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 .
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 .