This book is an elementary axiomatic development of the real number system. Its aim is to make available to the non-science student and to the teacher of secondary school mathematics the fundamental concepts that underlie the structure of algebra and arithmetic. For this purpose it adopts an unsophisticated approach to the general methods of modern mathematics and at a leisurely pace explores and develops the main properties of real numbers as logical consequences of a system of fundamental assumptions. Although in recent years several texts on this subject have been published or have been made available in translation, for various reasons they have failed to meet the urgent needs of teachers of mathematics for whom this material forms an essential mathematical background. Either these texts are highly sophisticated in viewpoint, as is Landau's Grundlagen der Analysis, and require an unusual competence in mathematics and extensive familiarity with mathematical ideas, or they treat the subject from a non-mathematical viewpoint, omitting its rational content, and contribute little more than a descriptive account of number notations and the evolution of the number concept. The average prospective teacher of mathematics finds such a treatise as Landau's much too difficult to read; and certainly he will not find the mathematical content he needs as a teacher in the popular treatments of the subject. It is to fill this void in textbook material that this book is written.

The essential features of this text are the abstract viewpoint it adopts at the outset toward the totality of real numbers as logical entities; the extensive set of axioms it selects for characterising the real numbers; the unsophisticated method of proof it uses throughout which is appropriate to the subject matter and to the maturity of the beginning student; and the completeness of development it obtains of the properties of real numbers at the elementary level.

To accomplish the aims of the text, I have selected the axiomatic approach because its broad and unified viewpoint allows consideration of extensive mathematical material and at the same time provides an effective means for teaching the detailed content. This method affords a clear formulation of what is to be proved and builds up a compact body of theorems as substance for constructing proofs. For the fundamental assumptions I have chosen a system sufficiently extensive to assure a complete characterisation of the totality of the real numbers. From the outset real numbers are the subject of inquiry. For logical as well as for pedagogical reasons, the study begins with the axioms that describe the arithmetic operations.

The educational advantages of characterising real numbers at the outset are clear when this approach is contrasted with the alternative one that starts with the Peano axioms of the natural numbers, for the Peano axioms are much too abstract and intangible for immature minds to grasp and apply. Moreover, sufficient time for mastering the logical difficulties inherent in developing the consequences of the Peano axioms must certainly leave little opportunity in a normal teaching program for extending the concept of number to the rationals and the irrationals.

This system of real number axioms is an exceptionally workable and efficient one, as is evident from the ease with which it develops an extensive body of algebraic and arithmetic results in elegant simplicity and clarity. Throughout the text I have followed a consistent procedure in constructing proofs in order that by repetitive application of the method the student can learn to apply it effectively for himself. Because the book is written for the beginner, inexperienced in algebraic proof, I have spared no effort to make clear what is to be proved and how a proof will be made, even to show that the proof is complete. When derivations are left to the student, ample suggestions are given for their proof.

To resolve the problem of writing the many reasons needed in a proof, the text employs an alphabetical code of references, based on abbreviations of descriptive names given to the axioms, definitions, and theorems.

Frequent exercises have been inserted to help the student fix or clarify a newly introduced idea or for the purpose of organised review. No answers have been supplied to the few exercises on numerical computation, as it is thought the student should check all such work for himself.

In order to develop the significant results of algebra and arithmetic in a single book at an elementary level, I have found it necessary to restrict the content of the text severely. For instance, I have found it quite inappropriate at the elementary level to introduce the terminology of modern algebra in any way, either in the body of the text or in footnotes. Also, space was not available to discuss alternative systems of axioms equally well suited for developing a logical structure of algebra, or even to introduce interesting examples of simple axiomatic systems as a desirable background for the study of abstract mathematics.

However, to point the way to further development of the ideas of this elementary text, I have supplied a short bibliography of references at the end. Of these, the more readily usable are Higher Algebra for the Undergraduate by Marie J Weiss, and A Survey of Modern Algebra by G Birkhoff and S MacLane. The material of Chapters 1 to 7 and part of Chapter 15 has been successfully taught over several years at Brooklyn College in a one-semester course at the freshman level. With varying emphasis different instructors have found the approach, methods, and material of the text highly rewarding for instruction.

To these earnest and enterprising colleagues, who by experimentation have proved the worth of this untraditional approach to fundamental mathematics and who have helped both themselves and their students find keen delight in intellectual pursuits, I express my deepest gratitude and appreciation. And even more thanks I give to the unspoiled students whose honest and critical reactions to the abstract material and its method of presentation have revealed to me its relevant substance and forced me to rely on simplicity and clarity of exposition.

