76 I The Expanding Universe of Numbers
Boolean algebra was given by Huntingdon [39]. For an introduction to Stone’s repre-
sentation theorem, referred to in§8, see Stone [69]; there are proofs in Halmos [30] and
Sikorski [66]. For applications of Boolean algebras to switching circuits see, for ex-
ample, Rudeanu [62]. Boolean algebra is studied in the more general context of lattice
theory in Birkhoff [6].
Dedekind’s axioms forNmay be found on p. 67 of [19], which contains also his
earlier construction of the real numbers from the rationals by means of cuts. Some
interesting comments on the axioms(N1)–(N3)are contained in Henkin [34]. Start-
ing from these axioms, Landau [47] gives a detailed derivation of the basic properties
ofN,Q,RandC.
The argument used to extendNtoZshows that any commutativesemigroupsat-
isfying the cancellation law may be embedded in a commutativegroup. The argument
used to extendZtoQshows that any commutativeringwithout divisors of zero may
be embedded in afield.
An example of an ordered field which does not have the Archimedean prop-
erty, although every fundamental sequence is (trivially) convergent, is the field∗Rof
hyperreal numbers, constructed by Abraham Robinson (1961). Hyperreal numbers are
studied in Stroyan and Luxemburg [70].
The ‘arithmetization of analysis’ had agradual evolution, which is traced in
Chapitre VI (by Dugac) of Dieudonn ́eet al. [22]. A modern text on real analysis
is Rudin [63]. In Lemma 7 of Chapter VI we will show that all norms onRnare
equivalent.
The contraction principle (Proposition 26) has been used to prove thecentral
limit theoremof probability theory by Hamedani and Walter [32]. Bessaga (1959) has
proved aconverseof the contraction principle: LetEbe an arbitrary set,f:E→Ea
map ofEto itself andθa real number such that 0<θ<1. If each iteratefn(n∈N)
has at most one fixed point and if some iterate has a fixed point, then a complete metric
d can be defined onEsuch that d(f(x′),f(x′′))≤θd(x′,x′′)for allx′,x′′∈E.A
short proof is given by Jachymski [40].
There are other important fixed point theorems besides Proposition 26.Brouwer’s
fixed point theoremstates that, ifB={x∈Rn:|x|≤ 1 }is then-dimensional closed
unit ball, every continuous mapf :B → Bhas a fixed point. For an elementary
proof, see Kulpa [44]. TheLefschetz fixed point theoremrequires a knowledge of al-
gebraic topology, even for its statement. Fixed point theorems are extensively treated
in Dugundji and Granas [23] (and in A. Granas and J. Dugundji,Fixed Point Theory,
Springer-Verlag, New York, 2003).
For a more detailed discussion of differentiability for functions of several variables
see, for example, Fleming [26] and Dieudonn ́e [21]. The inverse function theorem
(Proposition 27) is a local result. Some global results are given by Atkinson [5] and
Chichilnisky [14]. For a holomorphic version of Proposition 28 and for the simple way
in which higher-order equations may be replaced by systems of first-order equations
see, e.g., Coddington and Levinson [16].
The formula for the roots of a cubic was first published by Cardano [12], but it
was discovered by del Ferro and again by Tartaglia, who accused Cardano of breaking
a pledge of secrecy. Cardano is judged less harshly by historians today than previ-
ously. His book, which contained developments of his own and also the formula for