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Summary Newton's method for the evaluation of the zeros of analytic functions is generalized by the recurrence relation zi+i = Zi- (ki-li)fi\zi)/fli+1)(zi), where ki and k are determined so that ideally U + 1 = ki = k in the vicinity of a zero of order k. The process so defined is a second-order one and yields accurate approximations to zeros for all values of k, provided I,, + 1 = k{ = k.

We will show that this construction is impossible if a straightedge and compass are the only tools which we may use. We precede the proof with a couple of definitions and facts which will be needed. A complex number is algebraic if it is a zero of a polynomial with integer coefficients. Numbers which are not algebraic are called transcendental. The set of all algebraic numbers is a field. This implies, in particular, that for any algebraic number a and any natural number k the number ak iS also algebraic. Recall also that each constructible number, i.e., a number which can be constructed using a straightedge and compass only, is algebraic. A crucial result for us is the following theorem due to Lindemann. If x 7s 0 is an algebraic number, then ex is a transcendental number. We refer to [2, 3] for unexplained terminology and details.

We start with a special case. Consider an irreducible quintic polynomial f(X) = X + a1X + a2X + a3X + a4X + a5 with rational coefficients and with three real roots and one pair of complex conjugate roots. For example, f(X) could be X − 10X + 5. Question. If α is a root of f , then how many roots of f lie in the field Q(α)? The field Q(α) is obtained by adjoining the root α to Q. Thus Q(α) contains at least one root of f , and of course it can contain at most five roots of f . Answer. The number r(f) of roots of f in Q(α) is 1. We prove that, for an arbitrary irreducible polynomial f and root α, r(f) divides the degree of f . For the quintic under discussion, adjoining one of the real roots cannot possibly produce the nonreal roots, so r(f), being a divisor of 5, must be 1. An informal survey of books and colleagues indicates that the divisibility result “r(f) divides the degree” is not well known. In what follows, K is a field and, unless stated otherwise, all roots and field extensions are taken in a fixed algebraic closure K of K. When K = Q, we always take K inside the complex numbers so that we can speak of real roots and nonreal roots. Theorem 1. Let f(X) in K[X] be an irreducible polynomial, and let α be a root of f . Set

Fix integers n and r with n > 2 and r > 0. A section s E HO(Pn, TPn(r)) was called [GK] a meromorphic vector field of degree r (on pn); let Z := (s)0 be its zero locus; s was called nondegenerate if dim (Z) = 0 and Z is reduced, i.e. if s has only isolated zeroes and each zero has multiplicity one. It was proved in [GK] that a nondegenerate meromorphic vector field is uniquely determined by its zero locus and that its zero locus is stable in the sense of [Mu]. Set x(n, r) := cn(TPn(r)) = ((r + 1)n+l 1)/r (the last equality following from the Euler's sequence). It is a natural problem to know if, for a given integer y < x(n, r) and a given set Z of y points of pn, there is a nondegenerate meromorphic vector field of degree n vanishing on Z; we can ask also for the dimension of such vector fields. We work over an algebraically closed field K (but of course the interesting case is when K is the complex number field). For a real number u, let [u] be its integral part; set w(n, r) := [(h0(Pn, TPn(r)) 1)/n]; note that by the Euler's sequence we have ho(Pn, TPn(r)) = (n+ 1)((n+r+ 1)!/(n!(r+ 1)!))((n+r)! /(n!r!). Here are the main results of this paper.

Must a sequence { z k } \{ {z_k}\} of complex numbers be identically zero if ∑ f ( z k ) = 0 \sum {f({z_k}) = 0} for every entire function f f vanishing at the origin? Lenard’s example of a nonzero sequence of complex numbers whose power sums ("moments") all vanish is shown to give a negative answer to this question and to lead to a novel representation theorem for entire functions. On the positive side it is proved that if { z k } \{ {z_k}\} is in l p {l^p} where p > ∞ p > \infty , then vanishing moments imply { z k } \{ {z_k}\} is identically zero. Virtually the same proof shows that, on a Hubert space, two compact normal operators A A and B B with trivial kernels are unitarily equivalent if some power of each belongs to the trace class and tr ⁡ ( A n ) = tr ⁡ ( B n ) \operatorname {tr}({A^n}) = \operatorname {tr}(B^n) for all n n in a set of positive integers with asymptotic density one.

