Sign-Conserving Multiplication Theory
by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich, Germany)
Mathematical Journal
of the "Collegium" All World Academy of Sciences
Munich (Germany)
12 (2012), 10
Keywords: Fundamental, mega-overmathematics, power tower, commutative hyperoperation, exponentiation, tetration, negative base power theory, number scale transformation, general power-exponential function hyperefficiency theory, sign-conserving multiplication theory, chaos theory, fractal theory.
Introduction
Numbers with very small and very large absolute values [Wikipedia Large_numbers] are extremely important for real world modeling. Moreover, their role exponentially increases because of computer science evolution which requires the so-called scientific number representation, as well as the storage and handling of such numbers to avoid the permanent danger of "computing overflow".
In classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]), exponentiation as raising numbers to powers by Michael Stifel [1544], as well as power functions y = xn with constant exponents n and exponential functions y = ax with constant bases a are widely used. Some power-exponential functions with variable bases and variable exponents such as
y = xx
and also iterated (nested) exponentials (power towers)
a^b^c^... = a^{b^[c^(...)]}
with multiply (repeatedly) raising bases to powers so that power exponents are powers themselves are well-known, see also [Wikipedia Tetration]. Leonhard Euler [1777] introduced the notation
expa(x) = a^x = ax ,
which can be combined with function iteration notation fn(x) giving
expan(x) = a^a^...^a^x
(with a used n times on the right-hand side). He also showed that the infinite power tower
a^a^...
defined as the limit of
a^a^...^a
(with a used n times), converges for e-e ≤ x ≤ e1/e as n goes to infinity, which roughly gives the interval from 0.066 to 1.44. In particular, at a = 21/2 , this limit equals 2. Hans Maurer [1901] already used modern tetration notation
na = a^a^...^a (with a used n times on the right-hand side).
Donald Ervin Knuth [1976] introduced his up-arrow notation
a↑n = a^n = an ,
a↑↑n = a^a^...^a (with a used n times on the right-hand side),
a↑↑n(x) = expan(x) = a^a^...^a^x (with a used n times on the right-hand side)
interpreting super-powers and super-exponential functions via using m arrows in expression a↑m n(x). John Horton Conway [1996] chained arrow notation
a→n→2 = a^a^...^a (with a used n times on the right-hand side)
provides similar generalization via increasing the number 2 and, more powerfully, by extending the chain.
Albert Arnold Bennett [1915] proposed commutative hyperoperations sequence defined by the recursion rule
Fn+1(a , b) = exp(Fn(ln(a), ln(b))
beginning with
F0(a , b) = ln(e^a + e^b) = ln(ea + eb),
addition (I)
F1(a , b) = a + b ,
multiplication (II)
F2(a , b) = ab = eln(a) + ln(b) ,
a commutative form of exponentiation (III)
F3(a , b) = eln(a) ln(b) ,
F4(a , b) = e^{e^[ln(ln(a))ln(ln(b))]}
not to be confused with tetration [Wikipedia Hyperoperation].
Wilhelm Ackermann [1928] defined the function
φ(m , n , p)
resembling the hyperoperation sequence with reproducing such basic operations as addition, multiplication, and exponentiation at p = 0, 1, 2, respectively:
φ(m , n , 0) = m + n ,
φ(m , n , 1) = mn ,
φ(m , n , 2) = m^n = mn ,
φ(m , n , p) = m↑p-1 (n + 1)
for p > 2 with extending these basic operations using Knuth's up-arrow notation.
Reuben Louis Goodstein [1947] introduced the hyperoperations sequence of operations extending succession (the 0th) 1 + b , addition (the 1st) a + b , multiplication (the 2nd) ab , and exponentiation (the 3rd) ab and gave the extended operations beyond exponentiation the Greek names tetration (the 4th)
a↑↑b ,
pentation (the 5th)
a↑↑↑b = a↑3 b ,
hexation (the 6th)
a↑↑↑↑b = a↑4 b ,
etc., where each operation is defined by iterating the previous one.
All this is used for numbers with so-called very small and very large absolute values [Wikipedia Large_numbers].
But common approaches have many disadvantages:
1) investigating already available possibilities is much less efficient than concertedly creating new possibilities;
2) positive number bases only are usually considered;
3) bases between 0 and 1 are not efficiently used for representing numbers with so-called very small and very large absolute values;
4) a uniform number scale is not suitable for creating hyperoperation hierarchy;
5) known number scale transformations such as using logarithmic scales cannot provide suitably simultaneously representing numbers both with very small and very large absolute values of the both signs;
6) natural numbers (positive integers) of multiple (combined, composite) power exponents only are usually considered;
7) multilevel placing multiple power exponents brings many typesetting difficulties and misunderstanding, especially by text transformation via software including browsers;
8) already usual exponentiation ab is noncommutative and nonassociative, e.g.
23 = 8 ≠ 9 = 32,
2^3^4 = 2^(3^4) = 281 ≠ 212 = (2^3)^4,
because in
ab = eb ln(a)
the roles of a and b are very different;
9) a commutative form of exponentiation (III)
F3(a , b) = eln(a) ln(b) = aln(b)
by Albert Arnold Bennett [1915] provides noninteger values by natural a , b > 1 and growth much more slower than that of ab by great a , b , which is a very important disadvantage when applying this commutative form of exponentiation to representing great numbers;
10) individual quantities of operands and operation results are not considered at all.
Therefore, in classical mathematics, both power functions and exponential functions providing often useful high orders of growth especially by multiply (repeatedly) raising bases to powers have very bounded domains of definition and efficiency.
