This is an introduction to the basic parts of number theory culminating in quadratic extensions. The author uses an algorithmic approach following Gauss and Kroneckers’ dictum that everything defined should be managed by exact computations (manipulations) with positive integers. The book contains many examples (each chapter ends with examples or computer experiments, the answers given at the end of the book). To work preferably with positive integers and to avoid negative numbers is the first unusual element in the exposition. Hypernumbers (which are formal expressions of the form of a quadratic irrational) form the second nonstandard tool used here. The third unexpected feature of the book is that the author gives new names to familiar objects. Nevertheless, the book is written in a lucid and readable style and it can be recommended to students interested in learning classical parts of number theory and some of its applications in a less standard way.