User:Pmanderson/Bailey

Bailey notation is a form of numerical notation, primarily for the use of absurdly large numbers, which are otherwise ridiculously long to write in conventional ways. It was created by and named after Martin Bailey, in an attempt to describe Graham's number in a small amount of space, in a simple way.

Martin Bailey is a well-known mathematician and astrophysicist, and was reportedly told about graham's number by a friend. Intrigued, he did some research and found that no-one had thought of a sensible way of writing it down, other than G, or in complicated long notations such as Knuth's up-arrow notation. He decided to come up with the notation as an alternative for scientific notation. In order to make it different from other notations and/or used symbols, he supposedly pressed 'shift' on his computer keyboard and stabbed various random characters, until he came up with '?' and '&'.

Constructing Bailey notation
Bailey's notation requires a number to the power of itself by a certain number of times, eg. nn n, or n^^n. This would become 2?n. Using real numbers, 7^^^^7 would become 4?7.

For bigger numbers, where there are a large number of powers, the notation can be bracketed as follows:


 * (4?7)?7

In a further attempt to simplify the notation, Bailey introduced the '&' sign to indicate further 'self-powers', as follows: This only works if each of the powers in the nested brackets are the same, which can mean that to get specific numbers, decimals are often needed, however one can place powers in and after Bailey notated numbers, in ways such that numbers such as googol and googolplex can be made (see below).
 * (4?7)?7 = 4?7&2
 * ((4?7)?7)?7 = 4?7&3

Graham's number
Graham's number is widely considered the biggest number that has been seriously used in any mathematical formula or problem. It is constructed as follows:


 * G1 = 3^^^^3
 * G2 = 3^^^...^^^3 with G1 up arrows
 * G3 = 3^^^...^^^3 with G2 up arrows

and so on...

G (Graham's Number) = 3^^^...^^^ with G62 up arrows.

Following the pattern for Graham's number as above:


 * G1 = 3^^^^3 = 4?3
 * G2 = 3^^^...^^^3 with G1 up arrows = (4?3)?3
 * G3 = 3^^^...^^^3 with G2 up arrows = ((4?3)?3)?3

and so on...


 * G = (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((4?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3)?3.

To save space and time, however, the brackets can be substituted for the '&', as follows:


 * G1 = 3^^^^3 = 4?3
 * G2 = 3^^...^^3 = (4?3)?3 = 4?3&2
 * G3 - 3^^...^^3 = ((4?3)?3)?3 = 4?3&3
 * G = 3^^...^^3 = 4?3&63

Other numbers
While mainly used for large numbers such as Graham's Number, Bailey Notation can be used for small numbers as well. For instance:


 * 2?2 = 2^^2 = 2^(2^2) = 2^4 = 16.

The '&' symbol is only needed for bigger numbers. The smallest example of the '&' symbol is:


 * 1?2&2 = (1?2)?2 = (2^2)?2 = 4?2 = 2^^^^2 = 2^(2^(2^(2^2))) = 2^(2^(2^4) = 2^(2^16) = 2^65536 = 2.0035299304068464649790723515603 x 1019728 (in scientific notation).

Powers can also be used within and after Bailey notation, as in the following cases:


 * googol = 10100 = 10^10^2 = 1?102
 * googolplex = 10googol = 10^10^10^2 = 10^^10^2 = 2?102
 * Graham's Number^2 (theoretical) = (4?3&62)2
 * (102)^^(102) = 2?(102)

Note the different positions of the power in the notation. For googol and googolplex, there are no brackets at all, meaning that there is a power on the end of the chain, eg. 4^^^^^4^3 = 5?43. For squaring Graham's Number, Bailey Notation is used to find the number itself, and the whole product is squared. Therefore, the entire notation is bracketed, and the power comes after. For the fourth example, it is a squared nubmer itself that is being powered. Therefore the number and the power goes in the brackets, and as before the number of times it powers itself comes beforehand.

Of course, this can give rise to nested notations such as 4?(2?3)^5?2. expanded, this could be read as ((3^^3)^^^^(3^^3))^(2^^^^^2), or alternatively:


 * ((33 3 )(3 3 3 )(3 3 3 )(3 3 3 )(3 3 3 )   )2 2 2 2 2 2

This is obviously unconventional, as too many mistakes could be made, hence the reason for Bailey notation.

Rules
There are a few rules when working with Bailey Notation:
 * If the number preceding the '?' is 0, the number remains the same.
 * &minus;0?n = n
 * &minuns;0?5 = 5


 * If the number preceding the '?' is 1, the number is self-powered.
 * &minus;1?n = n^n
 * &minus;1?57 = 5757


 * If the number after the '&' is 0, the answer is invariably 1, as this is the same as raising to the power of 0.
 * &minus;n?x&0 = 1
 * &minus;65423?794567865&0 = 1


 * If the number after the '&' is 1, it is not written, as it is the same as without (similar to the power of one, or 1x = x)
 * &minuns;n?x&1 = n?x
 * &minus;4?2&1 = 4?2

Summary
Of course, with a notation like this, impossibly huge numbers are possible to notate in a small space. For example, 100?100&100 is so huge that if that many micrometres were stretched out, the universe would not be big enough to fit it! Therefore, this notation is mainly used by some astrophysicists (but only relatively small distances and weights et cetera and theoretical mathematiticians.

Refrences

 * www-users.cs.york.ac.uk
 * Large Number Theories, by Maxamillion Taylor and Jane P. Schäfer (published September 2007)