相关论文: Diophantine and Non-Diophantine Arithmetics: Opera…

Science and mathematics help people to better understand world, eliminating many inconsistencies, fallacies and misconceptions. One of such misconceptions is related to arithmetic of natural numbers, which is extremely important both for…

Arithmetic operations (addition, subtraction, multiplication, division), as well as the calculus they imply, are non-unique. The examples of four-dimensional spaces, $\mathbb{R}_+^4$ and $(-L/2,L/2)^4$, are considered where different types…

The results of difference sequences theory are applied to analytic function theory and Diophantine equations. As a result we have the equation which connects the $n$-th derivative of a function with the difference sequence for the values of…

All but a few digital computers used for scientific computations have supported floating-point and digital arithmetic of rather limited numerical precision. The underlying assumptions were that the systems being studied were basically…

In basic computational physics classes, students often raise the question of how to compute a number that exceeds the numerical limit of the machine. While technique of avoiding overflow/underflow has practical application in the electrical…

We claim that human mathematics is only a limited part of the consequences of the chosen basic axioms. Properly human mathematics varies with time but appears to have universal features which we try to analyze. In particular the functioning…

The differences between the sets in which ideal arithmetics takes place and the sets of floating point numbers are outlined. A set of classical problems in correct numerical evaluation is presented, to increase the awareness of newcomers to…

Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is…

All sciences need and many arts apply mathematics whereas mathematics seems to be independent of all of them, but only based upon logic. This conservative concept, however, needs to be revised because, contrary to Platonic idealism…

For a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd…

Being mathematics a natural language to Mankind and to physics, it must be constantly adapted to our necessities and our natural perception. Then, mathematical concepts are not absolute to reality. Although mathematical theories are…

Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are…

A paradigm for a global algebraic number theory of the reals is formulated with the purpose of providing a unified setting for algebraic and transcendental number theory. This is achieved through the study of subgroups of nonstandard models…

Natural phenomenon of coevolution is the reciprocally induced evolutionary change between two or more species or population. Though this biological occurrence is a natural fact, there are only few attempts to use this as a simile in…

Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not…

A famous problem posed by Diophantus was to find sets of distinct positive rational numbers such that the product of any two is one less than a rational square. Some sets of six such numbers are presented and the computational algorithm…

A coordinate system is a foundation for every quantitative science, engineering, and medicine. Classical physics and statistics are based on the Cartesian coordinate system. The classical probability and hypothesis testing theory can only…

The world of mathematics is often considered abstract, with its symbols, concepts, and topics appearing unrelated to physical objects. However, it is important to recognize that the development of mathematics is fundamentally influenced by…

The arithmetic-digital anomaly of $5\div 2 = 2.5$ has been observed several times in the past. We generalize it to an exponential Diophantine equation and inequality in the general number base, which is the object of our analysis. First, we…

The vast use of computers on scientific numerical computation makes the awareness of the limited precision that these machines are able to provide us an essential matter. A limited and insufficient precision allied to the truncation and…