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This is a textbook for beginning mathematics students. Knowledge of school mathematics is not presumed: it starts with the basics of counting. The underlying idea is that the best way to learn mathematics is by doing mathematics. For beginning students it is sometimes a problem to assume when looking for proof. For the exercises in this textbook, this situation does not occur: except for the introductory part all is built on Peano's axioms for the natural numbers, using the language of set theory only. The book starts explaining the way mathematics works: the use of intuitive set theory and the relation between language and mathematical entities. The common thread in the book is the construction of the number system all the way from the natural numbers, via the rationals and the reals to the complex numbers. For the student, the advantages of this approach are: One learns concepts which are fundamental for all of mathematics. The common thread offers a natural way for the introduction of these concepts. It helps to stay motivated during the course. One learns to think like a mathematician. One obtains insight into the way mathematics is built from simple ideas. It helps to decide whether one is fitted for a mathematics study. For the interested reader also the other possible completions of the rationals - the p-adic numbers - are constructed. The book contains more than just the construction of the number system: there is also attention for its use, especially in combinatorics, number theory and cryptography, leaving mathematical analysis to the many textbooks for analysis and calculus courses.

The world around us is saturated with numbers. They are a fundamental pillar of our modern society, and accepted and used with hardly a second thought. But how did this state of affairs come to be? In this book, Leo Corry tells the story behind the idea of number from the early days of the Pythagoreans, up until the turn of the twentieth century. He presents an overview of how numbers were handled and conceived in classical Greek mathematics, in the mathematics of Islam, in European mathematics of the middle ages and the Renaissance, during the scientific revolution, all the way through to the mathematics of the 18th to the early 20th century. Focusing on both foundational debates and practical use numbers, and showing how the story of numbers is intimately linked to that of the idea of equation, this book provides a valuable insight to numbers for undergraduate students, teachers, engineers, professional mathematicians, and anyone with an interest in the history of mathematics.

Mathematical Modeling describes a process and an object by use of the mathematical language. A process or an object is presented in a "pure form" in Mathematical Modeling when external perturbations disturbing the study are absent. Computer simulation is a natural continuation of the Mathematical Modeling. Computer simulation can be considered as a computer experiment which corresponds to an experiment in the real world. Such a treatment is rather related to numerical simulations. Symbolic simulations yield more than just an experiment. Mathematical Modeling of stochastic processes is based on the probability theory, in particular, that leads to using of random walks, Monte Carlo methods and the standard statistics tools. Symbolic simulations are usually realized in the form of solution to equations in one unknown, to a system of linear algebraic equations, both ordinary and partial differential equations (ODE and PDE). Various mathematical approaches to stability are discussed in courses of ODE and PDE.