Which Numbers are Real?

My first — and wrong — impression of the book was that it might be a modernized version of Edmund Landau’s classic Foundations of Analysis. After all, both books describe constructions of nothing else but several number systems. On closer inspection, though, I saw my error; the differences are significant. Landau constructs an expanding chain of number systems: natural numbers, integers, fractions, real and then complex numbers. The understanding of algebraic tools, say equivalence relations and classes, is taken for granted; the book is intended to be read (at least by a novice) from the beginning to the end. Henle, on the other hand, delves into the equivalence relations and equivalence classes at the beginning of the book. Integers and rationals are assumed to be known, but the constructions of rational numbers is well outlined. The number systems Henle describes are essentially different from Landau’s: real numbers, complex numbers, quarternions, constructive reals, hyperreals, and surreal numbers. Except for the first two chapters, which serve as the basis for the rest of the book, the remaining chapters are very much self-contained and could be read independently.

Having set up the mechanism of equivalence relations for generating number systems, Henle introduces the field and (linear) order axioms, and the (order) completeness axiom that are going to characterize the set of the reals, and rounds up the first chapter by linking completeness, Cauchy completeness, and the Archimedean property in an ordered field.

In the second chapter, the book builds the reals in two ways: with Cauchy sequences (following G. Cantor) and with the Dedekind cuts; shows each construction leads to a complete field, and proves that all such fields are isomorphic. The last section introduces continuity and differentiability of functions over the reals and establishes the Intermediate Value Theorem. These are the essential properties of the reals to juxtapose with the properties of the number systems developed in later chapters of the book.

The complex numbers introduced in the third chapter shed the linear order but maintain Cauchy completeness; they form the only multidimensional field over the reals; the field of complex numbers is algebraically closed: the Fundamental Theorem of Algebra holds for the complex numbers but not for the reals; complex differentiability implies conformality in the complex plane. Towards the end, the chapter briefly presents two other 2-dimensional number systems: the double and dual numbers.

The quarternions in the fourth chapter form a skew field (an algebraic structure with all the field properties except for commutativity of multiplication.) The quarternions offer a convenient presentation of geometric transformations in three dimensions. Complex quarternions relate to the Lorentz transformation, Minkowski separation, and special relativity.

The hyperreal and surreal numbers (the subjects of chapters six and seven, respectively) are two linearly ordered incomplete fields that, besides the reals, contain infinitely small and infinitely large numbers. The hyperreals, discovered by Abraham Robinson in 1961, vindicate the uncritical (albeit successful) employment of the infinitesimals by the founders of Calculus so venomously derided by Bishop George Berkeley. The surreals were found by John Horton Conway in the early 1970s. All introductions of the surreal numbers follow Conway’s original construction; this book is no exception.

The surreal numbers are an integral part of Conway’s theory of combinatorial games. Games are played by two players (Left and Right) and are identified by two sets of possible positions that are reached after one move by either player. Positions are nothing but games in their own right. The notations for the games thus remind of the Dedekind cuts, although the theory does not even assume the existence of the integers, much less the rational numbers. All games and numbers are defined inductively, starting with 0 = {|} — the game with no moves. Numbers are the games whose positions are also numbers such that no left position ever exceeds or equals any right position. Henle’s exposition does justice to this elegant and naturally playful theory. The Simplicity Theorem is especially appealing.

Chapter six on the hyperreal numbers is based on Infinitesimal Calculus, by (James) Henle and Kleinberg. The hyperreal numbers are defined as the equivalence classes of arbitrary sequences of the reals induced by an ultrafilter. The theories of both the real and hyperreal numbers are set in the framework of formal languages; both classes of numbers form structures in the same formal language. According to the Transfer Principle, any sentence in that language true for the reals is true for the hyperreals, and vice versa. The upside of the Transfer Principle is that it is often easier to prove a statement in the structure of the hyperreals than for the reals. It will be true for the reals as a consequence.

The question in the title of the book Which Numbers Are Real? is no doubt rhetoric. Among the number systems I mentioned so far only the hyperreals may in some sense substitute for the authentic system of the real numbers. Indeed, in some colleges the beginning calculus courses are based on the hyperreals. However, even though the hyperreals remove much of the difficulty associated with computing limits and differentiation, the upfront cost of introducing formal languages, the novelty (for unprepared high school graduates) of the concept and its abstractness seem to preclude such initiatives from spreading widely.

There is just one class of numbers (different from the reals) whose proponents promote as the only valid one. These are the constructive numbers dealt with in chapter five. As the author writes,

The constructive reals are the product of a radically conservative approach to mathematics. The constructivists take the integers as intuitively given, god-given as Kronecker said, and the one and only source of truth in mathematics. To preserve this truth, they insist that all mathematical statements should be verifiable by computations with integers. The key idea here is that of a computation, by which is meant an operation or sequence of operations that can be performed by a finite intelligence (you, me, for example, or a digital computer) in a finite number of steps.

Following this, Henle gives several examples of mathematical derivations acceptable to the constructivists; constructivist criticism of conventional mathematics; classical and constructivist logic. Then the constructive numbers are defined, followed by elements of constructive calculus. The constructive numbers are shown to be Cauchy complete; it is left as an exercise to prove that they possess the Archimedean property. Interestingly, in chapter one it was shown that together these two imply order completeness: every set bounded from above has a least upper bound. But obviously, the claim of completeness goes against constructivist philosophy. The apparent contradiction is resolved by pointing out that some methods used in chapter one do not stand up to the constructivist critique. This is all very interesting, although it seemed to me that, instead of making them more intuitive, constructivist constraints rather complicate mathematical constructions.

My final impression of the book is very positive. Although the subject of the book is just numbers, the collection of number systems outlined in the book serves to demonstrate the breadth and power of mathematical methods and surprising diversity of properties and proof techniques even within related topics. The book should be welcomed by mathematically inclined high school seniors, college undergraduates and their instructors.

Alex Bogomolny is a former associate professor of mathematics at University of Iowa. He lives in New Jersey, maintains a popular site Interactive Mathematics Miscellany and Puzzles, with a server somewhere in Michigan, and blogs at CTK Insights. In an extreme need you can tweet him at @CutTheKnotMath.