Math Articles

Here is our every growing collection of expert math articles that deal with all mathematics disciplines. These cover all areas of math and all skill ranges from algebra to advanced calculus.

Infinitesimals

What Are Infinitesimals – Advanced Version

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Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say…
integration and complex differentiation

An Overview of Complex Differentiation and Integration

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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…
lie group physics

When Lie Groups Became Physics

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Abstract We explain by simple examples (one-parameter Lie groups), partly in the original language, and along the historical papers of Sophus Lie, Abraham…
math classifications

Classification of Mathematics by 42 Branches

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 I often read questions about our classification scheme that we use on physicsforums.com to sort posts by science fields and subjects, what has…
history of numbers

Counting to p-adic Calculus: All Number Systems That We Have

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An entire book could easily be written about the history of numbers from ancient Babylon and India, over Abu Dscha'far Muhammad ibn Musa al-Chwarizmi (##\sim…
evariste galois

Évariste Galois and His Theory

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 * Oct. 25th, 1811  † May 31st, 1832 ... or why squaring the circle is doomed. Galois died in a duel at the age of twenty. Yet, he gave…
definition differences

Yardsticks to Metric Tensor Fields

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I asked myself why different scientists understand the same thing seemingly differently, especially the concept of a metric tensor. If we ask a topologist,…
pvsnp

P vs. NP and what is a Turing Machine (TM)?

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P or NP This article deals with the complexity of calculations and in particular the meaning of ##P\stackrel{?}{\neq}NP## Before we explain what P and…
Riemann Hypothesis History

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…
Hyperbola

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…
probabilities virus testing

Probabilistic Factors Involved in Disease and Virus Testing

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Introduction This Insight looks at the various probabilistic factors and related terminology involved in disease and virus testing.As we all know,…
blackboard

10 Math Things We All Learnt Wrong At School

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The title is admittedly clickbait. Or a joke. Or a provocation. It depends on with whom you speak, or who reads it with which expectation. Well, I cannot…
bayesian inference

How Bayesian Inference Works in the Context of Science

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Confessions of a moderate Bayesian part 3 Read part 1: How to Get Started with Bayesian Statistics Read part 2: Frequentist Probability vs Bayesian ProbabilityBayesian…
bayesian statistics part 2

Exploring Frequentist Probability vs Bayesian Probability

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Confessions of a moderate Bayesian, part 2 Read Part 1: Confessions of a moderate Bayesian, part 1Bayesian statistics by and for non-statisticianshttps://www.cafepress.com/physicsforums.13280237 Background One…
bayesian statistics

How to Get Started with Bayesian Statistics

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Confessions of a moderate Bayesian, part 1 Bayesian statistics by and for non-statisticianshttps://www.cafepress.com/physicsforums.13265286 Background I…
mary Somerville

Mathematician Mary Somerville Features in Google Doodle

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The Google Doodle for 2 February 2020 celebrated Mary Somerville, the Scottish polymath and science writer, and with Caroline Herschel, the joint first-ever…
writing proofs

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…
Maupertuis Principle

A Pure Hamiltonian Proof of the Maupertuis Principle

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Here is another version of proof of Maupertuis's principle. This version is pure Hamiltonian and independent of the Lagrangian approach.The proof…
The Sum of Geometric Series from Probability Theory

The Sum of Geometric Series from Probability Theory

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Here I present a simple (but to the best of my knowledge, new) derivation of the formula for the sum of the infinite geometric series. The derivation is…
lie algebra representations

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…
Lie Algebra Structure

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…
lie algebra basics

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,…
stock options math

Learn a Simplified Synthesis of Financial Options Pricing

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Financial options (the right to purchase ("call") or sell ("put") stock (or other assets)) at a fixed price at a future date have been around for a long…
hilbertspaces2

Learn the Basics of Hilbert Spaces and Their Relatives: Operators

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  Operators. The Maze Of Definitions. We will use the conventions of part I (Basics), which are ##\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}##,…
hilbertspaces

