# Which Numbers are Real?

Michael Henle
Edition: 1
Pages: 230
https://www.jstor.org/stable/10.4169/j.ctt5hhb1h

1. Front Matter
(pp. i-vi)
2. Introduction
(pp. vii-viii)

The real numbers are fundamental. Although mostly taken for granted, they are what make possible all of mathematics from high school algebra and Euclidean geometry through the calculus and beyond, and also serve as the basis for measurement in science, industry, and ordinary life. In this book we study alternative systems of numbers: systems that generalize and extend the reals yet stay close to the fundamental properties that make the reals central to so much mathematics.

By an alternative number system we mean a set of objects that can be combined using two operations, addition and multiplication, and that share...

(pp. ix-x)
4. ### I THE REALS

• [Part I Introduction]
(pp. 1-2)

What makes a number system a number system? In this book the real numbers serve as the standard with which other number systems are compared. To be called a number system a mathematical system must share most if not all of the fundamental properties of the reals.

What are the fundamental properties of the reals? We use a set of properties (or laws or axioms) that characterize the reals completely, meaning that any mathematical system with these properties is the same as the reals. Such a set of properties for a particular mathematical object is called a categorical axiom system....

• 1 Axioms for the Reals
(pp. 3-34)

In this section we describe a process that is used to construct many number systems and other mathematical systems as well. It is used to construct about half of the number systems in this book. This process uses the concept of an equivalence relation. Here is the definition:

Definition. LetSbe a set and ~ a relationship that may or may not hold between two elements ofS. A relation onSsatisfying these properties:

(a) For every a inS,a~a, — reflexivity

(b) Fora,binS, ifa~b, thenb~a,...

• 2 Construction of the Reals
(pp. 35-54)

We will prove the existence of the real numbers two times—by twice constructing a complete, linearly ordered field. Afterwards, we prove that all complete, linearly ordered fields are isomorphic, meaning that our axiom system for the reals is categorical.

The reader might well object to this chapter in the following terms: “I’ve used the reals all my life; their properties are familiar; I know how to calculate with them; I know the calculus; I know everything. Why should I bother to construct the reals, when I know the result in advance?”

How do we know the reals exist? Just...

5. ### II MULTI-DIMENSIONAL NUMBERS

• 3 The Complex Numbers
(pp. 57-76)

The complex numbers are the oldest and best-known extension of the reals. They have an elaborate theory and many important applications. Complex numbers were viewed with suspicion by mathematicians for many years and used only with reluctance because they seemed to have no basis in reality. True understanding of the complexes came after it was grasped that they are a two-dimensional number system. Interpreting the complexes as points in the plane gave them concreteness and opened the door to their exploration and application.

For convenience, we define the complexes as certain 2 by 2 matrices. This gives the theory a...

• 4 The Quaternions
(pp. 77-94)

It is disappointing that C is the only multi-dimensional field containing the reals. It would particularly be useful if some kind of multiplication could be defined in three-dimensional space making it a field. (We live, after all, in three-dimensional space.)

Unfortunately, this is impossible. However, if we give up one field axiom, we can proceed.

Definition. A skew field is a setStogether with operations of addition and multiplication satisfying all the axioms of a field except the commutative law of multiplication.

There is no skew field of three-dimensions, sad to say, but there is a four-dimensional skew field....

6. ### III ALTERNATIVE LINES

• 5 The Constructive Reals
(pp. 97-124)

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 within 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 orme, for example, or a digital computer) in a finite number of steps. By insisting that mathematics be computationally...

• 6 The Hyperreals
(pp. 125-170)

The hyperreal number system was invented comparatively recently (in the 1960s). What makes it unusual is that it contains infinitely small numbers: hyperreals so small that they are greater than 0 but less than 1/nfor allnin N.

Such numbers, called infinitesimals, have been around for a long time. Leibniz used them in his development of the calculus, and they are still taught in the form of the symbolsdxanddyused in differentiation and integration. Right from the beginning of the calculus (in the 1680s), the use of infinitesimals was criticized, and for years the calculus...

• 7 The Surreals
(pp. 171-204)

The surreals, like the hyperreals, are a relatively recent invention, discovered in the early 1970s by John H. Conway. Like the hyperreals, the surreals contain infinitesimally small numbers and infinitely large numbers. Unlike the hyperreals, the surreals are not a structure for the reals. Thus it is not true that every function and relation on the reals has an extension to or a meaning for the surreals.

A spirit of playfulness animates the surreals. They might be called hippie numbers after the “flower” children, who dropped out of society and lived communally in the Vietnam War period when the surreals...

7. Bibliography
(pp. 205-208)
8. Index
(pp. 209-218)