This monograph is intended as an exposition of some central results on irrational numbers, and is not aimed at providing an exhaustive treatment of the problems with which it deals. The term “irrational numbers,” a usage inherited from ancient Greece which is not too felicitous in view of the everyday meaning of the word “irrational,” is employed in the title in a generic sense to include such related categories as transcendental and normal numbers.

The entire subject of irrational numbers cannot of course be encompassed in a single volume. In the selection of material the main emphasis has been on those aspects of irrational numbers commonly associated with number theory and Diophantine approximations. The topological facets of the subject are not included, although the introductory part of Chapter I has a sketch of some of the simplest set-theoretic properties of the irrationals as a part of the continuum. The axiomatic basis for irrational numbers, proceeding say from the Peano postulates for the natural numbers to the construction of the real numbers, is purposely omitted, because in the first place the aim is not in the direction of the foundations of mathematics, and in the second place there are excellent treatments of this topic readily available.

Equivalence. We further restrict our forms in this chapter by requiring that their coefficients be p-adic integers, that is, in R(p). As the theory for F(p) led to results for forms with rational coefficients, so the theory for R(p) leads to results for forms with integer coefficients though, as we shall see, there is here an intermediate case which has no previous analogy. Suppose a transformation T with elements in R(p) takes a form f into a form g where both forms are in R(p) and have non-zero determinants df and dg, respectively, and a transformation S in R(p) takes g into f. Then

|T2|·df = dg and |S2|·dg = df

imply that |T|2|S|2 = 1. Since the determinants of T and S are p-adic integers, they must be units. Thus, following section 20, we call a transformation unimodular in R(p) or p-adically unimodular if its elements are in R(p) and its determinant a p-adic unit. Two forms are equivalent in R(p) or p-adically equivalent if one may be taken into the other by a unimodular transformation in R(p). Two p-adically equivalent forms represent the same p-adic integers for values of the variables in R(p). In this chapter, unless it is stated to the contrary, equivalence means p-adic equivalence and we denote it by the sign ≅. Two forms equivalent in R(p) are said to be in the same p-adic class.

Introduction. If f and g are forms with integer coefficients and n and m variables respectively (n > m) the results of the last chapter give us methods of finding whether or not some form in the genus of f represents g integrally. Corollary 44a or 44b may be used to show that the existence of such a representation depends on the solvability of the congruence f ≡ g (mod 8 | g | P) where P is the product of the odd primes in |g|·|f|, or on the existence of representations in R(p) for all p dividing 2|f||g|. When there is only one class in the genus of f, the same criteria serve to determine the existence of representations of g by the form f. However, when there is more than one class in the genus except for certain very special cases and asymptotic results there are no known criteria for existence of representations.

When it comes to determining the number of representations of g by f, known results, except for those in section 38 below, depend on analytic theory which is beyond the scope of this book. However we shall describe such conclusions.

For the case m = n two fundamental problems arise. First, the question of equivalence cannot in general be elegantly resolved in the ring of rational integers. Faced with such a problem one would first test for semi-equivalence by methods of the previous chapter; then employ a reduced form such as is described in theorem 23.