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The transreal numbers are a total number system in which every arithmetical operation is well defined everywhere. This has many benefits over the real numbers as a basis for computation and, possibly, for physical theories. We define the topology of the transreal numbers and show that it gives a more coherent interpretation of two's complement arithmetic than the conventional integer model. Trans-two's-complement arithmetic handles the infinities and 0/0 more coherently, and with very much less circuitry, than floating-point arithmetic. This reduction in circuitry is especially beneficial in parallel computers, such as the Perspex machine, and the increase in functionality makes Digital Signal Processing chips better suited to general computation. [ABSTRACT FROM AUTHOR]

This paper reports a study of numerical computation problems from a theoretical viewpoint. It shows that computer systems are not capable of computing of real numbers correctly due to the differences between the algebraic structures of real numbers and model numbers. These two classes of numbers are not isomorphic. From this study, we have learned that there are no machine errors or computation errors. If fact, one can view it is a human mistake by putting real valued problem onto a model number platform for computation. This paper proposes use of the concept of computer model numbers to approximate rough numbers for computation. Moreover, we revise an arbitrary initial compact interval to a shortest initial closed-open model interval for ordinary interval computation. This way, we can assure that the final resulting interval will be the shortest interval and that the computation will result in the greatest precision. [ABSTRACT FROM AUTHOR]

In the appendix all the mathematical background needed for complete understanding of the text is quickly reviewed. Depending on who you are, this chapter may be entirely superfluous, or it may be one of the most useful chapters in the entire book. You should probably at least look it over before starting to read Chapter 1. If most of the material in this chapter is unfamiliar to you, then you are probably not ready to continue reading. You should definitely consult it whenever you feel uncomfortable with the mathematics being used in any of the chapters, and it is written to be useful in a more general setting. Under no conditions should you read it in its natural place, after the last chapter; if you have already finished the book, you do not need it any more. [ABSTRACT FROM PUBLISHER]

This article discusses various issues related to the number theory. In the contemporary study of mathematics, number theory stands out as a peculiar branch for many reasons. Most of the development of mathematical thought is concerned with the identification of certain structures and relations in these structures. Number theory, however, is the study of the properties of those few systems that arise naturally, beginning with the natural numbers, progressing to the integers, the rational numbers, the real numbers, and the complex numbers. Rather than identifying very general principles, number theory is concerned with very specific questions about these few systems. Number theory is a field with great paradoxes. This purest of mathematical disciplines has served as the source for arguably the most important set of applications of mathematics in many years.

The article states that the eigenvectors corresponding to distinct eigenvalues are linearly independent. Eigenvalues of a real matrix appear as real numbers or complex conjugate pairs. A symmetric real matrix has all real eigenvalues. Model analysis is based on the state space representation. It also explores eigenvalues, eigenvectors, and transfer functions. Complex eigenvalues occur in conjugate pairs, and each pair corresponds to an oscillatory mode. The real component of the eigenvalues gives the damping, and the imaginary component defines the frequency of oscillation.

The article presents an encyclopedia entry for gamma distribution. A continuous random variable is said to have such distribution if its probability density can be written in the form f(t)=a(at)b-1e-at/Γ(b), where a and b are any positive real numbers and Γ(b) is the gamma function evaluated at b.

A definition of the term "Discrete Cosine Transform" (DCT) is presented. DCT refers to a Fourier-related transform which is similar to the discrete Fourier transform, but uses only real numbers. It operates on real data with even symmetry where the input and output data are shifted by half a sample.

The entry presents a definition of the term vector n-space. It is defined as a set of vectors of points, each with n components and rules for vector addition and multiplication by real numbers.

A definition of the term "payoff matrix" is presented. It stresses that the payoff matrix is an m x n matrix of real numbers for a zero-sum, two-person game.

The encyclopedia entry for the term Line is presented. It refers to the set of points where two values are points in a dimensional space and another is a real number.

A mathematical problem about number choices is presented.

A quiz about mathematics is presented.