Conway chained arrow notation, created by mathematician
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
, is a means of expressing certain extremely
large numbers
Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical ...
. It is simply a finite sequence of
positive integers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
separated by rightward arrows, e.g.
.
As with most
combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many app ...
notations, the definition is
recursive
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
. In this case the notation eventually resolves to being the leftmost number raised to some (usually enormous) integer power.
Definition and overview
A "Conway chain" is defined as follows:
* Any positive integer is a chain of length
.
* A chain of length ''n'', followed by a right-arrow → and a positive integer, together form a chain of length
.
Any chain represents an integer, according to the six rules below. Two chains are said to be equivalent if they represent the same integer.
Let
denote positive integers and let
denote the unchanged remainder of the chain. Then:
#An empty chain (or a chain of length 0) is equal to
#The chain
represents the number
.
#The chain
represents the number
.
#The chain
represents the number
(see
Knuth's up-arrow notation
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976.
In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called ''hyperoperati ...
)
#The chain
represents the same number as the chain
#Else, the chain
represents the same number as the chain
.
Properties
# A chain evaluates to a perfect power of its first number
#Therefore,
is equal to
#
is equivalent to
#
is equal to
#
is equivalent to
(not to be confused with
)
Interpretation
One must be careful to treat an arrow chain ''as a whole''. Arrow chains do not describe the iterated application of a binary operator. Whereas chains of other infixed symbols (e.g. 3 + 4 + 5 + 6 + 7) can often be considered in fragments (e.g. (3 + 4) + 5 + (6 + 7)) without a change of meaning (see
associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
), or at least can be evaluated step by step in a prescribed order, e.g. 3
4567 from right to left, that is not so with Conway's arrow chains.
For example:
*
*
*
The sixth definition rule is the core: A chain of 4 or more elements ending with 2 or higher becomes a chain of the same length with a (usually vastly) increased penultimate element. But its ''ultimate'' element is decremented, eventually permitting the fifth rule to shorten the chain. After, to paraphrase
Knuth, "much detail", the chain is reduced to three elements and the fourth rule terminates the recursion.
Examples
Examples get quite complicated quickly. Here are some small examples:
:
(By rule 2)
:
(By rule 3)
:Thus,
:
(By rule 4)
:
:
:
:
:
:
(By rule 4)
:
(see
Knuth's up arrow notation)
:
(By rule 4)
:
:
:
:
:
:
:
(see
tetration
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common.
Under the definition as rep ...
)
:
(By rule 6)
:
(By rule 3)
:
(By rule 5)
:
(By rule 4)
:= ''much larger than previous number''
:
(By rule 6)
:
(By rule 3)
:
(By rule 5)
:
(By rule 4)
:= ''much, much larger than previous number''
Systematic examples
The simplest cases with four terms (containing no integers less than 2) are:
*
: (equivalent to the last-mentioned property)
*
*
We can see a pattern here. If, for any chain
, we let
then
(see
functional powers).
Applying this with
, then
and
Thus, for example,
.
Moving on:
*
Again we can generalize. When we write
we have
, that is,
. In the case above,
and
, so
Ackermann function
The Ackermann function may be expressed using Conway chained arrow notation:
:
for
(Since
in hyperoperation)
hence
:
for
:(
and
would correspond with
and
, which could logically be added).
Graham's number
Graham's number
Graham's number is an immense number that arose as an upper bound on the answer of a problem in the mathematical field of Ramsey theory. It is much larger than many other large numbers such as Skewes's number and Moser's number, both of which ar ...
itself cannot be expressed concisely in Conway chained arrow notation, but it is bounded by the following:
Proof: We first define the intermediate function
, which can be used to define Graham's number as
. (The superscript 64 denotes a
functional power
In mathematics, function composition is an operation that takes two function (mathematics), functions and , and produces a function such that . In this operation, the function is function application, applied to the result of applying the ...
.)
By applying rule 2 and rule 4 backwards, we simplify:
:
(with 64
's)
:
:
:
(with 64
's)
:
(with 64
's)
:
(with 65
's)
:
(computing as above).
:
Since ''f'' is
strictly increasing,
:
which is the given inequality.
With chained arrows, it is very easy to specify a number much greater than
, for example,
.
:
:
:
which is much greater than Graham's number, because the number
is much greater than
.
CG function
Conway and Guy created a simple, single-argument function that diagonalizes over the entire notation, defined as:
meaning the sequence is:
...
This function, as one might expect, grows extraordinarily fast.
Extension by Peter Hurford
Peter Hurford, a web developer and statistician, has defined an extension to this notation:
All normal rules are unchanged otherwise.
is already equal to the aforementioned
, and the function
is much faster growing than Conway and Guy's
.
Note that expressions like
are illegal if
and
are different numbers; one chain must only have one type of right-arrow.
However, if we modify this slightly such that:
then not only does
become legal, but the notation as a whole becomes much stronger.
See also
*
Steinhaus–Moser notation
In mathematics, Steinhaus–Moser notation is a notation for expressing certain large numbers. It is an extension (devised by Leo Moser) of Hugo Steinhaus's polygon notation.
Definitions
: 20px, n in a triangle a number in a triangle means nn ...
*
Systematically creating ever faster increasing sequences
References
External links
Factoids > big numbersThe Book of Numbers by J. H. Conway and R. K. Guy
{{DEFAULTSORT:Conway Chained Arrow Notation
Mathematical notation
Large numbers
John Horton Conway