User talk:180.190.60.236/Mixed arrow notation

The table uses base-3 (trinary) form for use of hyper operators, so we even use the exponents

The total sum of it is ${\textstyle \sum _{i=1}^{n}3^{i}.}$

Number	1	2	3
Notation	↑	↓	↕

Exponents

Exponent

Notation	Equivalent to	Equality?	Decomposition
a↑b1, a↓b2, a ↕ b3	$a^{b}$	True	$a\cdot (\underbrace {a\cdot (\cdots (} _{b}a\cdot \underbrace {a)\cdots ))} _{b}$ $\underbrace {(((\cdots ((} _{b}a\cdot \underbrace {a)\cdot a)\cdots )\cdot a} _{b}$, $a^{\frac {b}{2}}\cdot a^{\frac {b}{2}}={\sqrt {a^{b}}}\cdot {\sqrt {a^{b}}}=({\sqrt {a\cdot a}})^{b}$ for $a>0$

Variable Notes

There are 3 arrangements in it

->exp(a+(2*pi*i*k)/(log(b)+2*pi*i*n)*(ln(b)+2*pi*i*n)))

->exp((log(b)+2*pi*i*n)/a)

->(log(a)+2*pi*i*k)/(log(b)+2*pi*i*n)

We use subscripts if they are more than one variable of the same letter (example: a,a, it changes to a_1,a₂)

On tetration, it is similar, but we use subscripts to avoid confusion with others and has a similar periodicity

Tetration

Tetration

Notation	Equivalent to	Equality?	Decomposition
a↑↑b1 (4), a↓↑b2 (5), a@↑b3 (6)	The power towers	True	$a^{\underbrace {a^{a^{\cdots ^{a^{a}}}}} _{b}}$
a↑↓b4 (7), a↓↓b5 (8), a@↓b6 (9)	Left but weak tetration	True	$\underbrace {((\cdots ((} _{b}a\underbrace {^{a})^{a})\cdots )^{a})^{a}} _{b}$ $a^{a^{b-1}}{\text{if}}|b|<1$Tetration if $b>0\land b<2$
a↑↕b7 (10), a↓↕b8 (11), a↕↕b9 (12)	As to exponents, except both left and right	True	$(a\updownarrow (b-1))^{a\updownarrow (b-1)}$

There are $3^{2}=9$ and $3^{3}=27$ arrangements of compositions

Examples

$10\uparrow \uparrow 2=10^{10}=10000000000$

Inverses

See Inverse functions, here

Extensions

See Extensions, here

Special cases

1. $a[{\text{Right associative tetration}}]2=a[{\text{Left associative tetration}}]2=a[{\text{Mixed tetration}}]2=a^{a}$
2. $a[{\text{Right associative tetration}}]1=a[{\text{Left associative tetration}}]1=a[{\text{Mixed tetration}}]1=a$
3. $a[{\text{Right associative tetration}}]0=a[{\text{Left associative tetration}}]0=a[{\text{Mixed tetration}}]0=\log _{a}{(a)}={\frac {log(a)}{log(a)}}=1$
4. $1[{\text{Right associative tetration}}]b=1[{\text{Left associative tetration}}]b=1[{\text{Mixed tetration}}]b=1$

Notes

$a\downarrow \downarrow b<a\uparrow \uparrow b<a\updownarrow \updownarrow b$

This seems to be a comparison of these functions, but seems to be like an ever-expanding universe