Learn the Basics of Hilbert Spaces and Their Relatives

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  Basics Language first: There is no such thing as the Hilbert space.Hilbert spaces can look rather different, and which one is used in…
dice

Learn About Intransitive Dice with a Twist

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Intransitive dice are sets of dice that don't follow the usual rules for "is better/larger than". If A<B and B<C, then A<C. If Bob runs faster…

A Journey to The Manifold SU(2): Differentiation, Spheres, and Fiber Bundles

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Part 2  Differentiation, Spheres, and Fiber Bundles Image source: [24]The special unitary groups play a significant role in the standard…
mathoperators

How to Tell Operations, Operators, Functionals, and Representations Apart

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 All these concepts belong to the toolbox of physicists. I read them quite often on our forum and their usage is sometimes a bit confusing.…
geometrysimple

When Simple Geometry Unveils Deep Math

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Introduction It is a remarkable fact that consideration of very elementary concepts in geometry often leads quickly into deep and unexpected mathematical…
representations

Linear Representations and Why Precision is Important in Math

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First of all: What is a representation? It is the description of a mathematical object like a Lie group or a Lie algebra by its actions on another space…
angledimensions

Can Angles be Assigned a Dimension?

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1. Some Background on Dimensional Analysis ... if you are not already familiar with it. 1.1 Dimensions Dimensional Analysis is a way of analyzing…
groupsandgeometry

Exploring the Relationship Between Group Theory and Geometry

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There is a very deep link between group theory and geometry. Sadly, this link is not emphasized a lot in most courses of group theory, even though it is…
computers

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…
logicp3

Scientific Inference: Balancing Predictive Success with Falsifiability

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  Bayes' Theorem: Balancing predictive success with falsifiability Despite its murky logical pedigree, confirmation is a key part of learning.…
encryption

Is It Possible to Design an Unbreakable Cipher?

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Is it possible to design an unbreakable cipher? Do methods of encryption exist that guarantee privacy from even the most capable and highly-resourced…
cipher

The Monographic Substitution Cipher: From Julius Caesar to the KGB

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A monographic substitution cipher works by replacing individual characters of plaintext with corresponding characters of ciphertext. It is perhaps the…
complexmath

Hear the Case for Learning Complex Math

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Resistance to complex math seems to never die out.  I see it frequently in PF posts.  Often it takes the form of challenges rather than questions. …
complexnumbers

Things Which Can Go Wrong with Complex Numbers

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At the first sight, there are many paradoxes in complex number theory. Here are some nice examples of things that don't seem to work:Example A [itex]-1=i^2=\sqrt{-1}\cdot\sqrt{-1}=\sqrt{(-1)(-1)}=\sqrt{1}=1[/itex]Example…
MillenniumPrize

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…
micro2

Learn Axioms for the Natural Numbers

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** Bloch Chapter 1.2The Peano system in Bloch has a special element ##1\in \mathbb{N}##. The intuitive idea here is that ##\mathbb{N} = \{1,2,3,...\}##.…
micro3

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…
physicsturnedmath

Trials and Tribulations of a Physicist who Became a Math Geek

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How did I go from the brink of changing my major from physics to ceramics (no more math) to the Math faculty of the Air Force Academy? How did I go from…
math obvious

Lessons From My Experience Teaching Math

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My #1 goal, when I teach a math class, is to convey a certain way of thinking about math. It's quite different from what my students have done before,…
subway

A Brachistochrone Subway Is Not a Cost-effective Idea

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It is apparent that a subway tunnel could be built without the need for supplied energy like electricity, assuming zero friction everywhere.  The tunnel…
abstract_algebra

Is Zero a Natural Number?

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Using: Anderson-Feil Chapter 1.1 Is zero a natural number? This is a pretty controversial question. Many mathematicians - especially those working in…
micro1

What is a Property Formally in Mathematical Logic

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** Hrbacek-Jech Chapter 1.2Hrbacek and Jech do not go into full detail about what a property is formally. This is a part of mathematical logic, but…