Related papers: Real numbers as infinite decimals -- theory and co…
In this expository article, the real numbers are defined as infinite decimals. After defining an ordering relation and the arithmetic operations, it is shown that the set of real numbers is a complete ordered field. It is further shown that…
In this paper we provide a complete approach to the real numbers via decimal representations. Construction of the real numbers by Dedekind cuts, Cauchy sequences of rational numbers, and the algebraic characterization of the real number…
In this paper, we chronologically recount several situations that have contributed to the development and formalization of the objects known as imaginary or complex numbers. We will begin by introducing the earliest documented knowing for…
Completing Loo-Keng Hua's approach to the real number system pioneered in 1962, this paper defines arithmetical operations directly on infinite decimals without appealing to any ordering structure. Therefore, the widespread belief that…
Infinitesimals are natural products of the human imagination. Their history goes back to the Greek antiquity. Their role in the calculus and analysis has seen dramatic ups and downs. They have stimulated strong opinions and even vitriol.…
In order to prove irrationality of \sqrt{2} by using only decimal expansions (and not fractions), we develop in detail a model of real numbers based on infinite decimals and arithmetic operations with them.
Exact representations of real numbers such as the signed digit representation or more generally linear fractional representations or the infinite Gray code represent real numbers as infinite streams of digits. In earlier work by the first…
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.…
We give a~detailed construction of the complete ordered field of real numbers by means of infinite decimal expansions. We prove that in the canonical encoding of decimals neither addition nor multiplication is {\em computable}, but that…
We present an explicit bijection between finite-decimal real numbers and natural numbers ($\mathbb{N} = \{1, 2, 3, ...\}$) using a systematic 4-tuple parametrization with closed-form mathematical formulas for enumeration. Our enumeration…
In recent publications in physics and mathematics, concerns have been raised about the use of real numbers to describe quantities in physics, and in particular about the usual assumption that physical quantities are infinitely precise. In…
While the prime numbers have been subject to mathematical inquiry since the ancient Greeks, the accumulated effort of understanding these numbers has - as Marcus du Sautoy recently phrased it - 'not revealed the origins of what makes the…
In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are…
A new computational methodology for executing calculations with infinite and infinitesimal quantities is described in this paper. It is based on the principle `The part is less than the whole' introduced by Ancient Greeks and applied to all…
This paper looks at how ancient mathematicians (and especially the Pythagorean school) were faced by problems/paradoxes associated with the infinite which led them to juggle two systems of numbers: the discrete whole/rationals which were…
In 1969, H. Davenport and W. M. Schmidt studied the problem of approximation to a real number \xi by algebraic integers of degree at most three. They did so, using geometry of numbers, by resorting to the dual problem of finding…
A recently developed computational methodology for executing numerical calculations with infinities and infinitesimals is described in this paper. The developed approach has a pronounced applied character and is based on the principle `The…
We explore the potential of Simon Stevin's numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis.
It is a ubiquitous opinion among mathematicians that a real number is just a point in the line. If this rough definition is not enough, then a mathematician may provide a formal definition of the real numbers in the set theoretic and…
We describe how to compute very far decimals of $$\pi$$ and how to provide formal guarantees that the decimals we compute are correct. In particular, we report on an experiment where 1 million decimals of $$\pi$$ and the billionth…