Cutler's bar notation

In mathematics, Cutler's bar notation is a notation system for large numbers, introduced by Mark Cutler in 2004. The idea is based on iterated exponentiation in much the same way that exponentiation is iterated multiplication.

Introduction

A regular exponential can be expressed as such:

$\begin{matrix} a^b & = & \underbrace{a_{} \times a \times\dots \times a} \\ & & b\mbox{ copies of }a \end{matrix}$

However, these expressions become arbitrarily large when dealing with systems such as Knuth's up-arrow notation. Take the following:

$\begin{matrix} & \underbrace{a_{}^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} & \\ & b\mbox{ copies of }a \end{matrix}$

Cutler's bar notation shifts these exponentials counterclockwise, forming ${^b} \bar a$. A bar is placed above the variable to denote this change. As such:

$\begin{matrix} {^b} \bar a = & \underbrace{a_{}^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} & \\ & b\mbox{ copies of }a \end{matrix}$

This system becomes effective with multiple exponent, when regular denotation becomes too cumbersome.

$\begin{matrix} ^{^b{b}} \bar a = & \underbrace{a_{}^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} & \\ & {{^b} \bar a}\mbox{ copies of }a \end{matrix}$

At any time, this can be further shortened by rotating the exponential counter-clockwise once more.

$\begin{matrix} \underbrace{b_{}^{b^{{}^{.\,^{.\,^{.\,^b}}}}}} \bar a = {_c} \bar a \\ c \mbox{ copies of } b \end{matrix}$

The same pattern could be iterated a fourth time, becoming $\bar a_{d}$. For this reason, it is sometimes referred to as Cutler's circular notation.

The Cutler Bar Notation can be used to easily express other notation systems in exponent form. It also allows for a flexible summarisation of multiple copies of the same exponents, where any number of stacked exponents can be shifted counter-clockwise and shortened to a single variable. The Bar Notation also allows for fairly rapid composure of very large numbers. For instance, the number $\bar {10}_{10}$ would contain more than a googolplex digits, whilst remaining fairly simple to write with and remember.
However, the system reaches a problem when dealing with different exponents in a single expression. For instance, the expression $^{a^{b^{b^{c}}}}$ could not be summarised in Bar notation. Additionally, the exponent can only be shifted thrice before it returns to its original position, making a five degree shift indistinguishable from a one degree shift. Some have suggested using a double and triple bar in subsequent rotations, though this presents problems when dealing with ten and twenty degree shifts.