Chained arrow notation

By John Horton Conway invented chained arrow notation is a mathematical representation for very large integers, similar to that of Donald Ervin Knuth developed arrow notation , which is to distinguish it.

notation

With the concatenated arrow notation, any number of natural numbers are written one after the other and concatenated with arrows, and such a chain represents a natural number.

It should be noted that a string of numbers can not be easily broken down into parts that are evaluated in itself because it is a -place operation and not the one after execution of tens: . ${\displaystyle n\geq 3}$ ${\displaystyle n}$ ${\displaystyle 2\rightarrow (3\rightarrow 2)\not =2\rightarrow 3\rightarrow 2\not =(2\rightarrow 3)\rightarrow 2}$

If a chain within another string is supposed to represent a number, it is clasped The chain consists of links 3: 3, 6 and , with the latter is a self-chain of the number is, therefore . ${\displaystyle 3\rightarrow 6\rightarrow (5\rightarrow 4)}$ ${\displaystyle (5\rightarrow 4)}$ ${\displaystyle 5^{4}=625}$ ${\displaystyle 3\rightarrow 6\rightarrow (5\rightarrow 4)=3\rightarrow 6\rightarrow 625}$

Note: The linked arrow notation ( ) is - especially when using variables  - not to be confused with the logic used notation for the implication and subjunction ( ), in which often the same simple arrow ( ) symbolizing the connective is used. ${\displaystyle n\rightarrow m}$ ${\displaystyle a\rightarrow b}$ ${\displaystyle \rightarrow }$

definition

The following should apply:

• ${\displaystyle n,m\in \mathbb {N} ^{+}}$
• ${\displaystyle A}$represents a partial chain . can for example correspond to. ${\displaystyle A\rightarrow n}$ ${\displaystyle 5\rightarrow 2\rightarrow n}$

The values ​​of chains are thus defined as follows:

1. An empty chain (with length 0) has the value 1
2. A chain of length 1 with the link has the value ${\displaystyle n}$ ${\displaystyle n}$
3. The value of a chain of length 2 is the power of its links:
   ${\displaystyle n\rightarrow m=n^{m}}$
4. If a chain with length has an end link with the value 1, this can be omitted: ${\displaystyle \geq 2}$
   ${\displaystyle A\rightarrow 1=A}$
5. With applies: ${\displaystyle n,m>1}$
   ${\displaystyle A\rightarrow n\rightarrow m=A\rightarrow (A\rightarrow (n-1)\rightarrow m)\rightarrow (m-1)}$
   ${\displaystyle A\rightarrow 1\rightarrow m=A}$

Alternative formulation of Rule 5: . The partial chain is noted a total of times and the link times.
${\displaystyle A\rightarrow n\rightarrow m=A\rightarrow (...(A\rightarrow (A)\rightarrow (m-1))...\rightarrow (m-1))\rightarrow (m-1)}$
${\displaystyle A}$ ${\displaystyle n}$ ${\displaystyle (m-1)}$ ${\displaystyle (n-1)}$

Example:
${\displaystyle A\rightarrow 3\rightarrow 5=A\rightarrow (A\rightarrow 2\rightarrow 5)\rightarrow 4=A\rightarrow (A\rightarrow (A\rightarrow 1\rightarrow 5)\rightarrow 4)\rightarrow 4=A\rightarrow (A\rightarrow (A)\rightarrow 4)\rightarrow 4}$

Inferences

${\displaystyle n}$, , As in the definition, is now also a part of the chain, a natural number. ${\displaystyle m}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle k}$

• ${\displaystyle A\rightarrow 1\rightarrow B=A}$ all chain links after a 1 are omitted
• ${\displaystyle 1\rightarrow A=1}$
• ${\displaystyle n\rightarrow m\rightarrow k=n\uparrow ^{k}m}$ with Knuth's arrow notation
• ${\displaystyle 2\rightarrow 2\rightarrow A=4}$ every chain, whose first two links are 2, has the value 4 (as well as ) ${\displaystyle 2+2=2\cdot 2=2^{2}=2\uparrow ^{n}2=4}$
• ${\displaystyle A\rightarrow 2\rightarrow 2=A\rightarrow (A)}$ if a chain ends in two twos, these can be replaced by the value of the chain in front of it (note: not ) ${\displaystyle A\rightarrow A}$

The calculation of a chain usually comes down to applying rule 5 to reduce the last link until it is 1 and can therefore be omitted. In this process, the penultimate link is usually increased enormously, and the more so the more complex the partial chain before the last two links is, because the full length of this is included in the calculation of the penultimate link. In this way the chain is shortened until it only contains two links and is thus due to the exponentiation.

Sample calculations

First a simple example:

${\displaystyle 2\rightarrow 3\rightarrow 2=2\uparrow \uparrow 3=2^{2^{2}}=2^{4}=16}$

Or:

${\displaystyle 2\rightarrow 3\rightarrow 2=2\rightarrow (2\rightarrow 2\rightarrow 2)\rightarrow 1=2\rightarrow (4)=2^{4}=16}$

Another tripartite example:

${\displaystyle 5\rightarrow 3\rightarrow 2=5\rightarrow (5\rightarrow 2\rightarrow 2)\rightarrow 1=5\rightarrow (5\rightarrow (5))=5\rightarrow (5^{5})=5^{5^{5}}=5^{3125}\approx 1{,}91101259794547752\cdot 10^{2184}}$

However, this example can easily be abbreviated with Knuth's arrow notation: ${\displaystyle 5\rightarrow 3\rightarrow 2=5\uparrow \uparrow 3=5^{5^{5}}}$

Hence a four-part example:

${\displaystyle 3\rightarrow 2\rightarrow 3\rightarrow 2=3\rightarrow 2\rightarrow (3\rightarrow 2\rightarrow 2\rightarrow 2)\rightarrow 1=3\rightarrow 2\rightarrow (3\rightarrow 2\rightarrow (3\rightarrow 2))}$

${\displaystyle =3\rightarrow 2\rightarrow (3\rightarrow 2\rightarrow 9)=3\rightarrow 2\rightarrow (3\uparrow ^{9}2)}$

${\displaystyle =3\uparrow ^{3\uparrow ^{9}2}2}$

The calculation is therefore based on the arrow operator of the order , which can no longer be represented meaningfully in exponential notation . ${\displaystyle 3\uparrow ^{9}2=3\uparrow ^{8}3=3\uparrow ^{7}3\uparrow ^{7}3}$

However, this calculation makes it very clear that the concatenated arrow notation is probably the shortest way to represent enormously large numbers.

This becomes clear just by looking at . ${\displaystyle 43\rightarrow 91\rightarrow 74\rightarrow 84\rightarrow 101}$

See also

• Arrow spelling
• Steinhaus-Moser notation
• Ackermann function
• Graham's number