{12}}\approx -0.662358979}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24869e2adc1e953a4622000822f53cbbfbf12899)
![{\displaystyle B={\frac {{\sqrt[{3}]{12{\sqrt {3}}+4{\sqrt {23}}}}-{\sqrt[{3}]{12{\sqrt {3}}-4{\sqrt {23}}}}}{4}}\approx 0.562279512}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0966f7f417a4073dc3cc576dbb741f9372bcf03f)
![{\displaystyle {\frac {\sqrt {6{\sqrt[{3}]{100-12{\sqrt {69}}}}+6{\sqrt[{3}]{100+12{\sqrt {69}}}}-12}}{6}}\approx 0.868836962}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55861dd0b21b0500ed65c0a28895fcd7259df875)
![{\displaystyle \psi ={\frac {2+{\sqrt[{3}]{116+12{\sqrt {93}}}}+{\sqrt[{3}]{116-12{\sqrt {93}}}}}{6}}\approx 1.465571232}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aec81b065441c0fba77b810f71d9f3a58ede686e)
![{\displaystyle A={\frac {4-{\sqrt[{3}]{116+12{\sqrt {93}}}}-{\sqrt[{3}]{116-12{\sqrt {93}}}}}{12}}\approx -0.232785616}](https://wikimedia.org/api/rest_v1/media/math/render/svg/063bd454329db0d54fb667fa2db9b260a3e3d48e)
![{\displaystyle B={\frac {{\sqrt[{3}]{348{\sqrt {3}}+108{\sqrt {31}}}}-{\sqrt[{3}]{348{\sqrt {3}}-108{\sqrt {31}}}}}{12}}\approx 0.792551993}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aac9894d9a057bcb6e62fac92e48f17c95491967)
![{\displaystyle {\frac {\sqrt {6{\sqrt[{3}]{108+12{\sqrt {93}}}}+6{\sqrt[{3}]{108-12{\sqrt {93}}}}}}{6}}\approx 0.826031358}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c22ad563d67dc43a54978fa4565f720a2ab97b6)
![{\displaystyle x={\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}\approx 1.839286755}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60eaf02cb37d54f3c3afca9ee831e74d50b4e102)
![{\displaystyle A={\frac {2-{\sqrt[{3}]{19+3{\sqrt {33}}}}-{\sqrt[{3}]{19-3{\sqrt {33}}}}}{6}}\approx -0.419643378}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbb22a14cde31808b46b2e98487301193d44edaf)
![{\displaystyle B={\frac {{\sqrt[{3}]{57{\sqrt {3}}+27{\sqrt {11}}}}-{\sqrt[{3}]{57{\sqrt {3}}-27{\sqrt {11}}}}}{6}}\approx 0.606290729}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd45497a34e0059aa05b87ae9f73a0c00b35ced8)
![{\displaystyle {\frac {\sqrt {3{\sqrt[{3}]{17+3{\sqrt {33}}}}+3{\sqrt[{3}]{17-3{\sqrt {33}}}}-3}}{3}}\approx 0.737352706}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abdf07aea2fc4513c166d7c112a4c1e5920528c7)