E-mail: [email protected] Las ecuaciones de la Física no relacionan sin más números, vectores o tensores de índole matemática, sino cantidades diádicas formadas con esos componentes vinculados a unidades diversas que indican cantidades de magnitudes naturales. Entonces, ¿por qué se opera con los entes diádicos de la Física como si fuesen elementos matemáticos puros?, ¿no supone esta ficción una aberración que envilece todo el conocimiento científico? Algunos autores han advertido de esta laguna crítica, que oculta a la Física un pilar tan fundamental. Pueden citarse preeminentes físicos como Clerk Maxwell o Max Planck, entre otros clásicos. Todos manifestaron a su manera los escrúpulos suscitados por la tradicional e injustificada forma de operar con las magnitudes físicas y sus unidades. Aquí se descubre, describe y resuelve tan notable paradoja de «aritmetización» de la Física y se construye un álgebra rigurosa y coherente para las cantidades de magnitudes. La Primera álgebra de magnitudes resuelve la hipótesis falsa del Sistema Internacional de Unidades, consistente en suponer negligentemente que las magnitudes físicas presenten estructura multiplicativa de grupo abeliano. No puede ser así, como se demuestra en este trabajo. Finalmente, se pone de manifiesto el camino lógico e inapelable que conduce del álgebra de magnitudes a los espacios «dismétricos», que se estudian con mayor profundidad en el segundo volumen de esta obra. La «dismetría» es una nueva y poderosa herramienta para representar con precisión los fenómenos físicos de un universo variable. Esta nueva Física acoge multitud de innovaciones, que sin duda sabrán apreciar muchos investigadores emprendedores. The equations of Physics do not simply relate numbers, vectors or tensors of a mathematical nature, but rather dyadic quantities formed with these components linked to various units that indicate quantities of natural magnitudes. So, why do we operate with the dyadic entities of Physics as if they were pure mathematical elements? Doesn't this fiction suppose an aberration that debases all scientific knowledge? Some authors have warned of this critical gap, which hides such a fundamental pillar from Physics. Pre-eminent physicists such as Clerk Maxwell or Max Planck, among other classics, can be cited. All of them expressed in their own way the scruples aroused by the traditional and unjustified way of operating with physical quantities and their units. Here such a remarkable «arithmeticization» paradox of Physics is discovered, described and solved and a rigorous and coherent algebra is constructed for the quantities of magnitudes. The First Algebra of Magnitudes resolves the false hypothesis of the International System of Units, consisting of negligently assuming that physical magnitudes have a multiplicative abelian group structure. It cannot be like that, as demonstrated in this work. Finally, the logical and unappealable path that leads from the algebra of magnitudes to the «dysmetric» spaces is revealed, which are studied in greater depth in the second volume of this work. «Dysmetry» is a powerful new tool for accurately representing the physical phenomena of a variable universe. This new Physics welcomes a multitude of innovations, which will undoubtedly be appreciated by many enterprising researchers.

In this book we develop step by step the FIRST ALGEBRA OF MAGNITUDES, the specific dyadic algebra for physical quantities, in order to rectify the sloppy hypothesis of «arithmetization» of Physics, normalized by the International System of Units in sections 2.1, 5.2 , 5.4.1 and 5.4.6 of his brochure SI, which is tolerated by a clueless scientific community. With dyadic algebra, full meaning is given to the meanings of the laws, equations and compound units of Physics, a sense that we all neglect today . As a culmination, the «DYSMETRIC» FORECAST is reached, with innumerable and far-reaching implications for the enrichment of physical models and the development of infinite innovations. In this way, the trap of «arithmetizing» Physics in which we all easily fall, even the most reputable and award-winning scientists, is ended. Except for one in the entire history of Physics, which was Newton, the only one who operated with magnitudes through the affinity of physical quantities with the elements of geometry, teaching us that, although Physics is not «arithmetizable», on the other hand it is it can be «geometrized». It seems incredible, but it is a grotesque fact that nowadays no one cares about what is really done when operating with physical magnitudes or what is the full meaning of the composite magnitudes or of the analytical formulations, which underlie all of Physics, for what no one should take a step without first having clarified this knowledge. On the contrary, it turns out that operations apparently as elementary as the multiplication of a meter by a kilogram have no arithmetic explanation, because no one identifies what the multiplier of that product is, which does not multiply numbers, but rather dyads or quantities of length and mass. Despite which, it seems that no one is bothered by such a ridiculous embarrassment. Can one call himself a physicist who cannot rigorously define this simple operation and does not care? Can a science be called Physics that lacks a coherent algebra to operate with its fundamental elements, the quantities of physical phenomena? The truth is that the defect is too gross not to take it into account. All this as a consequence of the fact that the current arithmetic hypothesis that postulates the abelian multiplicative group structure for the magnitudes is impossible. Such a structure is only valid for internal additive laws, it is not valid for external multiplicative laws. Obviously, this situation is shameful and pernicious for Physics, it is unsustainable and must be corrected as soon as possible. The dyadic algebra of magnitudes, in addition to giving meaning to the laws, equations, and compound magnitudes, reveals striking consequences, such as the non-existence of inverse elements of physical units, since heterogeneous multiplicative dyadic operations are not internal composition laws, but external. In turn, it naturally leads to «dysmetry», which makes it possible to represent the infinite physical realms of empty space and which radically transforms the vision of physical constants, incompatible in an absolute sense with «dysmetric» spaces, including the number pi and the speed of light.

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Geometric algebra has been presented in many different guises since its invention by William Kingdon Clifford shortly before his death in 1879. Our guiding principle is that it should be fully integrated into the foundations of mathematics, and in this regard nothing is more fundamental than the concept of number itself. In this book we fully integrate the ideas of geometric algebra directly into the fabric of matrix linear algebra. A geometric matrix is a real or complex matrix which is identified with a unique geometric number. The matrix product of two geometric matrices is just the product of the corresponding geometric numbers. Any equation can be either interpreted as a matrix equation or an equation in geometric algebra, thus fully unifying the two languages. The first 6 chapters provide an introduction to geometric algebra, and the classification of all such algebras. Exercises are provided. The last 3 chapters explore more advanced topics in the application of geometric algebras to Pauli and Dirac spinors, special relativity, Maxwell's equations, quaternions, split quaternions, and group manifolds. They are included to highlight the great variety of topics that are imbued with new geometric insight when expressed in geometric algebra. The usefulness of these later chapters will depend on the background and previous knowledge of the reader.Matrix Gateway to Geometric Algebra will be of interest to undergraduate and graduate students in mathematics, physics and the engineering sciences, who are looking for a unified treatment of geometric ideas arising in these areas at all levels. It should also be of interest to specialists in linear and multilinear algebra, and to mathematical historians interested in the development of geometric number systems.