Pentation

Pentation

Notation	Equivalent to	Equality?	Decomposition
a↓↓↓b1 (13), a↑↓↓b10 (22), a↕↓↓b19 (31)			(a↓↓↓(b-1))1 (13)↓↓a5 (8), (a↑↓↓(b-1))10 (22)↑↓b4 (7), (a↕↓↓(b-1))19 (31)↕↓b6 (9)
a↓↑↓b4 (16), a↑↑↓b13 (25), a↕↑↓b22 (34)			(a↓↑↓(b-1))4 (16)↓↑a2 (5), (a↑↑↓(b-1))13 (25)↑↑a1 (4), (a↕↑↓(b-1))22 (34)↕↑a3 (6)
a↓↕↓b7 (19), a↑↕↓b16 (28), a↕↕↓b25 (37)			(a↓↕↓(b-1))7 (19)↓↕a8 (11), (a↑↕↓(b-1))16 (28)↑↕a7 (10), (a↕↕↓(b-1))25 (37)↕↕ a9 (12)
a↓↓↑b2 (14), a↑↓↑b11 (23), a↕↓↑b20 (32)			a↓↓↑5 (8)(a↓↓↑(b-1)2 (14)), a↑↓4 (7)(a↑↓↑(b-1)11 (23)), a↕↓6 (9)(a↕↓↑(b-1)20 (32))
a↓↑↑b5 (17), a↑↑↑b14 (26), a↕↑↑b23 (35)			a↑↑1 (4)(a↑↑↑(b-1)14 (26)), a↓↑2 (5)(a↓↑↑(b-1)5 (17)), a↕↑3 (6)(a↕↑↑(b-1)23 (35))
a↓↕↑b8 (20), a↓↕↑b17 (29), a↕↕↑b26 (38)			a↑↕7 (10)(a↑↕↑(b-1)17 (29)), a↓↕8 (11)(a↓↕↑(b-1)8 (20)), a↕↕9 (12)(a↕↕↑(b-1)26 (38))
a↓↓↕b3 (15), a↑↓↕b12 (24), a↕↓↕b21 (33)			(a↓↓↕(b-1)3 (15))↓↓5 (8)(a↓↓↕(b-1)3 (15)), (a↑↓↕(b-1)12 (24))↑↓4 (7)(a↑↓↕(b-1)12 (24)), (a↕↓↕(b-1)21 (33))↕↓6 (9)(a↕↓↕(b-1)21 (33))
a↓↑↕b6 (18), a↑↑↕b15 (27), a↕↑↕b24 (36)			(a↓↑↕(b-1)6 (18))↓↑2 (5)(a↓↑↕(b-1)6 (18)), (a↑↑↕(b-1)15 (27))↑↑1 (4)(a↑↑↕(b-1)15 (27)), (a↕↑↕(b-1)24 (36))↕↑3 (6)(a↕↑↕(b-1)24 (36))
a↓↕↕b9 (21), a↑↕↕b18 (30), a↕↕↕b27 (39)			(a↓↕↕(b-1)9 (21))↓↕8 (11)(a↓↕↕(b-1)9 (21)), (a↑↕↕(b-1)18 (30))↑↕7 (10)(a↑↕↕(b-1)18 (30)), (a↕↕↕(b-1)27 (39))↕↕9 (12)(a↕↕↕(b-1)27 (39))

There are $3^{4}=81$ and $3^{3^{2}}=(3^{3})^{3}=19683$ arrangements of compositions, unlike Tetration

We won't take hexation because it is 3^4 is 81, so it would take long

On special cases, Underline is multivalued

Values

Besides 1, 0 and infinity, the pentation has special values

Special cases

A special case regarding is equal to the base raised to height of tetration

Tetration

a↑↑-1=0

a↓↓-1=1/0=Error (for somewhat \frac{log(b)}{log(x)} \text{at} x = 1)

a↑↑0=a↓↓0=a↕↕0 (for $b<0$ assuming b is an integer)=1

a↑↑1=a↓↓1=a↕↕1=a

1↑↑b=1↓↓b=1↕↕b=1*

0↑↑b (for $2n<2n\pm 1$)=0↓↓b (for $n>1$)=0↕↕b=1*

0↑↑b (for $2n\pm 1<2n\pm 2$)=0↓↓b (for $n<1$)=0*

Pentation

[Work in progress]

Higher Hyper-operators

If a notation ends in down-arrow (↓), then, it is left-associative

If a notation ends in up-arrow (↑), then, it is right-associative

If a notation ends in up/down arrows (↕), then, it would be a combination of both left and right associativity, ending up in a ↕ b = (a ↕ (b-1)) ↕ (a ↕ (b-1))

Evaluation

See here

Combining operators

Is it possible to combine a lower operator and higher operator, but even same level or higher operator and lower operator, but results are different

Aside from lower, and higher operators, we have the rules for combining

If the left argument is one, then simplify it into one

If the right argument is one, then retain the number

Same level of hyper operator

Left variant

(a↑↑b)↑↑c ~ a↑↑↓3

(a↑↑b)↓↓c

(a↑↑b)↕↕c

(a↓↓b)↑↑c

(a↓↓b)↓↓c ~ a↓↓↓3

(a↓↓b)↕↕c

(a↕↕b)↑↑c

(a↕↕b)↓↓c

(a↕↕b)↕↕c ~ a↕↕↓3

Right variant

a↑↑(b↑↑c) ~ a↑↑↑3

a↑↑(b↓↓c)

a↑↑(b↕↕c)

a↓↓(b↑↑c)

a↓↓(b↓↓c) ~ a↓↓↑3

a↓↓(b↕↕c)

a↕↕(b↓↓c)

a↕↕(b↑↑c)

a↕↕(b↕↕c) ~ a↕↕↑3

Both arguments

(a↑↑b)↑↑(c↑↑d) ~ a↑↑↕3

(a↑↑b)↑↑(c↓↓d)