Q(z) = ,Q?(&2 of an integral weight > 0; k is a positive integer; * is an embedding of K into C; r is an element of K0 such that Id2 is its only positive conjugate; 4D-b(* + Ap) + ,cpgp, where b and c, are non-negative integers, p is the complex conjugation, and {qp} is the set of all embeddings of K into C other than * and Ap. It will be shown that the series is convergent for sufficiently large Re (s) and can be continued to a meromorphic function on the whole plane. Now one of our main results will assert that the values ?D(4a) for certain integers , are algebraic numbers times wkPK(*i *)2k I, P (Tv, ,)-2c. where PK(q', A) is a complex number depending only on q and A. We shall actually prove such an algebraicity for the series defined in a similar way with a Hilbert modular form in place of Q (Theorem 9.2). If [K: QI = 2, 9) is essentially of the type

In a complete separable metrizable topological algebra, if the linear span of the set of all products of two elements has at most countable algebraic codimension, then it has finite codimension. In the present note we give a partial solution of a question posed to us by P. C. Curtis. Furthermore we give some closely related remarks. Let & be an algebra over the real or complex number field. By &2 we understand the linear span of all products of two elements. In the commuta- tive case this coincides with the linear span of all squares. The original question of P. C. Curtis was about &2; our theorem will contain both &2 and the linear span of squares as very special cases. We obtain completely satisfying results for separable complete metrizable topological algebras (local convexity is completely unnecessary for the proofs), but, unfortunately, every attempt to establish our results for nonseparable algebras has failed so far. Before we start on the theorem, let us give the motivation for the question of P.C. Curtis, who asked whether the square of a maximal modular ideal in a Banach algebra is closed if it has finite codimension in the ideal (the question seems to have been raised independently by other people; see (4)). If & is an algebra over the scalar field (real or complex numbers) and

GELMAN, ROCHEL. Logical Capacity of Very Young Children: Number Invariance Rules. CHILD DEVELOPMENT, 1972, 43, 75-90. Children 3-6 years of age, when given an identification task where number was redundant to length or density, solved the task on the basis of number. Surreptitious subtraction or addition elicited strong surprise as well as search behavior whereas displacements did not. Children who noticed the change in number or length and density gave unambiguous explanations of the nature of the intervening operations and were able to indicate how to reverse the effect. These findings are taken to show young children can treat small numbers as invariant. The results are discussed in terms of why children of the same age fail to conserve number in the standard conservation task and how complex number concepts might develop.

1. The resolvent, VI= x0 +.EX, + . x1 -I2x2 +* + E1-x 1, where E is a primitive n-th root of uiity, was introduced by Lagrange 2 in his memoirs devoted to the fundamental principles of the solutions of the cubic and quartic equations. Its entrance, however, into the field of geometry is very recent. If we represent any point P in the plane by the single complex number p, and if Mk (k 0, 1, * , n 1) represents the n vertices of a positivelyordered polygon, then when the co6rdinates pA of these vertices are subject to one and only one condition, namely that

In ? 2 we provide some criteria for the extension of 7. These are applied to prove stability theorems for families of complex structures (? 3 and ? 4), particularly complex structures uniformizable on bounded domains of the complex number space and of complex toral structures. In ? 1, basic definitions and facts are recollected for the convenience of the reader.

1. This paper, although mathematical in content, is motivated by quantumtheoretical considerations. The states of a quantum-mechanical system are usually described by vectors f of norm 1 in some Hilbert space A, and we assume explicitly that to every unit vector f corresponds a state of the system. This correspondence, however, is not one-to-one. In fact, the vectors which describe the same state form a ray f (in Weyl's terminology, cf. [13], p. 4 and p. 20),1 i.e. a set consisting of all vectors f = Tfo where fo is a fixed unit vector in & and r any complex number of modulus 1. (Every vector f in f will be called a representative of the ray f.) We have therefore a one-to-one correspondence between quantum states and rays, and every significant statement in Quantum Theory is a statement about rays. The transition probability from a state f to a state g equals (f, I)'2 where f, g are representatives of the rays f, g respectively. This suggests the introduction of the inner product of two rays by the definition

r 1/L r2/L rL ( L1)/L by rational numbers, r a suitable integer. Euler gave algorithms for L = 2,3,4,5. We will define an L X L matrix EL(r) and construct a complete set of eigenvalues and eigenvectors. Taking powers of this matrix and multiplying by a vector embraces Euler's algorithms. Rates of convergence are evident. This matrix provides an instructive example for Perron's theorem on positive matrices, anticipated in this instance by Euler. Let Z denote the integers, R the real numbers, and C the complex numbers. Let r > 1 be an integer. Define the L X L matrix

