The book is very good for people who want to be a high-school teacher of math, or be a mathematician. Even if you don't take a class with this book, read it on your own before taking real analysis. It will make your thought and logic complete and precise. A really nice training and practical preparation to do analysis.

The book is very simple and short. It deals with number system from natural to complex, gently. Simple things are usually not easy, though.

I took real analysis twice long time ago, but this book still improved my thinking of numbers very effectively.

I recommend this book to those who want to be precise and correct, no matter you are math or theoretical physics people.

And also for high-school students who want to know what pure mathematicians really do.

And also for independent thinkers of mathematical science, and would-be philosophers!

Landau's most known book is this little masterpiece. If you want to see everything about numbers proved, from the beggining, assuming just logical and set-theoretical principles and the five Peano axioms, you will find it here. You will see the proof of why 1+1=2, for instance, or why a+b=b+a. Usually people learn analysis with a lot of pictures and assumptions, and every once in a while one asks himself: how does it all begin? Because sometimes you see something which ought to be evident proved, and something which ought to be proved assumed. I recall that when I first met this book I became amazed and read it through with a lot of willing. It is difficult reading, so be prepared. That's because Landau wanted to follow the axiomatic Euclidean style in its most pure way. So the book is in the non-merciful telegram style of presenting everything in terms of "Axioms", "Definitions", "Propositions". Few books before and after strove to reach such pure and clear presentation of arithmetic. Thank God some one had once the patience to write such careful and complete text! In this book the words of Edgar Allan Poe are more than anywhere true: "What I here propound is true:-therefore it cannot die:-or if by any means it be now trodden down so that it die, it will 'rise again to the Life Everlasting'".

This book would more correctly be described as a book on analysis, because of the completeness of its proofs. It assumes you know just arithmetic of natural numbers, integers, rational, real and complex numbers-that is to say, just how to sum, multiply, subtract and divide. Edmund Landau wrote a masterpiece, because nothing is left without proof. You will not find a single step missing. From arithmetic to all the concepts of calculus, like differentiation, integration, infinite series and sequences, this book contains the mathematics that most mathematicians should know presented in a perfect way. It is not easy reading, though. Landau strives to reach the perfect axiomatic presentation, so like Euclid's "Elements" the book is the clear and beautiful presentation of a doctrine. I dare say that no book in analysis approaches Euclid's ideal of presentation better than Landau's, never in the past and never in the future. What you will find in this book is, in its best appearance, truth, that will not change even in a thousand years.

In his foreword to the student, Landau writes, "Please forget everything you learned in school; you haven't learned it." In the rest of the book he goes on to derive the properties of the natural, rational, real, and complex numbers from a set of five axioms. His treatment of the material is altogether rigorous. He makes no appeals to "common sense", but instead deduces theorem after theorem in a well-ordered, comprehensive system. The reader therefore comes to a good understanding of topics ignored in most math classes and hence builds mathematical confidence. This book is good for the student of math who hates to hear, "It just is that way; it doesn't matter why," and for the student of philosophy who wants a deeper understanding of mathematical reasoning as described by authors such as Betrand Russell. The only drawback is that the rigorous method is overwhelming at first. But once you trudge through the first few pages, everything else should come more easily.

Dover, please put this classic back into print!!
This does for numbers what Suppes (1960) did for ZF set theory.