Related papers: Origin of the numerals, Al biruni testimony
Number theory as a coherent mathematical subject started with the work of Fermat in the decade from 1630 to 1640, but modern number theory, that is, the systematic and mathematically rigorous development of the subject from fundamental…
There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been…
We discuss some examples that illustrate the countability of the positive rational numbers and related sets. Techniques include radix representations, Godel numbering, the fundamental theorem of arithmetic, continued fractions, Egyptian…
In this note, we provide bijective proofs of some identities involving the Bell number, as previously requested. Our arguments may be extended to yield a generalization in terms of complete Bell polynomials. We also provide a further…
Arabic language is one of the most popular languages in the world. Hundreds of millions of people in many countries around the world speak Arabic as their native speaking. However, due to complexity of Arabic language, recognition of…
A mathematically and culturally natural modification of Evart Bruins' explanation of the genesis of the numbers on the Old Babylonian tablet Plimpton 322 gives an economical accounting for the "missing pairs" in his reconstruction. When the…
This essay traces the history of three interconnected strands. Firstly, changes in the concept of number, secondly, the study of the qualities of number, which evolved into number theory, and thirdly, the nature of mathematics itself, from…
We retrace the recent history of the Umbral Calculus. After studying the classic results concerning polynomial sequences of binomial type, we generalize to a certain type of logarithmic series. Finally, we demonstrate numerous typical…
Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of…
This is the paper "Niels Henrik Abel and the birth of fractional calculus", Podlubny, I., Magin, R. L., Trymorush I., Fractional Calculus and Applied Analysis, vol.20, no.5, pp.1068-1075, 2017 (https://doi.org/10.1515/fca-2017-0057) with…
We give a simple proof and a multivariable generalization of an identity due to E. Alkan concerning a weighted average of the Ramanujan sums. We deduce identities for other weighted averages of the Ramanujan sums with weights concerning…
Mathematical notations around the world are diverse. Not as much as requiring computing machines' makers to adapt to each culture, but as much as to disorient a person landing on a web-page with a text in mathematics. In order to understand…
Arabic is a widely-spoken language with a long and rich history, but existing corpora and language technology focus mostly on modern Arabic and its varieties. Therefore, studying the history of the language has so far been mostly limited to…
There are a lot of intensive researches on handwritten character recognition (HCR) for almost past four decades. The research has been done on some of popular scripts such as Roman, Arabic, Chinese and Indian. In this paper we present a…
A numerical semigroup is an additive submonoid of the natural numbers with finite complement. The size of the complement is called the genus of the semigroup. How many numerical semigroups have genus equal to $g$? We outline Zhai's proof of…
Hilbert's Foundations of Geometry was perhaps one of the most influential works of geometry in the 20th century and its axiomatics was the first systematic attempt to clear up the logical gaps of the Elements. But does it have gaps of its…
In this article we introduce the concept of almost $\mathcal{P}$-numbers. We survey the existing results in literature for almost cyclic numbers and give characterizations for almost abelian and almost nilpotent numbers proving these two…
The Fibonacci number is the residue of a rational function, from which follows that Fibonacci number summation identities can be derived with the integral representation method, a method also used to derive combinatorial identities. A…
The induction principle for natural numbers expresses that when a property holds for some natural number a and is hereditary, then it holds for all numbers greater than or equal to a. We present a similar principle for real numbers.
Almost from the inception of Hilbert's program, foundational and structural efforts in proof theory have been directed towards the goal of clarifying the computational content of modern mathematical methods. This essay surveys various…