# Trending Articles in Mathematics

## Series in Mathematics: From Zeno to Quantum Theory

Introduction Series play a decisive role in many branches of mathematics. They accompanied mathematical developments from Zeno of Elea (##5##-th century BC) and Archimedes of Syracuse (##3##-th century BC), to the fundamental building blocks of calculus from the ##17##-th century on, up to modern Lie theory which is crucial for our understanding of quantum theory….

## Differential Equation Systems and Nature

Abstract “Mathematics is the native language of nature.” is a phrase that is often used when it comes to explaining why mathematics is all around in natural sciences, especially in physics. What does that mean? A closer look shows us that it primarily means that we describe nature by differential equations, a lot of differential…

## Beginners Guide to Precalculus, Calculus and Infinitesimals

Introduction I am convinced students learn Calculus far too late.   In my view, there has never been a good reason for this. In the US, they go through this sequence of Pre-Algebra, Algebra 1, Geometry, Algebra 2, Precalculus, Calculus 1, and Calculus 2.  But is this required?   Recently I came across two books that turned…

## What Are Numbers?

Introduction When doing mathematics,  we usually take for granted what natural numbers, integers, and rationals are. They are pretty intuitive.   Going from rational numbers to reals is more complicated.   The easiest way at the start is probably infinite decimals.  Dedekind Cuts can be used to get a bit more fancy.  A Dedekind cut is a…

## Introduction to the World of Algebras

Abstract Richard Pierce describes the intention of his book  about associative algebras as his attempt to prove that there is algebra after Galois theory. Whereas Galois theory might not really be on the agenda of physicists, many algebras are: from tensor algebras as the gown for infinitesimal coordinates over Graßmann and Banach algebras for…

## What Are Infinitesimals – Simple Version

Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be thrown away because they are negligible. That way, when defining the derivative, for example, you do not run into 0/0, but when…

## What Are Infinitesimals – Advanced Version

Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be thrown away because they are negligible. That way, when defining the derivative, for example, you do not run into 0/0, but when…

## The Art of Integration

Abstract My school teacher used to say “Everybody can differentiate, but it takes an artist to integrate.” The mathematical reason behind this phrase is, that differentiation is the calculation of a limit $$f'(x)=\lim_{v\to 0} g(v)$$ for which we have many rules and theorems at hand. And if nothing else helps, we still can…

## An Overview of Complex Differentiation and Integration

Abstract I want to shed some light on complex analysis without getting all the technical details in the way which are necessary for the precise treatments that can be found in many excellent standard textbooks. Analysis is about differentiation. Hence, complex differentiation will be my starting point. It is simultaneously my finish line because its…