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| | The author is called Irwin Wunder but it's not the most masucline title out there. To gather coins is one of the issues I love most. Minnesota is where he's been residing for many years. Hiring is my profession.<br><br>Look at my web page [http://www.uknewsx.com/what-you-want-to-do-when-confronted-with-candida-albicans/ http://www.uknewsx.com/what-you-want-to-do-when-confronted-with-candida-albicans/] |
| In [[mathematics]], '''Cutler's bar notation''' is a [[Mathematical notation|notation system]] for [[large number]]s, introduced by Mark Cutler in 2004. The idea is based on [[iterated exponentiation]] in much the same way that [[exponentiation]] is [[iteration|iterated]] [[multiplication]].
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| ==Introduction==
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| A regular [[exponentiation|exponential]] can be expressed as such:
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| :<math>
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| \begin{matrix}
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| a^b & = & \underbrace{a_{} \times a \times\dots \times a} \\
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| & & b\mbox{ copies of }a
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| \end{matrix}
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| </math>
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| However, these expressions become arbitrarily large when dealing with systems such as [[Knuth's up-arrow notation]]. Take the following:
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| :<math>
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| \begin{matrix}
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| & \underbrace{a_{}^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} &
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| \\
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| & b\mbox{ copies of }a
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| \end{matrix}
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| </math>
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| Cutler's bar notation shifts these exponentials counterclockwise, forming <math>{^b} \bar a</math>. A bar is placed above the variable to denote this change. As such:
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| :<math>
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| \begin{matrix}
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| {^b} \bar a = & \underbrace{a_{}^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} &
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| \\
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| & b\mbox{ copies of }a
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| \end{matrix}
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| </math>
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| This system becomes effective with multiple exponent, when regular denotation becomes too cumbersome.
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| :<math>
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| \begin{matrix}
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| ^{^b{b}} \bar a = & \underbrace{a_{}^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} &
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| \\
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| & {{^b} \bar a}\mbox{ copies of }a
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| \end{matrix}
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| </math>
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| At any time, this can be further shortened by rotating the exponential counter-clockwise once more.
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| :<math>
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| \begin{matrix}
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| \underbrace{b_{}^{b^{{}^{.\,^{.\,^{.\,^b}}}}}} \bar a = {_c} \bar a
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| \\
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| c \mbox{ copies of } b
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| \end{matrix}
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| </math>
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| The same pattern could be iterated a fourth time, becoming <math>\bar a_{d}</math>. For this reason, it is sometimes referred to as '''Cutler's circular notation'''.
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| ==Advantages and drawbacks==
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| 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 <math>\bar {10}_{10}</math> would contain more than a [[googolplex]] digits, whilst remaining fairly simple to write with and remember.
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| However, the system reaches a problem when dealing with different exponents in a single expression. For instance, the expression <math> ^{a^{b^{b^{c}}}}</math> 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.
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| ==See also==
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| *[[Mathematical notation]]
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| ==References==
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| *Mark Cutler, ''Physical Infinity'', 2004
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| *[[Daniel Geisler]], ''[http://www.tetration.org/ tetration.org]''
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| *R. Knobel. "Exponentials Reiterated." ''[[American Mathematical Monthly]]'' '''88''', (1981)
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| [[Category:Mathematical notation]]
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| [[Category:Large numbers]]
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The author is called Irwin Wunder but it's not the most masucline title out there. To gather coins is one of the issues I love most. Minnesota is where he's been residing for many years. Hiring is my profession.
Look at my web page http://www.uknewsx.com/what-you-want-to-do-when-confronted-with-candida-albicans/