May H Maria
Brooklyn, New York
September, 1958

According to the preface, this book is "an elementary axiomatic development of the real number system. Its aim is to make available to the non-science student and to the teacher of secondary school mathematics the fundamental concepts that underlie the structure of algebra and arithmetic." It seems to have been designed to fill the gap between Landau's Grundlagen der Analysis and the so-called "popular" treatments of the subject.

The book treats the ordinary operations of arithmetic, the order relation, a bit of high school algebra (more or less carefully presented), the positive integers, the continuity property of the real number system, and number notation. The exposition is painstakingly detailed.

One of the features of the text is an elaborate code which is used to refer to certain axioms, definitions, and theorems. For example, Tir is the Theorem on Irrational Numbers, which runs as follows: "If a non-zero rational number $r$ is combined with an irrational number $p$ by any one of the four operations of arithmetic, the result produced is an irrational number; in symbols, $r+p, r-p, p-r, rp, \Large\frac{r}{p}\normalsize , \Large\frac{p}{r}\normalsize$ are irrational numbers." According to the author's preface, "Experience in classroom teaching shows that the students use the code with alacrity and effectiveness in making full and concise proofs."

This reviewer feels that the book under review is a worthy addition to the literature; but on the whole he found the exposition somewhat clumsy. In a few places terms are used before they are explained (e.g. "empty set,") and in some places no explanation is offered where one is clearly required. (e.g. 0! is used, but never defined. Since 3! is defined, the reviewer presumes that no knowledge of factorials is assumed.) Functions are never mentioned, even though the use of functions could have simplified the treatment considerably. These objections, however, may possibly be regarded as minor. Finally, the exercises in the book are many in number and generally non-computational in nature.

The author's object is to build up the theory of real numbers on an axiomatic foundation without using the terminology of modern algebra or the symbolism of mathematical logic, to make the text available to a very wide class of students. The aim is an excellent one and has been very successfully accomplished. The book is very easy to read, sufficiently reliable for its purpose and containing a good many elementary results not readily available elsewhere, for instance a very clear proof of the validity of the familiar square root process of elementary arithmetic which is generally and mistakenly supposed to rest simply upon the corresponding algebraic process.

One primary difficulty in the axiomatic construction of the number system is to reconcile the natural numbers defined in the formalism with the natural numbers used in the descriptive text (and even those at the head of the page). Dr Maria does not draw the reader's attention to this problem, and I think that she should. To integrate the natural numbers defined axiomatically with the numbers of everyday usage what is needed is the incorporation of a theory of finite classes into the axiomatic development, or what comes to the same thing, the incorporation of a counting operator into the system (as for instance in the introduction to the reviewer's Recursive Number Theory).

The publishers bill The Structure of Arithmetic and Algebra as "A clear, concise treatment of the fundamental principles of modern mathematics." The statement is incorrect in nearly all phases. The book consists of two parts which we shall review separately. The first twelve chapters contain a reasonably clear and precise - but certainly not concise - axiomatic presentation of the real numbers and careful proofs of their more important properties. The first chapter muses at length about mathematical proofs but not until the second chapter does any mathematics appear. In the next eleven chapters the real numbers are defined by seventeen axioms and some of their properties are discovered; but no concern is given to existence. Every detail of every proof is tediously referred to the appropriate definition, axiom, or pre-proved theorem in the best tradition of plane Euclidean geometry. Throughout the book the reader is faced with the unfortunate choice of memorising one hundred and thirty-five idle, useless, and nearly meaningless abbreviations (such as TND which stands for "Theorem on Negative of a Difference") or skipping the whole thing. Since a proof is vitiated if the reader must look up every reference, the complete index of abbreviations is of use only for emergency aid. The problem of an adequate reference system in an elementary proof is eternal but it may be better solved orally by the teacher, rather than by a device in the textbook. The exercises demonstrate a further difficulty of the inordinate completeness of the text. Although the author has made a valiant effort, too often the interesting problem has been solved as a part of the text material and the exercises must be unnatural, trivial, or repetitious.

New ideas are introduced leisurely with adequate simple examples to aid the student to understand the principles involved. Too often, though, this degenerates into the philosophy that if an idea is worth stating once, it is better repeated five times.

The book is "modern" in spirit only in so far as it insists on rigour in algebra (and the demand for rigour is at least two thousand years old); it deliberately ignores the power of generalisation. For example, in successive sections the set of residue classes of the integers modulo a prime number and the rationals are shown to satisfy the field postulates, but the parallelism of the systems is neither pointed out nor exploited.

The latter part of the book is loosely connected melange. The first of three chapters concerns solutions of quadratic equations. Here again lack of generalisation causes trouble. Only the real numbers have been considered so the discriminant of the equation must be arbitrarily restricted to be nonnegative.