(a↑↑b)↑↑(c↕↕d)

(a↓↓b)↓↓(c↑↑d)

(a↓↓b)↓↓(c↓↓d) ~ a↓↓↕3

(a↓↓b)↓↓(c↕↕d)

(a↕↕b)↕↕(c↓↓d)

(a↕↕b)↕↕(c↑↑d)

(a↕↕b)↕↕(c↕↕d) ~ a↕↕↕3

Combining higher and lower hyper operator

Left variant

(a↑↑↑b)↑↑c

(a↑↑↑b)↓↓c

(a↑↑↑b)↕↕c

(a↓↓↓b)↑↑c

(a↓↓↓b)↓↓c

(a↓↓↓b)↕↕c

(a↕↕↕b)↑↑c

(a↕↕↕b)↓↓c

(a↕↕↕b)↕↕c

Right variant

a↑↑↑(b↑↑c)

a↑↑↑(b↓↓c)

a↑↑↑(b↕↕c)

a↓↓↓(b↑↑c)

a↓↓↓(b↓↓c)

a↓↓↓(b↕↕c)

a↕↕↕(b↑↑c)

a↕↕↕(b↓↓c)

a↕↕↕(b↕↕c)

Both arguments, with a middle lower

(a↑↑↑b)↑↑(c↑↑d)

(a↑↑↑b)↑↑(c↓↓d)

(a↑↑↑b)↑↑(c↕↕d)

(a↓↓↓b)↓↓(c↑↑d)

(a↓↓↓b)↓↓(c↓↓d)

(a↓↓↓b)↓↓(c↕↕d)

(a↕↕↕b)↕↕(c↑↑d)

(a↕↕↕b)↕↕(c↓↓d)

(a↕↕↕b)↕↕(c↕↕d)

Both arguments, with a middle higher

(a↑↑↑b)↑↑↑(c↑↑d)

(a↑↑↑b)↑↑↑(c↓↓d)

(a↑↑↑b)↑↑↑(c↕↕d)

(a↓↓↓b)↓↓↓(c↑↑d)

(a↓↓↓b)↓↓↓(c↓↓d)

(a↓↓↓b)↓↓↓(c↕↕d)

(a↕↕↕b)↕↕↕(c↑↑d)

(a↕↕↕b)↕↕↕(c↓↓d)

(a↕↕↕b)↕↕↕(c↕↕d)

Combining lower and higher hyper operator

Left variant

(a↑↑b)↑↑↑c

(a↑↑b)↓↓↓c

(a↑↑b)↕↕↕c

(a↓↓b)↑↑↑c

(a↓↓b)↓↓↓c

(a↓↓b)↕↕↕c

(a↕↕b)↑↑↑c

(a↕↕b)↓↓↓c

(a↕↕b)↕↕↕c

Right variant

a↑↑(b↑↑↑c)

a↑↑(b↓↓↓c)

a↑↑(b↕↕↕c)

a↓↓(b↑↑↑c)

a↓↓(b↓↓↓c)

a↓↓(b↕↕↕c)

a↕↕(b↑↑↑c)

a↕↕(b↓↓↓c)

a↕↕(b↕↕↕c)

Both arguments, with a middle lower

(a↑↑b)↑↑(c↑↑↑d)

(a↑↑b)↑↑(c↓↓↓d)

(a↑↑b)↑↑(c↕↕↕d)

(a↓↓b)↓↓(c↑↑↑d)

(a↓↓b)↓↓(c↓↓↓d)

(a↓↓b)↓↓(c↕↕↕d)

(a↕↕b)↕↕(c↑↑↑d)

(a↕↕b)↕↕(c↓↓↓d)

(a↕↕b)↕↕(c↕↕↕d)

Both arguments, with a middle upper

(a↑↑b)↑↑↑(c↑↑↑d)

(a↑↑b)↑↑↑(c↓↓↓d)

(a↑↑b)↑↑↑(c↕↕↕d)

(a↓↓b)↓↓↓(c↑↑↑d)

(a↓↓b)↓↓↓(c↓↓↓d)

(a↓↓b)↓↓↓(c↕↕↕d)

(a↕↕b)↕↕↕(c↑↑↑d)

(a↕↕b)↕↕↕(c↓↓↓d)

(a↕↕b)↕↕↕(c↕↕↕d)