The Extended Arrow Notation is a notation I coined myself to extend Up-Arrow Notation (Arrow Notation). Hope you guys like it.

## Basic Definition

This is the basic definition of Extended Arrow Notation.

a^(n)^b = a^^^...^^^b w/ n arrows (*^* is an arrow)

Easy, eh?

## Examples

10^(100)^10 = 10^^^...^^^10 w/ 100 arrows

or in full expansion:

10^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^10.

41^(11)^88 = 41^^^^^^^^^^^88

etc.

## Extensions

### Double Parenthesis

We can define an Double Parenthesis operator as a nesting of the normal definition:

a^((n))^b = a^(n^(n^(...(n^(n^(n)^n)^n)...)^n)^n)^b w/ n levels

### Triple Parenthesis

We can define triple parenthesis operator:

a^(((n)))^b = a^((n^((n^((n^((..((n^((n^((n))^n))^n))..))^n))^n))^n))^b w/ n levels

where a^((n^((n))^n))^b is a nest of the normal definition

### Multiple Parenthesis

We can define the 4 parenthesis operator as a nest of the triple parenthesis operator, the 5 parenthesis operator as a nest of the 4 parenthesis operator, etc.

The limit would be a^(((...(((n)))...)))^b

### Exponentiated Parenthesis

We can exponentiate parenthesis:

a^(^(n)^)^b = a^(((...(((n)))...)))^b w/ n parenthesis

We can also define the (^(())^) operator as a nest of the (^()^) operator, the (^((()))^) operator as a nest of the (^(())^) operator, etc.

### Multi Exponentiated Parenthesis

We can continue with double exponentiated parenthesis:

a^(^(^(n)^)^)^b = a^(^(((...(((n)))...)))^)^b w/ n parenthesis

We can continue with a^(^(^((n))^)^)^b as a nest of the previous number, and then we can have the (^(^((()))^)^) operator, etc.

We can continue with the triple exponentiating operator as a nest of parenthesis in the double exponentiating level, and then have a 4th level, 5th level, etc.

### Tetrated Parenthesis

We can tetrate parenthesis.

a^(^^(n)^^)^b = a^(^(^(^(...(^(^(^(n)^)^)^)...)^)^)^)^b w/ n levels

We can define the (^^(())^^) operator as a nest of the (^^()^^) operator, and then have (^^((()))^^), (^^(((())))^^), etc.

We can also have exponentiated tetrated parenthesis, double exponentiated tetrated parenthesis, multi exponentiated tetrated parenthesis, etc.

### Tetrated Exponentiated Tetrated Parenthesis

You did not wait that, did you?

We define:

a^(^^(^(^^(n)^^)^)^^)^b = a^(^^(^(^(^(...(^(^(^(n)^)^)^)...)^)^)^)^^)^b w/ n levels after (^^()^^)

Or something like that.

We can also have the (^^(^(^^(^(^^()^^)^)^^)^)^^) operator (using the other operators we made (for example, we can make a (^^(^(^^(^(^((()))^)^)^^)^)^^) operator)), etc.

### Tetrated Tetrated Parenthesis

We define:

a^(^^(^^(n)^^)^^)^b = a^(^^(^(^^(^(^^(^(^^(...(^(^^(^(^^(^(^^(n)^^)^)^^)^)^^)^)...)^^)^)^^)^)^^)^)^^)^b w/ n (^(^^()^^)^) operator levels after (^^()^^)

Or something like that.

We can also recycle our old operators to make operators like (^^(^^(^^(^^(^(^^(^(^^(^(^(^(^(((())))^)^)^)^)^^)^)^^)^)^^)^^)^^)^^) (in 4-level tetrated parenthesis (you get the point)), etc

### Multi-arrow Parenthesis

We can extend the notation to pentation defining (^^^(^(^^^()^^^)^)^^^) as a nest of (^(^^()^^)^) levels after (^^^()^^^) and (^^^(^^(^^^()^^^)^^)^^^) as a nest of (^^()^^) after (^^^()^^^). In fact, you get the point. We can extend it to hexation, heptation, octation, n-ation, etc.

### Extended Arrow Parenthesis

We can define:

a^(^(m)^(n)^(m)^)^b = a^(^^^...^^^(n)^^^...^^^)^b w/ m arrows.

We can also mix this extension with the other recycled operators (like this, for example: (^(((m)))^(n)^(((m)))^)).

This will give us to nesting of the ^()^ separators (i mean, things like (^(^()^()^()^)^(^()^()^()^)^(^()^()^()^)^))

### Double Arrow Extension (Nested Arrow Parenthesis)

We can define:

a^^(n)^^b = n nest of ^()^ operator, being n the value of all ()'s

We can also mix the ^^()^^ operator with our old operators (for example, the ^^(^^()^^(^^^^(^^^(^^(^(^^(^(^^(^(^(^(^(^(((((((((())))))))))^)^)^)^)^)^^)^)^^)^)^^)^^^)^^^^)^^()^^) operator)

### Multiple Arrow Extension

We can define the ^^^()^^^ operator as a nest of the ^^()^^ operator in ^^()^^ arrays. Then, we can continue with hexation, heptation, octation, n-ation, etc.

### Nested Arrow Extension

We can define the ^()^()^()^ operator as multiple arrows outside the main parenthesis (in the center). Then, we can have ^(())^()^(())^, and then recycle all of our old operators to make more possibilities. Eventually, we will get into things like ^()^()^()^()^()^()^()^, and that will start to be annoying.

The limit is nesting of ^()^.

### Beyond Nested Arrow Extension

These extensions have been made by other users and converted to reality.

If you have any extension for my notation, post it in the comment.

#### Two Parenthesis

This was suggested by googleaarex, and I converted it to reality, but stronger.

First, let define:

a^(n)(2)^b = a^(n)^(n)^(n)^...^(n)^(n)^(n)^b w/ 2^n-1 nestings of ^(n)^

a^((n))(2)^b = a^(n^(n^(...(n^(n^(n)(2)^n)(2)^n)...)(2)^n)(2)^n)(2)^b w/ n levels

etc.

We can reuse our old operators to get bigger numbers.

Next we have:

a^(n)(3)^b = a^(^(n^(^(n^(...(n^(n^(n)(2)^n)(2)^n)...)(2)^n)(2)^n)(2)^(n)^...^)^b w/ n levels (the last ^...^ space was made by me because i don't want to copy that)

etc.

Finally, we have the ^(n)((n))^ operator, etc.

(sorry, too lazy)

#### Three Parenthesis

We define:

a^(n)(1)(2)^b = a^(n)(^(n)(...(^(n)(2)^)...)(2)^)^b w/ n levels

We can also define the ^(n)(2)(2)^ operator, the ^(n)(1)(3)^ operator, etc.

#### Multiple Parenthesis

We can also have a four parenthesis operator as a nest of the third parenthesis, etc.

#### Curly Bracket

Note: This is from me.

We can define:

a^{n}^b = a^(b)(b)(b)...(b)(b)(b)^a w/ n parenthesis

We can have the ^{()}^ operator, etc.

#### Curly Bracket Operators

We define:

a^^b =

-- UNDER CONSTRUCTION --

## FGH

This page will be maked soon.