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Where Do Numbers Come From?

T. W. Körner
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Bill Satzer
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The author of several entertaining and well-regarded books (including The Pleasures of Counting and Fourier Analysis) writes here about numbers, where they arise and how they can be built on a rigorous foundation. He takes inspiration from Landau’s Foundations of Analysis, a text written for undergraduates and published in 1930. Landau himself said of his book that it was written “in merciless telegraphic style”. (He precedes this with the phrase “as befits such easy material”.) Körner’s style is considerably more relaxed, one he calls “homespun”. Of course, saying that does not make the book free of difficulty.
Although the book addresses number systems of many varieties, two goals stand out. The first is Körner’s desire to identify the “standard properties of the real numbers”, something he notes that other books and lecturers refer to and assume, but often do not explicitly work through or even identify. The other is to make it clear what properties of a number system are necessary to be able to do calculus.
The book has three parts. It begins with the rational numbers, continues with the integers, and treats real and complex numbers. It is steadily more difficult as it proceeds, especially with the development of the real numbers. Körner suggests that he wants to cater to relatively inexperienced mathematicians. Consequently, he designed the book so that readers could stop at certain points and come back later when they have more experience. He also includes introductions for less experienced readers to topics that more advanced students might be able to skip. (These include things like equivalence classes, isomorphism, induction, inductive definition, and countability). He does expect all his readers to be able to read proofs carefully and create proofs of their own.
Körner begins with historical anecdotes that illustrate how numbers have been used. He then proceeds to develop number systems axiomatically with careful definitions, theorems and proofs. His characteristic humor, digressions and historical insights are present throughout, but this is a rigorous treatment that seems – in some measure- to follow Dedekind’s original approach.
The book does not follow the path you might expect. It begins with the basic rules for the arithmetic of natural numbers and then goes directly to a formal introduction of the strictly positive rational numbers. Zero, negative numbers, and then the full set of rational numbers emerge in turn. Körner then goes back to create the natural numbers from the Peano axioms, and finally uses the rational numbers to construct the real numbers. 
One subject that gets special emphasis is what Körner calls the fundamental axiom of analysis. He provides several equivalent versions of this, one of which is the Bolzano-Weierstrass property. He notes that if we want to base calculus on limits, use the associated definition of continuous function, and demand that the intermediate value condition holds, then we need an ordered field that obeys that fundamental axiom. (In a later chapter he shows an ordered field constructed in a natural way from real polynomials that does not obey that fundamental axiom.)
Beyond the core topics, the reader is also introduced to many related subjects: finite fields and modular arithmetic, the Chinese remainder theorem, Bezout’s theorem, the fundamental theorem of algebra, coding theory and encryption. A whole chapter on the quaternions at the end provides another example of a skew field.
An amazing number of topics and intriguing detours are swirling about in this book. As fascinating as the detours are, they give the book an unfortunately fragmented feel. In the end, the whole never properly emerges from the parts.
Many exercises are provided. They range from easy to quite difficult, from routine to very clever. Sketches of solutions to almost all the exercises are provided on the author’s website.


Bill Satzer (, now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films and material science. He did his Ph.D. work in dynamical systems and celestial mechanics.