# Cutler's bar notation

__: Arithmetic notation system__

**Short description**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:

- [math]\displaystyle{ \begin{matrix} a^b & = & \underbrace{a_{} \times a \times\dots \times a} \\ & & b\mbox{ copies of }a \end{matrix} }[/math]

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

- [math]\displaystyle{ \begin{matrix} & \underbrace{a_{}^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} & \\ & b\mbox{ copies of }a \end{matrix} }[/math]

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

- [math]\displaystyle{ \begin{matrix} {^b} \bar a = & \underbrace{a_{}^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} & \\ & b\mbox{ copies of }a \end{matrix} }[/math]

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

- [math]\displaystyle{ \begin{matrix} ^{^b{b}} \bar a = & \underbrace{a_{}^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} & \\ & {{^b} \bar a}\mbox{ copies of }a \end{matrix} }[/math]

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

- [math]\displaystyle{ \begin{matrix} \underbrace{b_{}^{b^{{}^{.\,^{.\,^{.\,^b}}}}}} \bar a = {_c} \bar a \\ c \mbox{ copies of } b \end{matrix} }[/math]

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

## 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 summarization of multiple copies of the same exponents, where any number of stacked exponents can be shifted counterclockwise and shortened to a single variable. The bar notation also allows for fairly rapid composure of very large numbers. For instance, the number [math]\displaystyle{ \bar {10}_{10} }[/math] would contain more than a googolplex digits, while 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 [math]\displaystyle{ ^{a^{b^{b^{c}}}} }[/math] could not be summarized 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^{[who?]} have suggested using a double and triple bar in subsequent rotations, though this presents problems when dealing with ten- and twenty-degree shifts.

Other equivalent notations for the same operations already exist without being limited to a fixed number of recursions, notably Knuth's up-arrow notation and hyperoperation notation.

## See also

## References

- Mark Cutler,
*Physical Infinity*, 2004 - Daniel Geisler,
*tetration.org* - R. Knobel. "Exponentials Reiterated."
*American Mathematical Monthly***88**, (1981)

Original source: https://en.wikipedia.org/wiki/Cutler's bar notation.
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