Mathematics is full of examples that illustrate the principle of "extend and conquer." Put simply, mathematical results become clearer and more complete when the framework of the discussion is enlarged. The fundamental theorem of algebra is a good example. For it is usually difficult, if not impossible, to describe the number of integers or rational numbers that are solutions to a polynomial equation; but by enlarging the scope of the discussion to include complex numbers and by agreeing to count multiple solutions separately, we obtain the simple and elegant statement that every polynomial equation of positive degree n has exactly n complex solutions. This article discusses an example of the "extend and conquer" principle that arises in the representation theory of finite groupsthe Artin induction theorem. This theorem shows that every linear representation of a finite group over the field of rational numbers extends to a permutation representation of the group and, as such, provides a link between representing the elements of a finite group as linear transformations of a vector space and representing them as permutations of a set. But more importantly, when the Artin theorem is stated in terms of group characters, it provides a precise way to measure, numerically, how close the rational representations of the group are to being permutation representations. We denote this numerical measure by the symbol y. The purpose of this article is to discuss the history of y, its existence and known values, and to indicate briefly the local algebraic methods for determining this invariant.

(1985). Solvability of Infinite Systems of Polynomial Equations Over the Field of Complex Numbers. The American Mathematical Monthly: Vol. 92, No. 2, pp. 94-98.

As it says in the front of the ComputerMusic Journal number 4, there are many musicians with an interest in musical signal processing with computers, but only a few have much competence in this area. There is of course a huge amount of literature in the field of digital signal processing, including some first-rate textbooks (such as Rabiner and Gold's Theory and Application of Digital Signal Processing, or Oppenheim and Schafer's Digital Signal Processing), but most of the literature assumes that the reader is a graduate student in engineering or computer science (why else would he be interested?), that he wants to know everything about digital signal processing, and that he already knows a great deal about mathematics and computers. Consequently, much of this information is shrouded in mathematical mystery to the musical reader, making it difficult to distinguish the wheat from the chaff, so to speak. Digital signal processing is a very mathematical subject, so to make past articles clearer and future articles possible, the basic mathematical ideas needed are presented in this two-part tutorial. In order to prevent this presentation from turning into several fat books, only the main ideas can be outlined; and mathematical proofs are of course omitted. But keep in mind that learning mathematics is much like learning to play a piano: no amount of reading will suffice -it is necessary to actually practice the techniques described (in this case, by doing the problems) before the concepts become useful in the "real" world. Therefore some problems are provided (without answers) to give the motivated reader an opportunity both to test his understanding and to acquire some skill. Part I of the tutorial (this part) provides a general review of algebra and trigonometry, including such areas as equations, graphs, polynomials, logarithms, complex numbers, infinite series, radian measures, and the basic trigonometric functions. Part II will discuss the application of these concepts and others in transforms, such as the Fourier and z-transforms, transfer functions, impulse response, convolution, poles and zeroes, and elementary filtering. Insofar as possible, the mathematical treatment always stops just short of using calculus, though a deep understanding of many of the concepts presented requires understanding of calculus. But digital signal processing inherently requires less calculus than analog signal processing, since the integral signs are replaced by the easier-tounderstand discrete summations. It is an experimental goal of this tutorial to see how far into digital signal processing it is possible to explore without calculus.

Introduction. In a first course the case of two coupled linear differential equations tends to fall between two stools. The teacher's unrequited love for eigenvalues drives him into the complex domain, a maze in which he seldom finds the simple, real solutions of the original problem. And even if the complex numbers can be avoided he has difficulty returning through the coordinate transforms. It would seem that if the students had an adequate basis in algebra, everything would be easy. However, on the one hand, it is too much to include all that algebra. On the other hand, that particular subject is not something that can be used now and explained later. Hence, it is tempting to look for a simple, direct solution, which works in the real domain and only requires straightforward ideas.

