Conway chained arrow notation

The devised by John Horton Conway chained arrow notation in mathematics is similar to that developed by Donald Ervin Knuth arrow notation a type of notation to represent very large integers as short as possible.

Notation

In the chained arrow notation many natural numbers are arbitrarily consecutively written and linked with arrows. It should be noted that while there is no associativity, ie

In the chain, there are three members 3, 6 and the latter is a separate chain.

Designations

A distinction is that chained arrow notation as expressions thinks the other hand arrow notation says Knuth's arrow notation, for example.

The term chain several interlinked elements are understood. A member can be a further chain or a natural number.

Partial warps describe here a number of inter- chained links in the entire chain. Only the sequence is maintained, the position of the links in the chain does not matter. Is the considered chain is a possible substring. ( Although it is usually regarded more )

The term partial chain is thus needed to reduce to combine any number of limbs. It should be noted that the substring is not calculated separately in their use, but merely represents a short form.

Definition

, A is part of the chain, that is, may correspond, for example.

• ( The n the left is to be understood as monistic chain)

The substring A whole is listed n times, (m- 1) n -1 times. This account can be used to reduce the final element, until it can be reached and removed 1. However, since this usually is highly inconvenient following variant can be used:

Notes on the calculation

N, m, A, as in the definition of B is now a part of the chain, k is a natural number.

• That all links behind a 1 omitted.

This results in:

• With Knuth's arrow notation
• That each chain, whose first two elements are equal to 2, 4 Easy to understand, even as
• That ends a chain in two twos these two last terms can be replaced by the value of the previous chain. Note: not

The calculation of a chain runs mostly geared at producing at a member of a 1 or let it end a chain or partial chain on two deuces. Thus, the chain is simplified until there is only parenthetical substrings with three (or less) links which are then due to potentiation or the arrow notation.

Worked examples

First, an easy example:

Or:

Another tripartite example:

However, it can also shorten this example easily with Knuth's arrow notation:

Therefore, now a four -tier instance:

The attempt to represent that number even with orders of magnitude or even with power towers would be visible inappropriate.

This bill does, however, very well clear that the chained arrow notation can constitute the shortest enormous numbers.

This is now already clear by the mere contemplation of.