## Articles for: mathematics

### The History and Importance of the Riemann Hypothesis

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Riemann Hypothesis HistoryRH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line.
ERH: All zeros of L-functions to…

### The Extended Riemann Hypothesis and Ramanujan’s Sum

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Riemann Hypothesis and Ramanujan's Sum ExplanationRH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line.
ERH: All…

### The Amazing Relationship Between Integration And Euler’s Number

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We use integration to measure lengths, areas, or volumes. This is a geometrical interpretation, but we want to examine an analytical interpretation that…

### The Analytic Continuation of the Lerch and the Zeta Functions

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

### A Path to Fractional Integral Representations of Some Special Functions

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Introduction
This bit is what new thing you can learn reading this:) As for original content, I only have hope that the method of using the sets
$$C_N^n:…

### SOHCAHTOA: Seemingly Simple, Conceivably Complex

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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? A 10 Minute Introduction

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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? A 5 Minute Introduction

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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? A 5 Minute Introduction

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

### How to Write a Math Proof and Their Structure

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Proofs in mathematics are what mathematics is all about. They are subject to entire books, created entire theories like Fermat's last theorem, are hard…

### What is a Fibre Bundle? A 5 Minute Introduction

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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 is a Real Number? A 5 Minute Introduction

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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? A 5 Minute Introduction

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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…

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

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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? A 5 Minute Introduction

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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 Algebras: A Walkthrough The Representations

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

### Learn Lie Algebras: A Walkthrough The Structures

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

### Learn Lie Algebras: A Walkthrough The Basics

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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

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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

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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 Math

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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

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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

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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

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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?

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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…?

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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

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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…

### How I Overcame Learning Challenges That I Faced Studying Science

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