In this paper we generalize Hardy's inequality [3] for measures of analytic type and obtain a proof of the Littlewood conjecture [4] for the L' norm of exponential sums as a simple consequence. Let T be the circle group, Z the additive group of integers and M(T) the customary convolution algebra of Borel measures on T; for pe e M(T) and n G Z put M(%)= e-ino dp(O) Denote by H'(T) the classical space of all measures e e M(T) such that j(n) = 0 for all n < 0 and let C denote the complex numbers. We now state Hardy's inequality for measures of analytic type: THEOREM 1. If M C H'(T) then

tions the field K will be the field Q of algebraic numbers. We remark at this point that the results remain true if we take instead of the field of complex numbers C its p-adic analogue C for some fixed prime p and for K a corresponding subfield Kp of CP. We restrict ourselves to the complex case in order to avoid some minor complications appearing in the p-adic domain. These complications always arise when analytic functions come in. Our functions are defined globally in the case of complex numbers but only locally in the case when we deal with the p-adic domain. These difficulties can be avoided by a purely algebraic approach. If we took this approach the only condition on the ground field would be that it should be algebraically closed and of characteristic zero. But we prefer to avoid such an approach in order to keep the text understandable, also for those who are mainly interested in the applications in transcendence.

Log per capita real gross domestic product is modeled as a third-order autoregression with a pair of complex roots whose amplitude is smaller than the amplitude of the real root. The behavior of this terms series is interpreted in terms of these two amplitudes, the periodicity of the complex roots, and the standard deviation of the disturbance. Restrictions are evaluated and inference is conducted using the likelihood principle, applying Monte Carlo integration with importance sampling. These Bayesian procedures efficiently cope with restrictions that are awkward taking a classical approach. We find very little difference in the amplitudes of real roots between countries and of complex roots relative to within-country uncertainty. There are some substantial differences in the periodicities of complex roots, and the greatest differences between countries are found in the standard deviation of the disturbance.

Small ("spatial") free products of C*-algebras are constructed. Under certain conditions they have properties similar to those proved by Paschke and Salinas for the algebras C,*(GI * G2) where G1, G2 are discrete groups. The freeproduct analogs of noncommutative Bernoulli shifts are discussed. 0. Introduction. Let K be a field. Consider the category of unital algebras over K. It is well known that this category admits coproducts: free products of algebras [2]. Heuristically, the free product of algebras is the algebra generated by them, with no relations except for the identification of unit elements. If K = C, the complex numbers, and we consider unital * -algebras, we can easily define a * -operation on the free products. Let A, B be unital C*-algebras, and A * B their free product, which is a unital *-algebra. The question arises: in what ways may one define a pre-C* norm on A * B that extends the norms on A and B? Guided by analogy with tensor products, we expect to have a choice among many pre-C* norms, giving rise to many "C* free products" of A and B. One natural norm is 1c I I = sup{ 1IT(c)IH: ST * -representation of A * B). The * -representations of A * B are in 1-1 correspondence with pairs of * representations of A and B, which act on the same Hilbert space. Let A * B be the completion of A * B in this norm. It is easy to see that this construction defines a coproduct in the category of C*-algebras, and that A * B is the "biggest free product" of A and B, analogous to the biggest tensor product A 0 B. If G 1, G2 are discrete groups we obtain C*(G1) * C*(G2) C*(GI * G2) where G1 * G2 is the free product group, and this is analogous to the relation C*(G1) 0 C*(G2) C*(G1 X G2). This paper is motivated by the question: Is there a "smallest C*-product", A*B, in analogy to the smallest tensor product, A * B, satisfying a relation Cr*(GI * G2) Cr*(GI) * Cr*(G2) Received by the editors January 28, 1981 and, in revised form, May 19, 1981. 1980 Mathematics Subject Classification. Primary 46L05; Secondary 46L55. (1 982 American Mathematical Society 0002-9947/82/00001022/$04.25

0. The purpose of this paper is to prove some results about the characteristics of overdetermined systems of linear partial differential equations. Since these results are of an algebraic nature the context of this discussion will be algebraic. In ??5-7 we have included a brief sketch of the analytical applications. We will begin by describing our two main results. Let V and W be finite dimensional vector spaces over the complex numbers, and let g be a subspace of iom (V, W). If V* denotes the dual of V we can also regard g as a subspace of W 0 V*; and we will often shift from one interpretation of g to the other. Let U be a subspace of V*. We will say that U is noncharacteristic for g if g n w 0 U = 0 and characteristic otherwise. If X is a non-zero vector in V* we will say it is non-characteristic if the one-dimensional subspace {cw, c C C} is non-characteristic, and characteristic otherwise. We will show in ? 1 that certain invariants of the algebraic structure {V, W, g} can be described by a bigraded homology sequence

