Mathematics Articles

Mathematics as the study of “relationships” rather than “patterns” but they are obviously closely related(!). There is a field of mathematics called “category theory” that is just about as abstract as you can get (the textbook, in the preface, said category theory is often called “abstract nonsense” with no sense of that being derogatory at all).

A category has “objects” and “relations”. The collection of all sets is a category with sets as objects and functions between them as “relations”. The collection of topological spaces is a category with the topological spaces being the objects and continuous functions from one topological space to another being the relations.


lerch and zeta functions

The Analytic Continuation of the Lerch and the Zeta Functions

Introduction In this brief Insight article the analytic continuations of the Lerch Transcendent and Riemann Zeta Functions are achieved via the Euler's…
Integral Representations of Some Special Functions

A Path to Fractional Integral Representations of Some Special Functions

0. Introduction As for the reference material I have used the text Special Functions by Askey, Andrews, and Roy which covers much of the theorems here…

SOHCAHTOA: Seemingly Simple, Conceivably Complex

Preface My first experience with derivatives was seeing how they are obtained from the usual definition $$f'(x)=\underset{\text{$\Delta $x}\to 0}{\text{Lim}}\frac{f…
How to Find Potential Functions

How to Find Potential Functions? A 10 Minute Introduction

Definition/Summary Given a vector field ##\vec F(x,y,z)## that has a potential function, how do you find it? Equations $$\nabla \phi(x,y,z) = \vec F(x,y,z)$$…
What is a linear equation

What is a Linear Equation? A 5 Minute Introduction

Definition/Summary A first-order polynomial equation in one variable, its general form is [itex]Mx+B=0[/itex] where x is the variable. The quantities…
What are significant figures

What are Significant Figures? A 5 Minute Introduction

Definition/Summary Significant figures (commonly called "sig figs") are the number of figures (digits) included when rounding-off a number.For example,…
writing proofs

How Most Proofs Are Structured and How to Write Them

... or the answer to: "I have no idea where to start!" Proofs in mathematics are what mathematics is all about. They are subject to entire books, created…
What is a fibre bundle

What is a Fibre Bundle? A 5 Minute Introduction

Definition/Summary Intuitively speaking, a fibre bundle is space E which 'locally looks like' a product space B×F, but globally may have a different…
What are real numbers

What is a Real Number? A 5 Minute Introduction

Definition/Summary The real numbers are the most commonly encountered number system, familiar to the layman via the number line, and as the number system…
What is a parabola

What is a Parabola? A 5 Minute Introduction

Definition/Summary A parabola has many definitions, a classical one being, "A Parabola is the locus of all points equidistant from a given point (called…
Limit of a Function

What Is a Limit of a Function? A 5 Minute Introduction

Definition/Summary Limits are a mathematical tool that is used to define the 'limiting value' of a function i.e. the value a function seems to approach…
What is a Tangent Line

What is a Tangent Line? A 5 Minute Introduction

Definition/Summary The tangent to a curve in a plane at a particular point has the same Gradient as the curve has at that point.More generally, the…
lie algebra representations

Lie Algebras: A Walkthrough The Representations

  Part III: Representations  10. Sums and Products. Frobenius began in ##1896## to generalize Weber's group characters and soon investigated…
Lie Algebra Structure

Learn Lie Algebras: A Walkthrough The Structures

  Part II: Structures5. Decompositions.Lie algebra theory is to a large extend the classification of the semisimple Lie algebras…
lie algebra basics

Learn Lie Algebras: A Walkthrough The Basics

  Part I: Basics 1. Introduction. This article is meant to provide a quick reference guide to Lie algebras: the terminology, important theorems,…

How to Self Study Abstract Algebra

There are three big parts of mathematics: geometry, analysis, and algebra. In this insight, I will try to give a roadmap towards learning basic abstract…

An Interesting Ramsey Theory Riddle

Ramsey theory has its origins in a very nice riddle Consider a party of 6 people. Any two of these 6 will either be meeting each other for the first time…

How to Self Study Intermediate Analysis

If you wish to follow this guide, then you should know how to do analysis on ##\mathbb{R}## and ##\mathbb{R}^n##. See my previous insight if you wish to…

Intro to the Millennium Prize Problems

IntroductionIn this Insight, I will go over the background information for the Millennium Prize problems and briefly describe three of them. A future…

An Intro on Real Numbers and Real Analysis

It is important to realize that in standard mathematics, we attempt to characterize everything in terms of sets. This means that notions such as natural…

Learn About Matrix Representations of Linear Transformations

Let X and Y be finite-dimensional vector spaces. Let ##T:X\to Y## be a linear transformation. Let ##A=(e_1,\dots,e_n)## and ##B=(f_1,\dots,f_m)## be ordered…

Why Do People Say That 1 And .999 Are Equal?

Why do people say 1 and 0.999... are equal? Aren't they two different numbers?No, they really are the same number, though this is often very counterintuitive…

Is There a Rigorous Proof Of 1 = 0.999…?

Yes.First, we have not addressed what 0.999... actually means. So it's best first to describe what on earth the notation [tex]b_0.b_1b_2b_3...[/tex]…

The History and Concept of the Number 0

The goal of this FAQ is to clear up the concept of 0 and specifically the operations that are allowed with 0.The best way to start this FAQ is to look…
ADHD studying

How I Overcame Learning Challenges That I Faced Studying Science

Introduction For the past few days, during my summer break, I have been intensively self-studying mathematics (namely number theory) for several hours…