Hyperoperation

The Hyper- operator is a continuation of conventional mathematical operators of addition, multiplication and exponentiation. It is used for short representation of large numbers as power towers.

Derivation of the notation

Starting from the observations

One defines recursively a three-digit operator ( with )

And introduces the following designations:

Thus hyper1 the addition, multiplication and Hyper2 Hyper3 the potentiation. hyper4 is also known as Tetra -tion or super potency and can be written as follows:

General Understandably you could also say: Write the number a b times in a row and add each between the operator a level deeper.

The family has been extended for non- real numbers, because there are several " obvious" ways to do this, however, are not associative.

Knuth's arrow notation

See main article: Arrow notation

Another notation for the hyper operator was developed by Donald Knuth which is known as arrow notation. The definition is

Another notation used in place of the arrow sign. The definition applies just

This notation is used for the display of very large numbers, such as Graham's number.

Another extension

There is another way to get a more general definition of the specifications of the link, because it is also

• ,

Because the shortcuts are and commutative. This leads to the definition

This notation "collapses " but for; it results in contrast to hyper4 no power tower more:

How may differ for and suddenly? This is due to the associativity of a property that the operators and possess (see also body ), but lacks the power operator. ( In general ).

The other layers do not collapse in this way, which is why this family of operators, called "lower hyper- operators" of interest.

Examples

Addition

Multiplication

Potency

Tetra Transportation

It should be noted here that applies, see also with power tower.

External links (English)

• What Lies Beyond exponentiation?
• Lynz and the Clarkkkkson
• The Tetra Transportation Forum

• Potency (mathematics)