Top positive review
The closest thing there is to a mathematical encyclopedia
on June 28, 2016
This epic single volume spans all major areas of modern mathematical research. Each section was written by some of the most eminent mathematicians of our time, and covers topics in Algebra, Statistics, Topology, Computation, Combinatorics and so forth, and contains the basics of topics written in a relatively accessible but still rigorous manner. The explanations are concise, yet precise, and are designed for readers who are new to the topic.
Make no mistake, however, "relatively accessible" still means that this book requires some background in university-level mathematics. Though the book does contain an introductory chapter that goes over all prerequisite material, readers who are not used to thinking about advanced mathematics will likely find many sections to be quite a slog to read through, especially once the notation and the abstraction starts stacking up. But it would be unfair to fault the authors and editors for this -- writing rigourously yet concisely about Modular Forms, Orbifields, Differential Topology, etc. is an impossible task for people with no background at all. Indeed an illustrative example would be the overview of trigonometric functions, which would likely be a chapter in a high school textbook but is covered in the Companion in a single page.
To readers with a bit of mathematical background, however (perhaps a few proof-based courses in university), I reckon the Companion to be an invaluable reference for all areas of mathematics.