The next chapter introduces decimal notation for the real numbers and some elementary ideas of limit, and leads the student toward the idea of countability and uncountability but never quite reaches them.

The last chapter has an over-simplified and misleading short history of number notations, a lengthy discussion of the abacus, and much detail on number bases.

While most elementary algebra books have a deplorable lack of rigour and detail of proof, and at best pay lip service to the axioms of the number system, the solution does not lie with the other extreme. It is interesting to see how the real numbers can be placed on a logical basis and derived from a few axioms. The real danger is that a student, having read such a text, would be well aware of what was involved in a careful proof and hate every tedious detail of it.

This book is designed primarily for school-teachers of mathematics, as a corrective to the traditional approach to elementary mathematics, but it may be commended also to any general reader who desires a logical account of algebra and arithmetic. The treatment adopted is based on axioms for the real numbers, introduced in stages in the order of their increasing conceptual difficulty. A disadvantage of this approach is that the numbers 1, 2, 3,... turn up as elements of an abstract system, and such familiar phrases as "a set of $n$ objects" call for a good deal of explanation - perhaps far more than the author has given.

The author states the aim of this book is to "make available to the non-science student and to the teacher of secondary school mathematics the fundamental concepts that underlie the structure of algebra and arithmetic." Actually, the book does more and does it very well. It could easily be used as a text or for self-study, not only for non-science students and teachers of secondary school mathematics, but also for elementary school teachers, particularly those in the upper elementary grades and in junior high school. Elementary teachers would probably restrict themselves to Chapters 1-7 and 15.

The totality of real numbers is taken as logical entities. Then the main properties of the real numbers are investigated as logical consequences of a system of fundamental assumptions. There is no attempt to start with a minimal set of assumptions, but a sufficiently extensive system is chosen to assure a complete characterisation of the totality of real numbers.

The outstanding feature of the book is its logical development. It assumes little or no background, but does demand an inquiring mind and the willingness to do some work.

In the beginning chapters, to help those readers with very little experience with algebraic proof, the author analyses precisely what assertions are made by an axiom and what the axiom does not say. Examples using these definitions and axioms are then worked out in detail with reasons given for each step of a proof. (An alphabetical code of references is used, which, although somewhat cumbersome, makes the detailed examples possible.) Care has been taken to eliminate misunderstanding as, for example, by using a negative sign as a superscript to avoid confusion between the sign of the number and the operational symbol for subtraction.

The exercise lists, although adequate, could be improved by including some numerical problems in more of the lists.

In Chapter 1, sections 1.8, 1.9, and 1.11 might falsely lead an inexperienced reader to believe that background in logic is necessary to understand the logical development. However, this is not the case and concepts not completely clear at first reading should not keep anyone from beginning Chapter 2.

In summary, I feel this is an excellent, though not a popular, treatment of the real number system. Any teacher, elementary or high school, would benefit greatly by reading this book. Indeed, it would also serve as an excellent book for superior high school students by offering insight on the real number system as well as developing skill in abstract algebraic proof.

The author states that this book is "designed for the reader with a limited background in science and who wishes to understand the fundamental concepts that underlie the structure of algebra and arithmetic. It adopts a simple approach to the general methods of modern mathematics and at a leisurely pace explores and develops the main properties of real numbers as logical consequences of a system of fundamental assumptions."

Some essential features of the text appear to be: (1) the abstract viewpoint it adopts at the outset toward the totality of real numbers as logical entities; (2) the extensive set of axioms it selects for characterising the real numbers; (3) the unsophisticated method of proof it uses throughout, which is appropriate to the subject matter and to the maturity of the beginning student; and (4) the completeness of development it obtains of the properties of real numbers at the elementary level.

Some other characteristics of the book that appealed to this reviewer are: (1) the axiomatic approach, beginning with the study of the axiom that describes the arithmetic operations; (2) the consistent procedure in constructing proofs; (3) the use of an alphabetical code of references, based on abbreviations of descriptive names given to the axioms, definitions, and theorems; (4) the frequent exercises that have been inserted to help the student fix or clarify a newly introduced idea or for the purpose of organised review; and (5) the simplicity and clarity of exposition.

The style of the book is fascinating and stimulating. The first few chapters are written primarily for the beginning student of mathematics but the author has included enough content in this book to make it a worth-while study for advanced students also.

The book aims to provide an axiomatic structure of the real number system, elementary and "unsophisticated", for the use of mathematics teachers (in the USA) and students outside the actual mathematical subjects. The author approaches the matter with great seriousness and does not try to evade the difficulties, although for the sake of readability she does not use idioms from abstract algebra and leaves the question of the freedom from contradiction of the axiom system to mathematical logic.