where a, 6, c, d are complex numbers. This group acts naturally on the extended complex plane P. However we can extend the action so that LF(2, C) operates as a group of conformal transformations of the extended 3-dimensional space E= R3 U {I C}, which leave invariant the upper half-space H= {(xl, X2, X3) I X3 > O}. (Here, the extended complex plane is imbedded in Eas the set P= {(x1, X2, X3) I X3-O} U {a o }). The extension to E can be described as follows. If f e LF(2, C), then f can be expressed as a product of 2 or 4 reflections in circles (or lines). Replace each circle C by the sphere S which is orthogonal to P along C. Then the product of reflections in these spheres is the required extension of f. Note that this extension is unique. For suppose f and g are conformal transformations of E leaving H (and hence P) invariant, and suppose f and g coincide on P. Then h = fg-' leaves P pointwise fixed. Since h maps spheres into spheres, it leaves invariant each sphere orthogonal to P. By taking intersections of spheres, one easily sees that h is the identity transformation. We shall now give H the riemannian metric

(1970). An Inequality About Complex Numbers. The American Mathematical Monthly: Vol. 77, No. 2, pp. 180-182.

A. Tarski [4] has given a decision method for elementary algebra. In essence this comes to giving an algorithm for deciding whether a given finite set of polynomial inequalities has a solution. Below we offer another proof of this result of Tarski. The main point of our proof is accomplished upon showing how to decide whether a given polynomial f(x, y) in two variables, defined over the field R of rational numbers, has a zero in a real-closed field K containing R.1 This is done in ?2, but for purposes of induction it is necessary to consider also the case that the coefficients of f(x, y) involve parameters; the remarks in ?3 will be found sufficient for this point. In ?1, the problem is reduced to a decision for equalities, but an induction (on the number of unknowns) could not possibly be carried out on equalities alone; we consider a simultaneous system consisting of one equality f(x, y) = 0 and one inequality F(x) $ 0. Once the decision for this case is achieved, at least as in ?3, the induction is immediate. Entering into our considerations are the field R of rational numbers and an arbitrary real-closed field K: the argument proceeds uniformly for all K. Because of this, one gets for real-closed fields a principle analogous to the so-called "Principle of Lefschetz." This principle asserts that results of a certain kindthe kind occurring, for example, in A. Weil's Foundations of Algebraic Geometry (see [6; pp. 242-245])-which are true for the field of complex numbers automatically hold also for an arbitrary algebraically closed field of characteristic 0. The corresponding principle for real-closed fields, which we may call the "Principle of Tarski," says that any sentence of elementary algebra which holds in one real-closed field also holds in every real-closed field. In particular it is true that any polynomial f(xi, ... , xn) e K[x1 , ... , x ], K a real-closed field, has on any n-dimensional closed interval a maximum and a minimum. In ?6(b) we illustrate the principle by showing that if an algebraic variety defined over a real-closed field carries any points with coordinates in K, then it also carries one such point which is nearest to the origin. Our proof may have some bearing on the actual construction of a decision machine. Some remarks on this point are made in ?6(e). Thanks are due to Professor Tarski for valuable comments on the paper.

If f(t) is a function defined for co o, such that p(t)/p(2t) is uniformly bounded and I t j-' = O(p(t)) (I t I0o) for some integer n (n ? 0). Then the set D,(t) of all complex-valued functions g(t), continuous except for discontinuities of the first kind, such that limt-_+? p(t)g(t) exists, can be metrised by means of the norm 11 g(t) 11 = maxO,?t?,, p(t) I g(t) I; and every such function g(t) has a generalized Fourier transform of order n + 2, -y(t). We shall show that the following statements are equivalent. (a) There exists no interval in which y(t) is equal to a polynomial of degree at most n + 1. (b) For any continuous function f(t) with limt_+? p(t)f(t) = 0, and for any E > 0, there exist real numbers X3, complex numbers A i, and an integer k, such that

In [2] the authors asked if any two real closed subfields R, R' of the field of complex numbers C such that R(x/ (-1))R'(V(1)) =C are isomorphic. It is not difficult to see that the answer is negative. This is proved in the first part of the note. In the second we study the problem if any field which is not prime contains a proper subfield of countable (finite or infinite) codimension. 1. PROPOSITION 1. Let R be any real closed field of power continuum. Then the field R(V/(1)) is isomorphic to the field of complex numbers. PROOF. It is well known (see e.g. [3, p. 274]) that if the field R is real closed then the field R(V/(1)) is algebraically closed. Since any algebraically closed field of power continuum and of characteristic zero is isomorphic to the field of complex numbers hence if R is real closed of power continuum then R(J(1)) is isomorphic to the field of complex numbers. Thus the proposition is proved. Because there exist nonisomorphic real closed fields of power continuum (e.g. the field R and the real closure of the ordered field R(t), where 0< t

