Hyper operator

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The hyper operator is a continuation of the conventional mathematical operators of addition , multiplication and exponentiation . It is used to briefly display large numbers such as power towers .

Derivation of the notation

Based on the observations

define a three-digit operator recursively (with )

and introduces the following terms:

(It should be noted with this notation that the spelling of and does not represent a multiplication, i.e. every actually occurring multiplication with the explicit operator must be noted. Likewise, there is no exponentiation. The use of the notation , on the other hand, rules out such possible confusion.

Thus hyper1 is the addition , hyper2 the multiplication and hyper3 the exponentiation . hyper4 is also known as tetration or superpotency and can be noted as follows:


More generally understandable one could also say: Write the number - times in a row and insert the operator one step lower in between.

The family has been expanded for not for real numbers because there are several “obvious” ways to do this, but they are not associative .

Knuth's arrow notation

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

Another notation uses the symbol instead of the arrow . With the definition, the following applies


This notation is used to represent very large numbers such as Graham's number .

Another extension

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

  • ,

because the links are + and commutative . This gives the definition

However, this notation "collapses" for ; In contrast to hyper4, it no longer results in a power tower:

How can and suddenly differ for? This is due to the associativity, a property that the operators and have (see also body ), but which the power operator lacks. (Generally is .)

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






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

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