But the text is undeniably puzzling in places. The first axioms state that within the real numbers one must be able to use the four types of arithmetic (i.e. that the numbers constitute a number field), after which one encounters the definition: "A fraction is said to be in reduced form if its numerator and denominator have no common factor." One wonders; in any case, it is not whole numbers that are intended, because they are not introduced until much later. A little later, a theorem is established with proof, called the theorem on substitution in fractional forms, which states that if two fractions are equal to two other fractions, then the sum of the first two is equal to the sum of the last two; the reader gets the impression that fractions are not real numbers after all, and the justification given for including the theorem can only be misleading to the reader and does not touch on the core of the problem at all.

The axiom system is further developed with the ordering of the numbers and with an induction axiom, which gives the natural number series (and only then does the author allow herself to give, for example, the distributive law with an arbitrarily finite number of terms), and finally with a continuity axiom, which makes it possible to write down the real numbers as decimal fractions, finite or infinite.

While the beginning of the book sticks to abstract considerations, later it is filled with pedagogical sections (e.g. on the importance of a good order when setting up calculations) and technical sections on arithmetic, such as combinatorics and on the solution of the quadratic equation; when in the latter several different solution methods are implemented, and it is shown that the solutions satisfy, and it is then further stated that in arithmetic examples one should test both for practical and theoretical reasons, then it seems to be pushing the envelope too far. The mention of the positional system gives rise to a longer historical consideration of the Roman number system and arithmetic.

To promote readability, the text's references are given in the form of abbreviated terms of the type TONN=Theorem on Order of Natural Numbers, a total of about 150 (!). The system works better than one might fear.

The author is employed at Brooklyn College, where much has been done in recent years to improve the education of mathematics teachers for secondary schools in the United States, and the book must be seen in the light of these efforts. It is a quite interesting, although not entirely successful, attempt to provide insight into the real numbers.

In 1547 William Buckley, Englishman, wrote a treatise entitled The Rules of Arithmetic, which [treatise] is noted chiefly for a short discussion on combinations. Much of the book, however, is concerned with rules for writing down numbers, for addition and subtraction and the other arithmetic operations. An expatriate from Mars who left England in 1547 might accordingly be predisposed to wonder at the lack of progress of English science since Prof Maria is concerned to discuss many of the same topics as Buckley. Prof Maria is, in fact, the latest of a long line of mathematical logicians who are concerned to describe on what mathematical thought is based. She does not waste much space on what is a number but moves briskly through the operations of addition, subtraction, multiplication and division. There are chapters on rational numbers, and actual numbers, irrational numbers and the properties of integers. In the generalisation of the distributive law a little combinatorial theory is introduced.

This book may be interesting to the beginner in mathematical lore if he is prepared to recognise that in the end he will have to take almost everything here discussed for granted. For the statistician, who in many circumstances would be hard pressed to defend the logic of his operations, it is unlikely that this book will have much appeal.

"This book is an elementary axiomatic development of the real number system. Its aim is to make available... to the teacher of secondary school mathematics the fundamental concepts that underlie the structure of algebra and arithmetic." - Preface. It begins with postulates such as the commutative laws, which are in some ways more convenient starting places than the Peano postulates. The properties of zero and of additive inverses are derived in Chapter 3 in a way reminiscent of the construction of negative numbers in some texts on modern algebra. Mathematical induction and the natural numbers are not reached until Chapter 8 - an indication of the slow thoroughness of the development. This is an excellent book for readers who wish an extremely detailed but elementary treatment. Its only motivating devices are "internal." (For example, the need for a new axiom "to identify in the real number system other elements that are non-rational" is discussed at length.) The book neglects, however, the "external" motivation of axiomatics which comes from introducing a wide variety of number systems, finite and infinite, commutative and otherwise.

Theorems about the real numbers in their full generality, including irrationals, are proved in chapter 12, from a continuity axiom on the existence of least upper bounds. Limits of sequences are discussed, and decimal expansions two chapters later; giving a substantial overlap with "advanced calculus" texts.

One gentle criticism: This reviewer believes that as teachers of mathematics we would do well to employ language with more significance and less bleak austerity than that now current. Consider, for example: "Here mathematics meets the problem of how to talk about its abstract entities, of what to say about them as mere blanks. What mathematics does in fact is to study not the entities themselves but relationships between a system of entities". Instead of speaking of the relationships of systems of blanks, should we not stress, for example, the importance of studying the structure of symbolic, detached languages; without undue regard, that is, to their content? Can we not find more constructive descriptions of the nature of mathematics?

Be that as it may, teachers who wish to spell out in detail the basic logical interrelationships of the real numbers will enjoy reading this book.