The Princeton Companion manages to be so much more than your typical popular mathematics book. While obviously a thousand pages could never hope to include everything that has been written on this vast subject, where this text shines is its uncanny ability, relative to its page limit, to paint a compelling picture of the modern mathematics landscape (emphasis on modern; if you don't know what I'm talking about read the preface) that is both thorough and also motivating. More precisely, this book gives an overview of essentially all of the most important areas of active research mathematics, while striking a balance between being too glib versus overly dry and verbose. If you're looking for the former see pop math books galore. The latter being something like a graduate mathematics textbook or monograph assuming all sorts of advanced prerequisites that might take semesters or even years to understand. Clearly it would be impracticable to attempt to include that level of detail here.<br><br>This is not so with The Companion. To give a concrete example, consider this definition of a scheme given by the book in its chapter on Algebraic Geometry: "Roughly speaking, a scheme is an algebraic set where we also keep track of the multiplicities and of the directions they occur in". On the one hand this lacks the formalism that would be necessary for an Algebraic Geometer. But it is also about as good as one could expect in a book this size, and indeed the concepts leading up to this, algebraic sets and multiplicities, are adequately explained without handwaving.<br><br>And this is the real virtue of the book: it provides an intuitive understanding of concepts, similar to an introductory textbook on a particular mathematics topic like say linear algebra that might forgo the abstract definition of a vector space over a field for the sake of efficiently providing very concrete examples over R or C. This can be done without talking about bases or dimension or everything that you would learn in a graduate level course. The book does something similar in its chapter on Algebraic Numbers, focusing on quadratic number fields specifically for most of the chapter until the very end when it becomes more appropriate to generalize the concept afterwards.<br><br>Overall a profound and inspiring mathematics book. I haven't seen anything else quite like this book before and I've been a passionate reader of mathematics for over a decade. If you have any interest in math do yourself a favor and purchase a copy of this book for yourself. And then you can purchase the books in the "Further Reading" sections once you're ready to learn even more about these topics. This book is a gateway drug to math you've been warned.