Cutler's bar notation

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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.

Advantages and drawbacks[edit]

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.

See also[edit]

References[edit]