Special Algebra for Special Relativity: Second Edition: Proposed Theory of Non-Finite Numbers

Description

Algebra without an actual infinity is proposed for applied mathematics.The first four chapters present an all-number mathematical structure for Special Relativity. The fifth chapter restricts a measurable quantity to finite precision by limiting place-value digits to a maximum before and after the decimal point. For example, a physically real square of unit length on a side has only finite precision for each unit length and, likewise, the diagonal also has only finite precision. The ideal of unit length by use of the integer one is not geometrically possible in the physical world, and neither is the ideal of a perfectly precise irrational number.The finite imprecision larger than the measurable rational number is the division reciprocal of the finite imprecision smaller than the rational number. In Special Relativity the larger and smaller imprecision are added to the time-space hyperbolic angle using a Lorentz Transformation. The small magnitude imprecision is trivial. The large magnitude imprecision models electromagnetism because Maxwell's Equations are derived from the Dirac Equation. Precision improves with time.The mathematics for predicting electric current density from the Dirac Spinor results in the electromagnetic field force density invariant, and it includes the empirically derived energy density and Poynting Vector. The union of those empirically derived models is new and suggests quantities in our geometric world actually do have finite precision, and that finite precision numbers apply to the more modern theories of physics. A proposed axiom replaces Cantor's Continuum Hypothesis so that real numbers do not apply. The unknown and unknowable place-value digits smaller than a rational number are related to Schrodinger's Cat.Visualizations and exercises help comprehension. 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.