Wiki Googologie
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Wiki Googologie

L'hyperopération ou les hyper opérateurs sont des extensions des opérateurs binaires standards addition, multiplication et exponentiation, ainsi que de la fonction successeur unaire. La multiplication est une addition répétée, et l'exponentiation est une multiplication répétée, il est donc naturel de les étendre davantage — l'exponentiation répétée est appelée "tétration," par exemple. Lorsqu'il est utilisé sur les entiers positifs, chaque hyperopérateur croît beaucoup plus rapidement que le précédent ; comme les nombres générés deviennent très grands, les hyperopérateurs sont considérés comme googological. Reuben Goodstein (1947)[1] a inventé la dénomination des hyperopérateurs en commençant par le quatrième.

Les bases

When we say "a × b", we mean "b copies of a added together":

a × b =

For example, 4 × 3 = 4 + 4 + 4.

When we say "ab," we mean "b copies of a multiplied together":

\[a^b = \underbrace{a \times a \times \cdots \times a \times a}_b\]

For example, 4^3 = 4 × 4 × 4. This is the limit of standard mathematical notation.

We can takes this a step further, however. We can define a new function, "\(^ba\)" (pronounced "power tower of a's with b terms high") which means "a tetrated to b":

\[^ba = \underbrace{a^{a^{a^{.^{.^.}}}}}_b\]

where there are b as, solved from the top down (which creates bigger numbers). This is called tetration. Since the function does not have much use in most areas in mathematics, there is no standard way of notating this.

The next natural step is pentation, which is repeated tetration; hexation, which is repeated pentation; heptation, which is repeated hexation; and so forth. The terms sexation, septation, etc. are also used, but they are considered nonstandard, being Latin prefixes and not Greek.

Notations

A common notation for tetration is ba (to-the-a b). The notation was introduced by Goodstein and popularized by Rudy Rucker; its use is mainly restricted to situations where only tetration and none of the greater hyper operators are used.

The most popular notation for the general hyper-operators is Donald Knuth's notation des puissances itérées, where n-ation is represented by n-2 arrows.

a + b, ab, a↑b, a↑↑b, a↑↑↑b, ...

It forms the basis for the notation des flèches chaînées. It is perhaps most well-known due to its appearance in the definition of notation des puissances itérées.

In ASCII settings, the caret symbol ^ frequently replaces the arrows.

a+b, ab, a^b, a^^b, a^^^b, ...

or in some computer language, * sign is duplicated as

a+b, a*b, a**b, a***b, a****b, ...

Another notation duplicates the + symbol:

a+b, a++b, a+++b, a++++b, ...

Goodstein himself used G(a,b,c) for b↑ac.

Before Goodstein defined the naming of hyperoperation in 1947, Wilhelm Ackermann defined a function of hyperoperation in terms of higher-order primitive recursion in 1928 as a 3 variable function ψ(a,b,n).[2][3] although the modified version later defined by Robinson is now popular and widely known as the fonction d'Ackermann.

Robert Munafo used superscript circled numbers:

ab, ab, ab, ab, ...

Pseudocode

// Upper hyper operators
function hyper(a, b, n):
    if n = 1:
        return a + b
    result := a
    repeat b - 1 times:
        result := hyper(a, result, n - 1)
    return result

// Lower hyper operators
function hyper_lower(a, b, n):
    if n = 1:
        return a + b
    result := a
    repeat b - 1 times:
        result := hyper_lower(result, a, n - 1)
    return result

Références

  1. Goodstein, R. (1947). Transfinite Ordinals in Recursive Number Theory. The Journal of Symbolic Logic, 12(4), 123-129. doi:10.2307/2266486
  2. Ackermann, W. (1928),"Zum Hilbertschen Aufbau der reellen Zahlen". Mathematische Annalen. 99: 118–133. doi:10.1007/BF01459088
  3. en:User_blog:Kyodaisuu/Reading Ackermann's paper
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