Special Algebra for Special Relativity

A special algebra is proposed for applied mathematics. A special algebra is developed specifically for discovering new theory inside the present theory of Special Relativity. In the book's climax, the special algebra unites Maxwell's Equations into the Dirac Equation to form a mathematical model of physics that combines the dynamics of a photon with the dynamics of an electron. An electron projects itself as a photon, per the application of the special algebra. The special algebra that...
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A special algebra is proposed for applied mathematics. A special algebra is developed specifically for discovering new theory inside the present theory of Special Relativity. In the book's climax, the special algebra unites Maxwell's Equations into the Dirac Equation to form a mathematical model of physics that combines the dynamics of a photon with the dynamics of an electron. An electron projects itself as a photon, per the application of the special algebra. The special algebra that leads to this climax is derived from a proof of irrationality of an irrational number. The place-value digits of an irrational number that are beyond a specific finite maximum in count are each unknown and unknowable, analogous to Schrödinger's Cat. Through use of the special algebra, the double existence of Schrödinger's Cat leads to the double existence of the electron/photon particle. The special algebra is derived from a proposed axiom that replaces Cantor's Continuum Hypothesis, from which the real numbers were derived. The mathematics is simple enough to be understood by a high school student who has taken first year level college math and physics classes (and is familiar with trigonometry and logarithms, complex numbers, matrix multiplication, geometric-unit-vectors, and partial differential equations). Visualizations and examples help the reader comprehend each subtle feature in the algebra. Each chapter has exercises so that the reader can check their comprehension. The book was written to be quickly read and easily understood, and the book includes the mathematical details. SPECIAL ALGEBRA FOR SPECIAL RELATIVITY o Derives a proper replacement for infinity for use in applied mathematics o Proposes a new theory for electromagnetism o Pushes hypercomplex number algebra to the extremes o Combines anti-matter, electromagnetism, matter-waves, and time-space Preface - The real numbers, defined per Cantor's Continuum Hypothesis, are replaced by a new set of numbers. The new set of numbers are defined per a proposed new axiom that is consistent with the proofs that specific numbers are irrational. By use of the mathematical operation of the Lorentz Transformation, the new set of numbers is applied to the existing mathematical model of the Dirac Equation to result in Maxwell's Equations. Unlike the real numbers, the new set of numbers has no dependency on an actual infinity. In base two, the new set of numbers has known or knowable place-value digits after the decimal point that extend to a finite count of place-value digits. Beyond that count, the place-value digits are unknown and unknowable. An unknown and unknowable place-value digit in base two is an analogy to Schrödinger's Cat. For example, in geometric space, the right triangle with two unit length sides has a finite imprecision to each unit length side because the quantity of zeros after the decimal point is a finite count. The quantity of zeros is not infinite per Cantor's theory of infinite sets, and is not the ultimate quantity of zeros as is assumed for an integer through the process of truncation at the decimal point. The length of the hypotenuse is similarly imprecise. The unknown and unknowable place-value digits after the decimal point cannot all be/become zero if replaced randomly with a one or a zero because there is no end to their count, and, therefore, a division reciprocal exists for the unknown and unknowable place-value digits. The division reciprocal is the proper replacement for an actual infinity because it may be used in applied mathematics. The division reciprocal is used in the Lorentz Transformation to model motion at the speed of light. The claim of applicability of the new set of numbers to applied mathematics is based on the correct calculation of the measured electromagnetic force density, by use of the complex conjugate of the Dirac Spinor.