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.

Tag Archive for: mathematics

vector spaces

Vector Spaces: Concepts, History, and Applications Guide

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The Concept A vector space is an additively written abelian group together with a field that operates on it.Intuitive picture vs abstract definition Vector…
complex numbers views

Views On Complex Numbers

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Abstract Why do we need yet another article about complex numbers? This is a valid question and I have asked it myself. I could mention that I wanted…
Lambert W Function in Finance

The Lambert W Function in Finance

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Preamble The classical mathematician practically by instinct views the continuous process as the "real" process, and the discrete process as an approximation…
infinity

Why Division by Zero Is Impossible: 10 Mathematical Reasons

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A division by zero is primarily an algebraic question. The reasoning therefore follows the indirect pattern of most algebraic proofs:What if it was…
Epsilontic limits and continuity

Epsilontic – Limits and Continuity

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Abstract I remember that I had some difficulties moving from school mathematics to university mathematics. From what I read on PF through the years, I…
Differential Equation Systems and Nature

Differential Equation Systems and Nature

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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…
calc precalc

Beginners Guide to Precalculus, Calculus and Infinitesimals

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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…
what are numbers

What Are Numbers?

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Introduction When doing mathematics,  we usually take for granted what natural numbers, integers, and rationals are. They are pretty intuitive.   Going…
math classifications

Mathematics Fields & Applications: A Structured Guide

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Introduction I often read questions about our classification scheme that we use on physicsforums.com to sort posts by science fields and subjects, what…
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…
Riemann Hypothesis History

The History and Importance of the Riemann Hypothesis

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Riemann Hypothesis History The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics, specifically in the field…
Riemann Hypothesis

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…
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…
lerch and zeta functions

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…
Integral Representations of Some Special Functions

The Orin Fractional Calculus

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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

SOHCAHTOA: Seemingly Simple, Conceivably Complex

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What is SOHCAHTOA SOHCAHTOA is a mnemonic acronym used in trigonometry to remember the relationships between the sides and angles of right triangles.…
How to Find Potential Functions

How to Find a Potential Function of a Vector Field

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

First-Order Linear Equation: Definition & Solutions

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Definition / Summary This article summarizes the first-order (linear) polynomial equation in one variable, its solution, and natural extensions to matrices…
What are significant figures

Significant Figures: Rounding Rules & Examples for Science

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Definition / Summary Significant figures (commonly called "sig figs") are the digits in a number that are considered when rounding a value to reflect…
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…
What is a fibre bundle

Fibre Bundles Explained: Definition, Examples & Visualization

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A fibre bundle is a mathematical structure consisting of a total space that looks locally like a simple product of two spaces but may have a different…
What are real numbers

Real Numbers Explained: Definition, Axioms & Properties

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Real numbers are the complete set of values on the continuous number line, denoted ℝ, and they include both rational numbers (fractions and repeating…
What is a parabola

What Is a Parabola? Equations, Focus, and Real Uses

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A parabola is a U-shaped curve where every point is equidistant from a fixed point called the focus and a fixed line called the directrix. Its standard…
Limit of a Function

What Is a Limit of a Function? Definition, Laws & Examples

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A limit describes the value a function or sequence approaches as its input gets arbitrarily close to a specific point, without requiring the function to…
What is a Tangent Line

Tangent Line Equations: Circles, Curves & Surfaces Explained

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A tangent line or tangent hyperplane to a curve or surface at a given point shares the same gradient, or first-order linear approximation, as that curve…
lie algebra representations

Learn Lie Algebras: A Walkthrough – The Representations

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  Part III: Representations  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: StructuresDecompositions.Lie algebra theory is to a large extend the classification of the semisimple Lie algebras which…
lie algebra basics

Learn Lie Algebras: A Walkthrough – The Basics

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

Intuitive Black‑Scholes Options Pricing Explained Simply

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Introduction Financial options — the right to purchase (call) or sell (put) stock or other assets at a fixed price on a future date — have been around…
selfstudy

Self-Study Guide: How to Learn Abstract Algebra Step by Step

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Self-studying abstract algebra requires only precalculus-level math and basic familiarity with mathematical proofs. A practical path runs from proof-writing…
computers

Ramsey Theory Explained: The Party Riddle and R(m,n)

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Ramsey's theorem guarantees that in any sufficiently large gathering, a specific pattern of mutual relationships must appear, no matter how the relationships…
selfstudyanalysis

Self-Study Analysis: Metric Spaces, Measure Theory & More

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Direct answer: After finishing analysis on real numbers and n-dimensional real space, the recommended path for self-study is Carothers' Real Analysis for…
MillenniumPrize

Millennium Problems: Poincaré, P vs NP, Riemann

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IntroductionWhat this Insight Covers In this Insight, I will go over the background information for the Millennium Prize problems and briefly describe…
micro3

Why Mathematics Defines Numbers and Functions as Sets

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Direct answer: Standard mathematics defines numbers, functions, and other objects as sets because the 19th-century collapse of certainty in Euclidean geometry…
lineartransformations

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…
999equals1

Why 1 = 0.999…: Four Simple Proofs Explained

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Yes, 1 and 0.999... are exactly the same real number, not merely two numbers that happen to be extremely close. This surprises many people because 0.999...…
999

Is 1 Equal to 0.999…? Rigorous Proof Explained

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Yes, 1 equals 0.999... exactly, not approximately. This is a proven mathematical fact once the notation 0.999... and the concept of an infinite sum are…

The Concept of Zero: History, Rules, and Why 0/0 Fails

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Zero is neither positive nor negative, is defined as the additive identity so that x + 0 = x for every number x, and produces 0 when multiplied by any…
ADHD studying

Overcoming Challenges of Self-Studying Number Theory

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Introduction During my summer break I spent several hours each day self-studying mathematics—primarily number theory—even though I had no prior experience…