Hence classical mathematics cannot (and does not want to) regard (adequately solve and even consider) very many typical urgent problems in science, engineering, and life. This generally holds when solving valuation, estimation, discrimination, control, and optimization problems, as well as in particular by measuring very inhomogeneous objects and rapidly changeable processes [Encyclopaedia of Physics 1973]. This is also very important for chaos theory (Ilya Prigogine [1993, 1997]) and fractal theory (Benoît Mandelbrot [1975, 1977, 1982]).
Mega-overmathematics by Lev Gelimson [1987-2012] based on its uninumbers, quantielements, quantisets, and uniquantities with quantioperations and quantirelations provides universally and adequately modeling, expressing, measuring, evaluating, and estimating general objects. This all creates the basis for many further developing, extending, and applying mega-overmathematics fundamental sciences systems. Among them is fundamental commutative exponentiation science including, in particular, negative base power theory which defines raising a negative number to a power and present sign-conserving multiplication theory which provides commutative hyperoperation hierarchy.
Principal Ideas
Possible ideas are very natural:
to consider general power-exponential functions as unary operations;
to commutatively extend them to make them applicable to any quantities of operands;
to further generalize such functions via operand role individualization.
Sign-Conserving Multiplication
In classical mathematics (Nicolas Bourbaki [1949], Arnold Kaufmann, Maurice Denis-Papin, Robert Faure [1964], G. A. Korn, T. M. Korn [1968], I. N. Bronstein, K. A. Semendjajew [1989], [Encyclopaedia of Mathematics 1988]), usual multiplication is well-known. If all the factors to be multiplied are positive, then their product is positive, too, which is natural. If some factors to be multiplied are positive whereas the remaining factors to be multiplied are negative, then their product is positive when the number of negative factors is even whereas their product is negative when the number of negative factors is odd. This seems to be rather artificial than natural. Its origin (source) is that in the (real or complex) numbers, multiplication of real numbers distributes over addition in the well-known concepts of a ring and a field which both are commutative.
Nota bene: Both in mathematical logic and in set theory, the both distributive laws (of multiplication over addition and of addition over multiplication) hold. In mathematical logic, namely in Boolean algebra, logical disjunction ∨ plays the role of addition whereas logical conjunction ∧ plays the role of multiplication, and for any sentences (propositions that may be true or false) A , B , and C , we have both
(A ∨ B) ∧ C = (A ∧ C) ∨ (B ∧ C)
and
(A ∧ B) ∨ C = (A ∨ C) ∧ (B ∨ C).
In set theory, namely in set algebra, unification ∪ plays the role of addition whereas intersection ∩ plays the role of multiplication, for any sets A , B , and C , we have both
(A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C)
and
(A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C).
In real or complex algebra, for any numbers A , B , and C , multiplication distributes over addition
(A + B)C = AC + BC
whereas addition does not distribute over multiplication because generally (as a rule)
AB + C ≠ (A + C)(B + C).
Therefore, already in the well-known concepts of a ring and a field which both are commutative, one of the the both distributive laws (of multiplication over addition and of addition over multiplication) does not hold. Hence it is possible and at least equally natural to additionally consider completely nondistributive rings and fields along with well-known rings and fields which are partially nondistributive.
Nota bene: Common multiplication naturally leads to noncommutative common power and exponential functions well-defined by negative bases if and only if exponents are integer whereas sign-conserving multiplication naturally leads to commutative sign-conserving power and exponential functions well-defined by any real-number bases and exponents.
The fundamental ideas of sign-conserving multiplication of real numbers are as follows:
the modulus (absolute value) of the sign-conserving product (as a result of sign-conserving multiplication) of real numbers equals the modulus (absolute value) of the usual product of these numbers;
the value of the sign function of the sign-conserving product of real numbers vanishes if and only if the value of the sign function of the usual product of these numbers vanishes, i.e. if and only if at least one of these numbers vanishes;
the value of the sign function of the sign-conserving product of real numbers equals 1 if and only if all these numbers are positive;
the value of the sign function of the sign-conserving product of real numbers equals -1 if and only if at least one of these numbers is negative and none of these numbers vanishes.
To denote sign-conserving multiplication, simply use the parenthesis " either instead of a multiplication sign if it is implicit (i.e. omitped) or to the left of a multiplication sign (e.g. × , • , Π , etc.) if it is explicitly used.
Analytically, for any index set J , all indices j ∈ J , and any indexed real numbers building a set {aj | j ∈ J}, their sign-conserving product
"Πj∈J aj = min(sign aj | j ∈ J) |Πj∈J aj|
so that
|"Πj∈J aj| = |Πj∈J aj|
and
sign(Πj∈J aj|) = sign|Πj∈J aj| min(sign aj | j ∈ J) = min(sign|aj| | j ∈ J) min(sign aj | j ∈ J).
Example: For any real numbers a , b , and c ,
a"b"c = a "× b "× c = a "• b "• c
= min(sign a , sign b , sign c) |abc|
so that
|a"b"c| = |abc|
and
sign(a"b"c) = sign|abc| min(sign a , sign b , sign c)
= min(sign|a|, sign|b|, sign|c|) min(sign a , sign b , sign c).
Nota bene: Introducing an additional factor, e.g.
sign|abc| = min(sign|a|, sign|b|, sign|c|),
is here necessary to provide
sign(a"b"c) = 0
if at least one of these numbers a , b , and c vanishes whereas then
a"b"c = 0
due to vanishing the factor |abc| which is absent in sign(a"b"c).
Basic Results and Conclusions
1. Sign-conserving multiplication theory is advanced on the base of the proposed ideas.
2. Sign-conserving multiplication theory is suitable for creating hyperoperation hierarchy.
3. Sign-conserving multiplication theory in mega-overmathematics by Lev Gelimson [1987-2012] is universal and very efficient.
References
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