# 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 a1,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

$10\uparrow \uparrow 2=10^{10}=10000000000$ #### 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. $1[{\text{Right associative tetration}}]b=1[{\text{Left associative tetration}}]b=1[{\text{Mixed tetration}}]b=1$ 