The Zero Delusion v0.99x

with roots
z ≈ −1.19136
z ≈ 0.26902 ± 1.15872ı
z ≈ 1.07653 ± 0.52796ı
z ≈ −0.74986 ± 0.93908ı
A computable representation of this set of algebraic numbers is precisely its
minimal polynomial in canonical form.
Conversely, a minimal polynomial exists for all a ∈ Ǎ [140]. The degrees of
an algebraic number and its minimal polynomial coincide; for example, degree
0 does not exist, rational roots have degree 1 (e.g., x − 7 /2), x 2 − x − 1 and the
golden ratio have degree 2, and the polynomial (2) and any of its seven roots
have degree 7. Note that we can distinguish between rational numbers per se
and rational roots of polynomials. For example, whereas the code of the rational
number 7 /2 is {(21/2)} 3
, 7 /2 in {1 − (21/2)} 3
denotes the algebraic value that
annuls x − 7 /2. We can use both in an algebraic expression, e.g., 7 /2 = x − 7 /2.
Another caveat is that if we want the minimal polynomial to be "identically
zero" [?], the expansion of a polynomial root in PN does not terminate, i.e., is an
endless calculation. To construct an algebraic number explicitly, we must relax
the definition of "root" so that nullifying a minimal polynomial is calculable in
Ǎ despite being undecidable in C. Then, a root is a value z ∈ Ǎ that makes its
minimal polynomial P (z) vanish identically (e.g., z = 4 /5 is a root of z 2 − 16 /25)
or approximately as an element of the sequence converging asymptotically to
z, where the pair of vertical bars indicates modulus (e.g., z = −1.19136
lim
|P (z)|→0
is a root of [2], and z = −1.19135785807 is even a "better" one). The inaccuracy
of Ǎ is intrinsic to natural phenomena.
Although Ǎ has Lebesgue (R-based Euclidean) measure nil as a subset of
C, the property "algebraic" occupies nearly all the possibilities of the complex
plane; the algebraic numbers are dense in the complex numbers, and these are
algebraic numbers "almost everywhere" (or "almost surely"). Then, what is the
use of complex numbers that are not algebraic? [190] Transcendental numbers,
i.e., those that belong to C \ Ǎ, such as ln ( 1 + √ 2 ) and 3 + πı, are de facto
virtual and indefinite entities; they are as inaccessible as 0 or ∞ because their
value is unthinkably far away.
These properties point to the nonzero algebraic numbers in canonical notation,
with limiting values 0 and ∞, as a solid basis for building a universal
information processing and propagation system. We can alternatively turn to
the mathematical structures that allow division by zero, such as the R-line and
the C-plane extended with "infinite points". For instance, consider ˆR ≡ R∪{∞}
implemented as a vast circle containing a pair of dual values, namely 0 and ∞,
so that every element has a reciprocal, i.e., 1 /x is a "total function". There is no
order relation because infinity is de facto ±∞, which does not admit comparison
with the rest of the elements. We have two additive constants because r + 0 = r
and r +∞ = ∞, 0 0 is so undefined as ∞ ∞ or ∞ 0 , and new indefinite expressions
appear, such as ∞ + ∞, ∞ − ∞, 0 · ∞, ∞ · 0, and ∞ /∞. The topology is that
of a circle, but the arithmetic of intervals, especially those involving 0 or ∞, is