Conway chained arrow notation

Summary

Conway chained arrow notation, created by mathematician John Horton Conway, is a means of expressing certain extremely large numbers.[1] It is simply a finite sequence of positive integers separated by rightward arrows, e.g. .

As with most combinatorial notations, the definition is recursive. In this case the notation eventually resolves to being the leftmost number raised to some (usually enormous) integer power.

Definition and overviewEdit

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 five (technically four) 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:

  1. An empty chain (or a chain of length 0) is equal to  
  2. The chain   represents the number  .
  3. The chain   represents the number  .
  4. The chain   represents the number   (see Knuth's up-arrow notation)
  5. The chain   represents the same number as the chain  
  6. Else, the chain   represents the same number as the chain  .

PropertiesEdit

  1. A chain evaluates to a perfect power of its first number
  2. Therefore,   is equal to  
  3.   is equivalent to  
  4.   is equal to  
  5.   is equivalent to   (not to be confused with  )

InterpretationEdit

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), or at least can be evaluated step by step in a prescribed order, e.g. 34567 from right to left, that is not so with Conway's arrow chains.

For example:

  •  
  •  
  •  

The fourth 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 second rule to shorten the chain. After, to paraphrase Knuth, "much detail", the chain is reduced to three elements and the third rule terminates the recursion.

ExamplesEdit

Examples get quite complicated quickly. Here are some small examples:

 

  (By rule 1)

 

  (By rule 5)
Thus,  

 

  (By rule 3)
 
 
 
   
 

 

  (By rule 3)
  (see Knuth's up arrow notation)

 

  (By rule 3)
 
 
 
 
 
 
  (see tetration)

 

  (By rule 4)
  (By rule 5)
  (By rule 2)
  (By rule 3)
= much larger than previous number

 

  (By rule 4)
  (By rule 5)
  (By rule 2)
  (By rule 3)
= much, much larger than previous number

Systematic examplesEdit

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 functionEdit

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 numberEdit

Graham's number   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.)

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 functionEdit

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 HurfordEdit

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.[2]

See alsoEdit

ReferencesEdit

  1. ^ John H. Conway & Richard K. Guy, The Book of Numbers, 1996, p.59-62
  2. ^ "Large Numbers, Part 2: Graham and Conway - Greatplay.net". archive.is. 2013-06-25. Archived from the original on 2013-06-25. Retrieved 2018-02-18.

External linksEdit

  • Factoids > big numbers
  • Robert Munafo's Large Numbers
  • The Book of Numbers by J. H. Conway and R. K. Guy