Negative Base Power 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), 11

Keywords: Fundamental, revolution, mega-overmathematics, exponential, counterexample, negative base power theory, sign-conserving power function, ill-defined, direction-conserving complex-base real-exponent power function, direction-adding complex power function.

Introduction

John Wallis [1656] extended power exponents from positive integers to rational numbers.

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]), such a finite pure number operation as raising a negative number to a power is well-defined for even positive integer exponents only. See counterexamples

(-1)³ = -1 ≠ 1 = [(-1)⁶]^1/2 = (-1)^6/2 ,

(-1)^1/3 = -1 ≠ 1 = [(-1)²]^1/6 = (-1)^2/6 .

Therefore, in classical mathematics, both power functions and exponential functions have very bounded domains of definition.

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]).

It is urgent to define raising a negative number to a power.

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, in particular, negative base power theory.

Negative Base Power Theory Fundamentals

If a < 0 and we want to regard the real (non-imaginary) numbers R only, then we may consider by

m ∈ Z = {0, ±1, ±2, ...}

(the integers) and

n ∈ N = {1, 2, ...}

(the positive integers) that

a^{(2m + 1)/(2n)} := a^{2(2m + 1)/(4n)}

giving real sense to a^b by any irrational b , too. Using modulus (absolute value) |a| gives the same results but is much less natural because

a ≠ |a|,

(2m + 1)/(2n) = 2(2m + 1)/(4n).

Mega-overmathematics by Lev Gelimson [1987-2011c] naturally introduces many further (also uncountable) quantioperations and quantirelations. Among them is sign-conserving power function

a"^b = |a|^b sign a

defined by any real numbers a ≠ 0 and b , as well as by a = 0 and any b > 0. Then we have, e.g.,

a"² = a² sign a ,

(-1)"³ = -1 = [(-1)"⁶]"^1/2 = (-1)"^6/2 ,

(-1)"^1/3 = -1 = [(-1)"²]"^1/6 = (-1)"^2/6 .

Nota bene: Fundamental, advanced, applied, and/or computational mathematical considerations also belong to fundamental, advanced, applied, and/or computational overmathematics, respectively, if and only if such considerations directly and explicitly use namely overmathematical very fundamentals revolutionarily replacing the inadequate very fundamentals of classical mathematics.

Examples:

1. Sign-conserving raising a specific nonnegative number to a real power belongs to mathematics but not to overmathematics because simply raising this specific nonnegative number to this real power in classical mathematics brings the same result. Hence replacing simply raising a specific nonnegative number to a real power with sign-conserving raising a specific nonnegative number to a real power brings nothing new. Therefore, raising a specific nonnegative number to a real power does not require overmathematical sign-conserving raising to a real power.

2. Sign-conserving raising a specific negative number to a real power belongs both to mathematics and to overmathematics because simply raising any negative number to any real power in classical mathematics is always ill-defined [Encyclopaedia of Mathematics 1988] because there are infinitely many irrational power exponents arbitrarily near to the given real power exponent so that for them, such a power is indefinite at all. All the more, for any even integer power exponent considered isolated, such a power is well-defined but brings the opposite result:

a^b = |a|^b ,

a"^b = |a|^b sign a = -|a|^b (a < 0).

For any odd integer power exponent, such a power (also considered isolated) is always ill-defined:

a^2z+1 = -|a|^2z+1 ≠ |a|^2z+1 = a^2(2z+1)/2 = [a^2(2z+1)]^1/2 (a < 0; z = 0, ±1, ±2, ...).

Hence replacing simply raising any negative number to any real power with sign-conserving raising this negative number to this real power always brings a new result. Therefore, raising any negative number to any real power does require namely overmathematical sign-conserving raising to a real power.

3. Sign-conserving real power function as whole belongs both to mathematics and to overmathematics because simply raising any negative number to any real power in classical mathematics is always ill-defined [Encyclopaedia of Mathematics 1988] because there are infinitely many real power exponents arbitrarily near to the given real power exponent so that for them, such a power is indefinite at all. All the more, for any even integer power exponent, such a power is well-defined but brings the opposite result. For any odd integer power exponent, such a power is ill-defined. Hence replacing simply raising any negative number to any real power with sign-conserving raising this negative number to this real power always brings a new result. Therefore, raising any negative number to any real power does require namely overmathematical sign-conserving raising to a real power.

Notata bene:

1. For any polarly represented complex (also imaginary) power base

a = re^iφ

where r is a nonnegative number (modulus, or polar radius), unique polar argument φ belongs to half-opened segment [0, 2π[ (0 included but 2π excluded), and

i² = -1,

naturally generalize the sign function with direction function

dir a = e^iφ = cos φ + i sin φ

and the above sign-conserving real power function with direction-conserving complex-base real-exponent power function

a"^b = |a|^b dir a = r^b dir a

with a complex power base a and a real power exponent b .

2. For any polarly represented complex (also imaginary) power base

a = re^iφ

where r is a nonnegative number (modulus, or polar radius), unique polar argument φ belongs to half-opened segment [0, 2π[ (0 included but 2π excluded), and

i² = -1,

as well as for any complex (also imaginary) power exponent

b = c + di

where c and d are real numbers,

further naturally generalize the above direction-conserving complex-base real-exponent power function

a"^b = |a|^b dir a = r^b dir a

with direction-adding complex power function

a"^b = a"^c+di = |a|^c+di dir a = r^c+di e^iφ = r^cr^di e^iφ = r^ce^{id ln r} e^iφ = r^ce^{i(d ln r + φ)} .

Nota bene: Use " in a"^b if necessary only.

Negative base power theory in mega-overmathematics by Lev Gelimson [1987-2012] is universal and very efficient.

References

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