Bowers's operators
This array notation was created by Jonathan Bowers, and is known as the Bowers's operators.[1][2] It was created to help represent very large numbers, and was first published to the web in 2002.
Functionality
Let , the hyperoperation (see Square bracket notation, this is the same as , they are just different notations of hyperoperation). That is
The function means , i.e. is equal to for every ∈ .
Tetrentrical operators
The first operator is and it is defined:
Bowers calls the function "m expanded to n".
Thus, we have
⋮
The number inside the brackets can change. If it's two
⋮
Bowers calls the function "m multiexpanded to n".
Operators beyond can also be made, the rule of it is the same as hyperoperation:
Bowers continues with names for higher operations:
is "m powerexpanded to n"
is "m expandotetrated to n"
⋮
The next level of operators is , it to behaves like is to .
This means:
⋮
and beyond will work similarly.
Bowers continues to provide names for the functions:
is "m exploded to n"
is "m multiexploded to n"
is "m powerexploded to n"
is "m explodotetrated to n"
⋮
and beyond will follow similar recursion. Bowers continues with:
is "m detonated to n"
is "m pentonated to n"
⋮
For every fixed positive integer , there is an operator with sets of brackets. The domain of is , and the codomain of the operator is .
Another function means , where is the number of sets of brackets. It satisfies that for all integers , , , and . The domain of is , and the codomain of the operator is .
Pentetrical operators and beyond
Bowers generalizes this towards 5+ entries with the following ruleset:
- if there are 0 entries (like Conway chained arrow notation, the value of the empty chain is 1)
- if there are only 1 or 2 entries
- if the last entry is 1
- if the second entry is 1
- if the 3rd entry is 1
- if none of the above rules apply
Bowers does not provide operator names for and beyond, but he describes a notation for up to 8 entries:
"a,b, and c are shown in numeric form, d is represented by brackets (as seen above), e is shown by [ ] like brackets, but rotated 90 degrees, where the brackets are above and below (uses e-1 bracket sets), f is shown by drawing f-1 Saturn like rings around it, g is shown by drawing g-1 X-wing brackets around it, while h is shown by sandwiching all this in between h-1 3-D versions of [ ] brackets (above and below) which look like square plates with short side walls facing inwards."
Numbers like TREE(3) are unattainable with Bowers's operators, but Graham's number lies between and .[3]
References
- ↑ https://sites.google.com/site/largenumbers/home/4-1/extended_operators
- ↑ "Array Notation". www.polytope.net. Retrieved 2018-02-07.
- ↑ Elwes, Richard (2010). Mathematics 1001: Absolutely Everything That Matters in Mathematics in 1001 Bite-Sized Explanations. Buffalo, New York 14205, United States: Firefly Books Inc. pp. 41–42. ISBN 978-1-55407-719-9.