The new math has arrived!


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 and need for the unit imaginary number and thus, the complex number system, etc.

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 and methods is affected.

Comparatively, revised arithmetic requires three number systems instead of an infinite number out of which only 13 have been invented to date (i.e., no complex [2-D] or hypercomplex number systems:  4-D, 8-D, 16-D, 32-D, 64-D, 128-D, 256-D, 512-D, 1024-D, etc) and three binary operations instead of six (i.e., no inverse binary operations:  subtraction, division, evolution) yet revised algebra based upon it 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.

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 but the revised system is vastly superior to the conventional system in accordance with Occam's razor.

MSC 03C62  models of arithmetic & set theory

A holistic set of introductory essays not included within paper I for thematic reasons is "beyond model I & II".

The essay "project overview (model I)" gives a clear idea of the purpose of this project.

Several essays "minimal completeness and maximal applicability", "comparing numerical systems", "an unnatural history", "unclear foundations of math", "the non-absoluteness of mathematical proofs", "an unconventional approach" and "representations with rectangular coordinate systems" give considerably more detailed explanations.

If you have a bit of a sense of humor toward math, please check-out "in search of intelligent life" and "appendix II- accommodating extremely large numbers with higher, revised binary operations".

"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

An automatic installation program conveniently places all of the educational materials of this project within reach.  It runs under all modern NT versions of MS Windows (for IBM-compatible computers).


symmetry.exe  (3559 KB)

This master file contains safe compressed files which, in turn, contain the entire project featuring 4 items:

     One paper (including 13 quality, color graphs)- 
         240 pages.

paper- model I

     Three demo programs.

unary operations

revised arithmetic- model I

revised logarithms

latest revision
 January 2, 2023

Graph Viewing Cabinet


Containing 13 quality, color graphs of functions, unary operations and binary operations for model I.


set of all 13 graphs

model I



Extended Real Number Continuum (circular)

model I

Extended Real Number Continuum (linear)

model I

Extended Real Number Continuum (dual I)

model I

Extended Real Number Continuum (dual II)

model I


model I


model I

Opposition & Reciprocation

model I


model I

Revised Multiplication

model I

Revised Involution

model I

Revised Power Functions

model I

Revised Exponential/Logarithmic Functions (x y axes)

model I

Revised Exponential/Logarithmic Functions (x y axes)

model I


           ISIS- Symmetry
           International Symmetry Association
            Number Theory Web
    Dr. Mark Burgin
Diophantine & Non-Diophantine Arithmetics