For every contrac io on ILBERT SPACE A SUBSPACE IS DEFINED ON WHIC THE OPERATOR WAS UNITARY. This analysis is considered to study this subspace and compare it with the spectral decomposition of the operator. (Author)

A sequenceX = {X(n) } of complex numbers is said to be a multiplier of H' into the sequence space 11 if Xf= {X(n)f(n) } Ell' for every f(z) E= f(n)znCHl. The space of all such multipliers is denoted (HI', 11). The only important known result about (H', 11) is the inequality of Hardy [5, p. 236]: {1/(n+1)} E(H1, 11). Other similar multiplier spaces have been completely characterized: an elementary sufficient condition for (H', 12), proved by Hardy and Littlewood [4], is also necessary, and the spaces (H', l) for 2 ? q ? ?O can be described similarly. Also (HP, 11) has been determined recently for O

1. In [10] (p. 650) we have proved a uniformization theorem for zero-dimensional valuations on an algebraic surface, over an algebraically closed ground field K (of characteristic zero). In the present paper we generalize this theorem to algebraic varieties, and on the basis of this generalization we obtain a solution of the problem of local uniformization in the classical case (i.e. when K is the field of complex numbers). The exact formulation of the generalized theorem, in its strongest form, will be given in A III and A IV. However, to begin with, we state here the following theorem which is literally a repetition of our theorem for surfaces, with the surface replaced by a variety, and which will be included in our final result: THEQREM U1. The Uniformization Theorem in invariantive form: Given a field 2 of algebraic functions of r independent variables, over an algebraically closed ground field K of characteristic zero, and given a zero-dimensional valuation B of 2, there exists a projective model V of 2 on which the center of B is at a simple point P. This theorem is in effect entirely invariantive in nature: it refers exclusively to the field 2 and to the valuation B of 2. It asserts the existence of uni

Let [a, b] be a bounded interval with a>O. Under what conditions on the sequence of exponents {A,,} can every function in LP[a, b] or C[a, b] be approxi mated arbitrarily closely by linear combinations of powers xAn? What is the distance between xA and the closed span Sc(xAn)? What is this closed span if not the whole space? Starting with the case of L2, C. H. Muntz and 0. Szasz considered the first two questions for the interval [0, 1]. L. Schwartz, J. A. Clarkson and P. Erdos, and the second author answered the third question for [0, 1] and also considered the interval [a, b]. For the case of [0, 1], L. Schwartz (and, earlier, in a limited way, T. Carleman) successfully used methods of complex and functional analysis, but until now the case of [a, b] had proved resistant to a direct approach of that kind. In the present paper complex analysis is used to obtain a simple direct treatment for the case of [a, b]. The crucial step is the construction of entire functions of exponential type which vanish at prescribed points not too close to the real axis and which, in a sense, are as small on both halves of the real axis as such functions can be. Under suitable conditions on the sequence of complex numbers {An} the construction leads readily to asymptotic lower bounds for the distances dk=d{xAk, Sc(xAn, nAk)}. These bounds are used to determine Sc(xAn) and to generalize a result for a boundary value problem for the heat equation obtained recently by V. J. Mizel and T. I. Seidman.

*I wish to thank "the referee" for pointing out that this article lies in the theory of inversive geometry. This article is a specialization to the real numbers of a method used by Riemann on the complex numbers.* By a geometric method we obtain a one-to-one correspondence between the set of real numbers and the set of all points, except one, on a circle of unit diameter. We show that if the circular correspondents of the two real numbers lie on a horizontal (vertical) line then their sum (product) is zero (one). For the product of two positive numbers, one less than and one greater than 1, we determine a circular bound. The circle is embedded in a Cartesian plane in a "natural" manner. It is shown that a line that passes through the origin intersects the circle in the point that corresponds to its slope. On a horizontal real number line, place a circle with unit diameter so that it is tangent to the line at zero. Denote the point on the circle that is diametrically opposite zero by K. We now consider the rays issuing from K which intersect the line, and consequently also intersect the circle in a second point (distinct from K).