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Abstract
In accordance with formalism, one of the two most widely accepted
foundations for modern mathematics, an experimental axiomatic system
having a variant number theory is admissible for study if it is
self-consistent. Nonetheless,
any given "revised" system is without exceptional theoretical
value or applicability unless it is comparatively advantageous to the
"conventional" system.
This
unconventional work initially involves the creation of a revised
multiplication in which the revised product of two negative, real number
factors equals a negative real number, contrary to conventional
multiplication. This
precludes the existence of the unit imaginary number and thus, the
complex number system.
By
a method analogous to how conventional involution is built upon
conventional
multiplication, likewise is revised involution built upon revised
multiplication. Although
addition is identical under both systems, with two of its three binary
operations revised, a revised arithmetic exists and consequently, a
revised algebra. Further
ramifications include a revised analytic geometry, revised analytic
trigonometry and revised calculus.
In fact, every branch of mathematics that is wholly or partially
based upon numerical definitions is affected.
Comparatively,
revised arithmetic requires three number systems instead of seven (i.e.,
no complex or hypercomplex number systems:
quaternion, octonion, sedenion) and three binary operations
instead of six
(i.e., no inverse binary operations:
subtraction, division, evolution)
yet maintains all comparable problem-solving capabilities.
In
revised algebra, a binomial, linear equation to any degree is
solvable since
after revised cross-multiplication, it is reducible to the original,
first degree equation. In
conventional algebra, a binomial, linear equation to the fifth degree or
higher is generally impossible to derive solutions for.
Through it all, any valid algebraic equation that is solvable by
the conventional complex number system (as well as including/beyond the
sedenion number system) is likewise solvable by the revised real number
system.
Ultimately,
the two numerical systems are fully isomorphic in describing the same
underlying mathematical reality as it exists independent of any
contrasting, arbitrarily-invented, mathematical languages of
interpretation.
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"Symmetry establishes a ridiculous and wonderful cousinship between objects, phenomena and theories outwardly unrelated:
terrestrial magnetism, women's veils, polarized light, natural selection, the theory of groups, structure of space, vase designs, quantum physics, scarabs, flower petals, X-ray interference patterns, cell division in sea urchins, equilibrium positions of crystals, Romanesque cathedrals, snowflakes, music, the theory of
relativity
..."
- Hermann Weyl
mathematical physicist
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