Abstract: In this correspondence we define a "nonadjacent form" for integers in an arbitrary radix r > 1 . This form is proved to be unique, and the arithmetic weight of an integer is shown to be equal to the number of nonzero terms in the form. Two algorithms are presented for the computation of this form. If r = 2 , our form coincides with the well-known modified binary nonadjacent form.

Abstract: As computers become capable of executing more arithmetic operations simultaneously, the question of compiling for such machines becomes more important.

Abstract: This paper presents the statistical results of tests of the accuracy of certain arithmetic systems in evaluating sums, products and inner products, and analytic error estimates for some of the computations. The arithmetic systems studied are 6-digit hexadecimal and 22-digit binary floating point number representations combined with the usual chop and round modes of arithmetic with various numbers of guard digits, and with a modified round mode with guard digits. In a certain sense, arithmetic systems differing only in their use of binary or hexadecimal number representations are shown to be approximately statistically equivalent in accuracy. Further, the usual round mode with guard digits is shown to be statistically superior in accuracy to the usual chop mode in all cases save one. The modified round mode is found to be superior to the chop mode in all cases.

Abstract: Algorithms are described for the basic arithmetic operations and square rooting in a negative base. A new operation called polarization that reverses the sign of a number facilitates subtraction, using addition. Some special features of the negative-base arithmetic are also mentioned.

Abstract: In this note, we show how interval arithmetic can be used to give a solution to the linear programming problem which is guaranteed to be on the safe side of the true solution, where roundoff error is taken into account.

Abstract: Pipelining an arithmetic process is a well known technique for improving the computation speed of the arithmetic algorithm. In the letter is proposed a pipeline version of the array for the extraction of square roots of binary numbers. It is shown that a significant speed improvement (on a throughout basis) can result by this modification of the conventional logic arrays.

Abstract: : Examples are presented, and sometimes analysed in detail, to reveal the unpleasant implications for scientific computation of flaws in the design of the arithmetic unit and in the supervisory software associated with it Attempts to axiomatize floating point arithmetic are discussed and the reasons why they are irrelevant It is shown that Interval Arithmetic can be misleading or helpful depending on the way it is used Some factors affecting a choice of radix base are elucidated

Abstract: A well-known product, referred to as the Dirichlet convolution product, is generalized to arithmetic functions defined on an order in a Cayley division algebra. Factorization results for orders, multiplicative functions and analogues of the Moebius inversion formula are discussed.

Abstract: In this paper I shall argue that the presumption of infinitude may be excised from the area of mathematics known as natural number theory with no substantial loss. Except for a few concluding remarks, I shall restrict my concern in here arguing the thesis to the business of constructing and developing a first-order axiomatic system for arithmetic (called ‘FA’ for finite arithmetic) that contains no theorem to the effect that there are infinitely many numbers. The paper will consist of five parts. Part I characterizes the underlying logic of FA. In part II ordering of natural numbers is developed from a restricted set of axioms, induction schemata are proved and a way of expressing finitude presented. A full set of axioms are used in part III to prove working theorems on comparison of size. In part IV an ordinal expression is defined and characteristic theorems proved. Theorems for addition and multiplication are derived in part V from definitions in terms of the ordinal expression of part IV. The crucial final constructions of part V present a new method of replacing recursive characterizations by strict definitions. In view of our resolution not to assume that there are infinitely many numbers, we shall have to deal with the situation where singular arithmetic terms of FA may fail to refer. For I know of no acceptable and systematic way of avoiding such situations. As a further result, singular-term instances of universal generalizations of FA are not to be inferred directly from the generalizations themselves. Nevertheless, (i) (x)(y)(x + y = y + x), for example, and all its instances are provable in FA.

Abstract: : The paper examines the concept of an associative processor whose arithmetic operations are based on residue arithmetic computations. Particular emphasis has been placed on the development and evaluation of arithmetic algorithms. Preliminary solutions are presented in a number of problem areas. One of the main results is that a residue arithmetic associative processor (RAAP) offers potentially large speedup in multiplication time over conventional associative processors. Improvement factors tabulated over a wide range of input bit lengths and epth of associativity are presented for various arithmetic algorithms. Interesting arithmetic problems such as input/output conversions and addition and multiplication algorithms are treated. The processor architecture for system control of a large number of residue fields is examined. Considerations of microprogramming arithmetic algorithms in the RAAP are described. (Modified author abstract)

Abstract: One of the most important types of digital electronic system is the digital computer. Indeed, it may be argued that much of the pressure for the development of cheaper and faster digital system elements has come from the rapidly expanding computer industry. In this chapter, we consider digital systems for performing arithmetic operations in the binary number system. Such systems of course are important components of digital computers. Readers not familiar with the basic arithmetic operations in the binary number system may wish to refer to appendix A before proceeding with this chapter.

Abstract: The appearance of hexadecimal floating-point arithmetic systems has prompted a continuing discourse on the relative numerical merits of various choices of base. Until lately this discourse has centered around the static properties of the floating-point representation of numbers, and has primarily concerned only binary and hexadecimal representations. Recent events may change this discourse considerably. A third numerically attractive alternative for the choice of base has been proposed, and a comparison of the dynamic numerical properties of floating-point arithmetic systems has been completed. This paper surveys these recent events and summarizes our current knowledge of the numerical characteristics of floating-point arithmetic systems.