The real numbers form an ordered body. This, and thus the study of inequalities, is the starting point of this chapter. The central concept is the convergence of sequences. This in turn leads to the concept of completeness and to the definition of \(\mathbb{R}\) as a complete ordered body. Completeness is examined from various angles, which also results in natural constructions of \(\mathbb{R}\) from the rational numbers. The transition to complex numbers is then a small step. However, trigonometric functions are already used here in anticipation of Volume 2 for the polar coordinate representation. In the treatment of series, the concept of summability offers considerable methodological advantages. It is therefore consistently used. The continuity of real and complex functions on subsets of \(\mathbb{R}\) or \(\mathbb{C}\) is extensively discussed, including the peculiarities in compact definition areas. As an application, one obtains the classic proof of the Fundamental Theorem of Algebra. The chapter concludes with the introduction of real exponential and logarithmic functions.