VOL. IV (XXI) 2009 - Departamentul de Filosofie si Stiinte ale ...

76 ON WITTGENSTEIN AND THE IMAGINARY NUMBER
Therefore, I ask you to notice the asymmetry, which seems to remind
us of its similarity to “Möbius band (strip)”. We take, for instance √–1048576,
which is in fact √–(2 20 ), in order to follow the algorithms of the removal from
the radical, of this negative number:
√–1,048,576 = √[–1,024(– 2 1,024)] = –32√–1,024 = –32√[–32(– 2 32)] =
(–5.6568..)( –32)√–32 = = 181.019336√–32 = 181.019336√[–5.656...( –
2 5.656...)] = ....... so on and so forth,
as it is highlighted through this example this extraction is never
consumed, so that what comes from under the radical, will tend squarely
more and more to 1,024, and what it remains perpetually under the radical is
exactly the perfect infinitesimal. This one sqarely misses for the number that
tends to 1,024, so that if these two parts multiply with themselves they will
have as a result 1,024. This sqarely perfect infinitesimal would tend to the
limit constituted by incommensurable number 1.(0)1. In other words, using
the given example, what comes from under the radical, could be called the
“progressum ad 1,024” number, and the number which remains under the
radical could be “regressum ab 1,024”. Algebraically, it can be represented in
the following way:
Finally having the relation:
√–n =
= √[(√–n)(√– 2 n)] =
= –(√√n) [√(√–n)] =
= –(√√n) {√√[(√–n)(√– 2 n)]} =
= –(√√√n) –(√√n) [√√(√–n)] =
= –[(√√√n) –(√√n)] [√√(√–n)] =
= +[(√√√n)(√√n)] [√√(√–n)] =
= +[(√√√n)(√√n)] {√√√[(√–n)(√– 2 n)]} =
= –[(√√√√n)(√√√n)(√√n)] [√√√(√–n)] =
etc.
√–n =
= [–n (2ˆ–∞) ] ·....................· [–n (2ˆ–2) ] (√{–n [2ˆ(–∞+1)] }) =
= {– + nˆ[(2 (∞ – 1) – 2 0 )/ 2 ∞ ]} {–nˆ[(2 0 /2 1 ) ∞ ]}.
Trough these relations, we can easily observe the bizarre aspects of
this number, which is simultaneously quasi-quantitas and quasi-qualitas.
That is why I used different aspects for the representation of these algebraic
expressions, in order to distinguish the numeric part which sublimes itself
continuously (the gray part) from the one which de-sublimes continuously
(the black part), i.e. regressum – ab, from progressum ad.