"Number Theory IV" is a comprehensive academic book that delves into the complex topics of number theory. Over a period of more than six years, this work was authored by N.1. Fel'dman and Yu. V. Nesterenko. It offers in-depth insights into the transcendence of numbers and the approximation of algebraic numbers. The authors have integrated valuable contributions from colleagues who reviewed the manuscript and suggested numerous improvements. This book is not only the result of intensive research but also a tribute to collaboration within the scientific community. It is aimed at students and professionals seeking a deeper understanding of number theory and wishing to trace the development of this discipline. The English translation by Neal Koblitz makes it accessible to a broader audience, allowing for the comprehension and discussion of the complex concepts of number theory.