Messen und Zählen. Eine Einfache Konstruktion der Reellen Zahlen
The aim of this book is to provide a rigorous foundation of the real number system. The first step is a treatment of natural numbers and their properties, which are stated as axioms (only the last chapter outlines the possibility of a construction of natural numbers on a set-theoretical basis). The real numbers are then defined as infinite sequences of decimal digits. At this point, it is possible to introduce the ordering of real numbers and prove the supremum property. The operations of addition and multiplication are first defined for numbers with finite decimal expansion (by shifting the decimal point, the problem is reduced to addition or multiplication of natural numbers); by means of a limit process, they are extended to all real numbers. The book also discusses additional interesting topics, in particular the definition of powers with real exponents, exponential and logarithm functions, Egyptian fractions, computer implementation of arithmetic operations and the uniqueness of real numbers up to isomorphism. The text is elementary and contains numerous remarks on the history of the subject.