Front Matter

Modern natural sciences (physics, chemistry, biology,…) are based on the understanding of Nature through models. The validity of a model is, in turn, measured by its capacity to accurately describe and predict natural phenomena in the simplest manner possible.

The experiments designed to test the validity of a givenmodelmust, therefore, be readily reproducible in different laboratories. To precisely define the results that will be observed in different laboratories, modern science is led to express the results in numbers that represent the measured quantities of the experiments. Thus, two separate laboratories with identical experimental setups can compare their results by simply comparing the numbers obtained by measuring the pre-established quantities associated with the experiments.

For example, if we are studying a moving physical object under given experimental conditions, we may wish to measure the distance between its initial position and its position at different times afterward. We will need real numbers to represent these distances and their respective times. In particular, as we aim to study why the object moved as it did, we will be led to develop mathematical models that will include real functions of position versus time to compare with and to analyze the measured results. The analysis of the measured results through the mathematical models will necessarily include the application of the properties of real functions.

Natural sciences are thus unavoidably led to incorporate numbers and functions. Functions and their properties becomemore than just useful tools, they become an essential intrinsic part of themodels with which we describe, understand, and attempt to predict nature. Three real functions that appear again and again in physics in a wide range of systems are the exponential, the logarithmic, and the trigonometric functions.

This book builds upon three previous books: Logic for Physicists, Set Theory for Physicists, and Real and Complex Numbers for Physicists. After presenting a brief review of logic, set theory, and isomorphism in the first chapter of this book, a brief review of natural numbers and integers in the second chapter, a brief review of rational numbers in the third chapter, and a brief review of real numbers in the fourth chapter. Working toward the exponential function, we present and derive properties of real number exponentiation with rational number exponents. Based on the properties of exponentiation with rational exponents, we present and derive properties of themore general case of real number exponentiation with real number exponents.We then define the number “e” and study general properties of the exponential function. Based on the properties of exponentiation with real number exponents, we present and derive properties of logarithmic functions in general and of the natural logarithmic function “ln(x),” in particular.We then present and discuss elements of geometry and analytic geometry, working toward trigonometric functions. Finally, we present and derive general properties of trigonometric functions and, in particular, properties of the trigonometric functions of a sum.

I wish to thank my family for their support, and I am grateful to the many students and colleagues that I have interacted with over the years.

Dr. Pereyra pursued his undergraduate studies in Physics at the Universidad Central de Venezuela, Caracas, where he graduated in 1991. He pursued his graduate studies in Physics at the University of Maryland at College Park, where he obtained his M.S. in 1995 and Ph.D. in 1997. Currently, he is an Associate Professor in Astrophysics at the Physics and Astronomy Department of the University of Texas Rio Grande Valley. His research has been largely in the development of computational models of physical systems.