Banica T. Principles of Mathematics 2025

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Banica T. Principles of Mathematics 2025

Textbook in PDF format This is an introduction to mathematics, with emphasis on geometric aspects. We first discuss numbers, counting, fractions and percentages, and their basic applications. Then we get into plane geometry, with a study of triangles and trigonometry, followed by coordinates and complex numbers. We then go into functions and analysis, with the basics of the theory explained, followed by exponentials, logarithms and more trigonometry, and with the derivatives and integrals discussed too. Finally, we provide an introduction to vector calculus, space geometry and basic mechanics. The present book is an introduction to this, foundational mathematics, by using the above traditional learning scheme, namely numbers and basic algebra, to start with, followed by basic geometry and trigonometry, followed by functions and basic analysis, and with some vector calculus material, which is more advanced, at the end. The book, meant to be accessible to anyone knowing and loving basic mathematics from school, and by this I mean we will start from 0 or almost, but we will go quite fast, with the goal of learning a maximum of things, is organized in 4 parts, as follows: Part I - Numbers. We discuss here numbers, fi rst with a philosophical discussion regarding numeration bases and notations, then with a look into basic arithmetic, and basic counting too, followed by real numbers and what can be done with them, including sequences and series, and with an introduction to more advanced aspects at the end. Part II - Geometry. Here we discuss geometry, fi rst with the basic study of triangles, their various centers, and other things that can be said about them, then with a look into angles and trigonometry, and then with more advanced aspects, namely real coordinates, and complex numbers too, and fi nally conics and other basic algebraic curves. Part III - Functions. Here we discuss functions, fi rst with a classical study of the polynomials and their roots, in general and in low degree too, followed by the standard modern theory of continuous functions, meaning basics, intermediate value theorem, derivatives of fi rst, second and higher order, and then integration and basic applications. Part IV - Vectors. We discuss here vectors, fi rst with a general introduction to space geometry, featuring tetrahedra, polyhedra and related topics, then with a discussion of basic linear algebra and matrix theory, followed by a discussion of multivariable functions and their analysis, and with an introduction to basic mechanics at the end