Cutler's bar notation

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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.

1 Introduction
2 Advantages and Drawbacks
3 See also
4 References

[edit] 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.

[edit] Advantages and Drawbacks

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 shfted counter-clockwise and shortened to a single variable. The Bar Notation also allows for farily 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.