# That sums it up

Elements of Logic Via Numbers and Sets - Elementary Number Theory - Introductory Mathematics - Mathematical Methods for Physics and Engineering - Basic Linear Algebra
March 13, 1998

Springer (London) has produced the first titles in its Springer University Mathematics Series, "SUMS". Five titles are available now, with more to follow. The books reviewed constitute a set of reasonably priced texts that will appeal to students studying mathematics on its own or, in some cases, as part of a science or an engineering programme.

Paul Matthews's Vector Calculus is an excellent introduction to the subject. The book is written without frills and adopts a "this is how to do it" approach. The material covered ranges from an introduction to vectors through vector calculus and the integral theorems to a discussion of Cartesian tensors. The final chapter gives an indication of the scope of applicability in electromagnetism, solid and fluid mechanics. The material has been class tested, and this comes out in the explanatory approach to the worked examples. The end-of-chapter summaries are particularly useful in highlighting important concepts.

One could point to omissions from the text, but to do so would be to miss the point. Students who master the content of this book will be capable of picking up additional material from the literature.

T.S. Blyth and E.F. Robertson's Basic Linear Algebra is a comprehensive introduction to the field. As expected, the material is standard and this is not a criticism. There is full coverage of matrices, including eigenvalues and eigenvectors. The introduction to vector spaces is clear and has many examples, which will be appreciated by students as they encounter what may well be their first major abstract concept. There is a lot of material in this book, and end-of-chapter summaries would have been helpful. Also, a very brief indication of the range of application of the subject might serve to increase the readership.

Geoff Smith's Introductory Mathematics: Algebra and Analysis is a delight. The book adopts a gentle style with discursive commentary throughout and contains real humour in places. This is, though, a book with a serious purpose, namely to introduce the fundamental ideas of algebra and analysis with precision and rigour.

Taking the view that to ask where numbers come from is at least as dangerous a question as to ask where babies come from, the book begins with a treatment of sets, showing clearly the problem of set definition. We learn that S means "add this lot up'' and are warned about "confusion opportunities''.

Complex numbers are introduced from first principles, and all through the treatment of the exponential and hyperbolic functions, the reader is persuaded of the importance of doing and writing mathematics with due regard for rigour.

Elementary linear algebra is included, with a treatment of vectors, matrices and determinants. Linear independence is considered, but eigenvalues and eigenvectors are left out, and this is appropriate. Group theory is introduced via a discussion of permutation groups leading on to abstract groups. The need for rigour is stressed again when sequences and series are considered, particularly the dangers arising from sloppy arguments in dealing with limits and infinite sums. Analysis is considered to be "calculus with attitude'', and here is the introduction to the feared "e-d technology''. This is done with sympathy and enables the concepts of continuity, limit and differentiability to be handled simply but with rigour.

There is a list of further reading, and the book is supported by a website, which also contains more examples. D.L. Johnson's Elements of Logic via Numbers and Sets aims to deal with the change of approach from that taken in schools to that required in universities. The emphasis is on proof-based formal logic with the underlying theme of number. The writing style is friendly, but the quote from Clint Eastwood at the start of chapter two, "If you want to play the game, you'd better know the rules'', makes the purpose of the book clear.

The book opens by looking at numbers and methods of proof of assertions about them: contradiction, contraposition and induction in its various forms.

Propositional logic is dealt with at the start of chapter two, which goes on to consider predicate calculus. The account of syllogisms could be slightly clarified, and it is unfortunate that a figure-labelling error occurs at a crucial point. The validity or otherwise of various syllogisms is demonstrated using Lewis Carroll diagrams, although a more transparent approach is seen later. The chapter on sets and operations on them introduces proof of the set-theoretic laws using truth tables, perhaps better thought of as membership tables. A tautology could perhaps have been usefully defined at this point. The chapter closes with a brief taster of the idea of topology on a set.

Later chapters are devoted to relations on sets and maps between them, and they give an insight into the notation and ideas used to describe modern pure mathematics. It is probably by design, and is a part of the training process intended by the author, that when the Peano axioms for the natural number system are being considered, the style changes to a somewhat staccato "theorem-proof-theorem-proof" form. The final chapter discusses cardinal numbers, including the Cantor-Schroeder-Bernstein theorem, a theorem that "has all the hallmarks of a top-class named theorem''! A brief bibliography is included.

Gareth and Mary Jones's Elementary Number Theory is a fuller and deeper text than the others reviewed, and it contains a complete undergraduate course in the subject. Number theory became unfashionable earlier in the century, but, as the authors point out, the application to cryptography now forms the basis of many secure communication systems. Aspects of the subject have thus followed the well-trodden path from abstract pure mathematics to a high revenue-earning application.

Concern is expressed about students' lack of familiarity with the need for, and the technique of, proof. The book encourages experimentation, but there is emphasis throughout on proof construction. The first three chapters, which cover divisibilty, prime numbers and congruences, are taken at a gentle pace and are appropriate for an introductory course. Chapter four deals with congruences with a prime power modulus, and in the chapter on Euler's function there is an explanation of public-key cryptography. The pace picks up at this point, and a basic knowledge of group theory is needed to master the material on the group of units. A chapter on the Riemann Zeta function makes the link from analysis to number theory and the pace reduces slightly, allowing a fuller explanation of some of the more complex derivations.

Sums of squares are dealt with, and included in the discussion is a section on quaternions. The final chapter sets Fermat's last theorem in its historical context, and gives a flavour of some of the modern developments that led to its proof.

Mathematical Methods for Physicists and Engineers, by K.F. Riley, M.P Hobson and S.J. Bence, published by Cambridge University Press is completely different in concept, aiming to be a comprehensive handbook over a wide field.

The opening chapters are designed to review material in the A-level syllabus, and the clarity of writing throughout makes it suitable as a textbook for undergraduate use, although rather more worked examples would be valuable.

The dual target of providing a book for physicists and engineers is very difficult to achieve, and this book will probably be found most useful by the former category of reader.

Having said this, the treatment of ordinary differential equations, including a most readable introduction to Green's functions, is very thorough, and there is a good introduction to partial differential equations. As with Matthews's book, the treatment of tensors is particularly clear. The introduction to integral equations is well judged, containing just sufficient of the theory to support the techniques being described.

Supporting the view that this a book primarily for physicists, there is a substantial chapter on group theory, including non-Abelian groups. There is also an extensive treatment of representation theory and its application to molecular bonding and normal modes of vibration. The chapter on probability contains the standard material and includes a description of the important continuous and discrete distributions.

There are omissions that engineers would recognise. The idea of frequency response and signal decomposition, which would link with the work on Fourier analysis, is not considered. There is no discussion of state-space for a linear system, a concept of great value to control engineers, and others. The problem of the solution of systems of linear equations is always present, and we seem never to have N (reliable) equations in N unknowns, but often many more. Thus, a treatment of least-mean-square methods would have been valuable, perhaps related to singular-value decomposition.

Notwithstanding these comments, this is a book that in view of its content and its modest softcover price, will find its way on to many bookshelves as a reference work. As with the texts in the SUMS series, hints and solutions are provided for the exercises.

Nigel Steele is head of mathematics, University of Coventry.

## Elements of Logic Via Numbers and Sets

Author - D. L. Johnson
ISBN - 3 540 76123 3
Publisher - Springer
Price - £14.95
Pages - 174

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