Virtual Calculus - Part I

An order theoretic and algebraic framework for the extended real numbers is established which includes extensions of the usual difference to expressions involving $-\infty$ and/or $+\infty$, so-called residuations. Based on this, definitions and results for directional derivatives, subdifferentials and Legendre--Fenchel conjugates for extended real-valued functions are given which admit to include the proper as well as the improper case. For set-valued functions, scalar repre...

The usual $\epsilon,\delta$-definition of the limit of a function (whether presented at a rigorous or an intuitive level) requires a "candidate $L$" for the limit value. Thus, we have to start our first calculus course with "guessing" instead of "calculating". In this paper we criticize the method of using calculators for the purpose of selecting candidates for $L$. We suggest an alternative: a working formula for calculating the limit value L of a real function in terms of i...

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 related to the non-standard analysis) is used to work with finite, infinite, and infinitesimal numbers \textit{numerically}. This can be done on a new kind of a computer - the Infinity Computer - able to work with all these types of numbers. T...

Treating divergent series properly has been an ongoing issue in mathematics. However, many of the problems in divergent series stem from the fact that divergent series were discovered prior to having a number system which could handle them. The infinities that resulted from divergent series led to contradictions within the real number system, but these contradictions are largely alleviated with the hyperreal number system. Hyperreal numbers provide a framework for dealing wit...

The Compositional Integral is defined, formally constructed, and discussed. A direct generalization of Riemann's construction of the integral; it is intended as an alternative way of looking at First Order Differential Equations. This brief notice aims to: familiarize the reader with a different approach to integration, fabricate a notation for a modified integral, and express a startling use for infinitely nested compositions. Taking inspiration from Euler's Method for appro...

There exist many applications where it is necessary to approximate numerically derivatives of a function which is given by a computer procedure. In particular, all the fields of optimization have a special interest in such a kind of information. In this paper, a new way to do this is presented for a new kind of a computer -- the Infinity Computer -- able to work numerically with finite, infinite, and infinitesimal numbers. It is proved that the Infinity Computer is able to ca...

Neutrosophic Analysis is a generalization of Set Analysis, which in its turn is a generalization of Interval Analysis. Neutrosophic Precalculus is referred to indeterminate staticity, while Neutrosophic Calculus is the mathematics of indeterminate change. The Neutrosophic Precalculus and Neutrosophic Calculus can be developed in many ways, depending on the types of indeterminacy one has and on the methods used to deal with such indeterminacy. In this book, the author presents...

We present a characterization of the completeness of the field of real numbers in the form of a \emph{collection of several equivalent statements} borrowed from algebra, real analysis, general topology, and non-standard analysis. We also discuss the completeness of non-Archimedean fields and present several examples of such fields. As an application, we exploit the characterization of the reals to argue that the Leibniz-Euler infinitesimal calculus in the $17^\textrm{th}$-$18...

In this article it is proven the existence of integration of indefinite integrals as infinite derivative's series expansion. This also opens a new way to integrate a definite integral.

Abraham Robinson's framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based ...