,
From course of mathematics the
principle of a construction of operations of "multiplication" and
"exponentation" is known:
. (1.1)
In 1987 considered properties of operation "tetration" mentioned in the popular scientific literature (originally in my publications the name "superdegree") appears:
, (1.2)
where 
tetration 
number 
(label of the author).
Numerically at whole it is easy to check up, that the given operation is
noncommutative:
.
Therefore, there are completely various two inverses of operations designated so:
, 
.
The datas of a label are taken by analogy to the radical and log:
and 
.
Originally was decided to try to receive an
assotiation for a numerical evaluation of the superradical of positive whole tetrations.
The given task can be decided, determining a tetration through a degree and to take
advantage of one of methods of a numerical solution of the equation 
, for example of half division.
The task of deriving of the formula of an evaluation of the superradical posed.
Some thousands of years back were open the iterative formula for an evaluation of a square root:
, (1.3)
where 
some initial
approximation of a square root from 
.
Criterion of reaching of the given exactitude 
is the condition:
,
where 
, 
value on 
of iteration.
There was is offered a hypothesis about existence of the similar iterative formula for the superradical of the second tetration:
(1.4)
The given assotiation was obtained by logic reasonings:
.
to apply 
. and . As the expression was
transformed in , was decided to try to
change a rank of operations on the average arithmetical, i.e. to replace it on average
geometric and . The obtained iterative formula (1.4)
has appeared a converging sequence at 
.
(In the following paragraphs another will be reduced fast converging an iterative sequence
for values of argument 
).
The autumn of 1987 undertook attempt to consider
values of function 
at 
(
set of integers), in particular 
. Was decided to accept from (1.2) 
. Then to that is
equal 
? From the definition (1.1) it is possible to notice, that 
, i.e. 
(
a set of natural numbers). The given definition
can be expanded for 
etc.:
. (1.5)
Similarly from (1.2) is obtained,
at :
(1.6)
Values of expressions (1.5) and (1.6) at 
:
, 
, 
.
From expression (1.6) it is easy
to receive a value of a tetration at negative tetrations. For example, at 
:
,
,
.
Problem: " to that is equal 
, at ? "
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