Elements of Higher Mathematics

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.