جبر ویژه برای نسبیت ویژه: ویرایش دوم: نظریه پیشنهادی اعداد نامتناهی

Applied mathematics algebra without positive actual infinity - Maxwell’s Equations unite with the Dirac Equation to combine electron dynamics with photon dynamics because, by use of the algebra, an electron projects itself as a photon. Electron/photon double existence derives from Schrцdinger’s Cat because each place-value digit of a real number beyond a finite maximum in count is unknown and unknowable, analogous to the cat being both alive and dead inside its unopened box. The algebra is derived from a proposed axiom that replaces Cantor’s Continuum Hypothesis. - Empirically derived energy density and the Poynting Vector unite in the force density invariant as one mathematical model. That unity suggests quantities in our geometric world actually do have finite imprecision, and that the new algebra applies to more modern theories of physics. - 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). - One particle at two places violates a preconceived notion that that isn’t possible. The one particle is material (fermion electron) and, its opposite, force (boson photon). Take this radical notion further by supposing perceived reality results from numbers, alone from objects, interacting by becoming more precise with respect to each other, to form patterns we see as the Dirac Equation and other mathematical models of physics. The universe is fundamentally numbers. - Preface - The first four chapters of Special Algebra for Special Relativity 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 count before and after the decimal point. - For example, each side of a unit square has small magnitude finite imprecision added to it. The finite imprecision adder is a non-finite type of number because it isn’t knowable. - Finite imprecision larger than a measurable quantity is the division reciprocal of small magnitude imprecision. In Special Relativity large magnitude imprecision is added to time-space hyperbolic angle “alpha” (that relates to speed by “v = c*tanh(alpha)”) using a Lorentz Transformation. Large magnitude imprecision models electromagnetism by uniting Maxwell’s Equations with the Dirac Equation. Precision improves with time to cause measurable dynamics. - Electromagnetic field force density components are calculated using the same process by which electric current density components are calculated. Included are energy density and Poynting Vector components. - Uniting those three empirically derived electromagnetic phenomena into one mathematical model is new and that success suggests quantities in our geometric world actually do have finite imprecision and suggests finite imprecision numbers should also apply to more modern theories of physics. 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. Contents Preface Finite Precision for Numbers Chapter 1 - Numbers 1.2 Geometric-Vectors 1.3 Quaternions 1.4 Translation Back to Geometry 1.5 Singular-Label-Numbers 1.6 Exercises - Numbers Chapter 2 - Particles 2.2 Inertial Reference Frame 2.3 The Unspecified-Speed-Parameter 2.4 Compound-Label-Numbers and Components 2.5 Adding Hyperbolic Angles 2.6 Energy, Time Dilation, Length Contraction 2.7 Space-Like and Time-Like Invariants 2.8 Electric Current Density 2.9 Motion Faster than Light 2.10 Anti-Matter 2.11 Distributed Material Theory 2.12 Exercises - Particles Chapter 3 - Fields 3.2 All-Number Notation 3.3 Gauges and Super-Potentials 3.4 Lorentz Transformation 3.5 Biot-Savart Law 3.6 Electric Energy-Momentum of an Electron 3.7 Maxwell's Wave Equation 3.8 Forces Using Geometric-Vector Notation 3.9 Force Density Invariant 3.10 Area and Volume Differential Operators 3.11 Exercises - Fields Chapter 4 - Waves 4.2 Development of the Dirac Equation 4.3 Solutions to the Dirac Equation 4.4 Particle Properties 4.5 Two Alternative Arrangements 4.6 Lorentz Transformation of a Dirac Spinor 4.7 Exercises - Waves Chapter 5 - Proposed Theory 5.2 Cantor's Theory of Infinite Sets 5.3 Algebra Field for Local-Real Numbers 5.4 Lorentz Transformation with Non-Finite #s 5.5 Dirac Equation Development 5.6 Force Density Using the Complex-Conjugate 5.7 Spin of a Photon 5.8 Exercises - Proposed Theory Appendix A - Octonions and Sedonions Appendix B - Spooky Action at a Distance Appendix C - Discovering an Abstraction The Storybook Glossary Index Back Cover