Tag Archive for: mathematics
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Views On Complex Numbers
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70 Comments
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…
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The Lambert W Function in Finance
Preamble
The classical mathematician practically by instinct views the continuous process as the "real" process, and the discrete process as an approximation…
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Why Division by Zero is a Bad Idea
A division by zero is primarily an algebraic question. The reasoning therefore follows the indirect pattern of most algebraic proofs:
What if it was allowed?
Then…
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Epsilontic – Limits and Continuity
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…
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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…
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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…
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What Are Numbers?
Introduction
When doing mathematics, we usually take for granted what natural numbers, integers, and rationals are. They are pretty intuitive. Going…
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Classification of Mathematics by 42 Branches
I often read questions about our classification scheme that we use on physicsforums.com to sort posts by science fields and subjects, what has…
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Évariste Galois and His Theory
* 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…
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The History and Importance of the Riemann Hypothesis
Riemann Hypothesis History
The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics, specifically in the field…
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The Extended Riemann Hypothesis and Ramanujan’s Sum
Riemann Hypothesis and Ramanujan's Sum ExplanationRH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line.
ERH: All…
![Hyperbola](https://www.physicsforums.com/insights/wp-content/uploads/2022/03/Hyperbola-1.png)
The Amazing Relationship Between Integration And Euler’s Number
We use integration to measure lengths, areas, or volumes. This is a geometrical interpretation, but we want to examine an analytical interpretation that…
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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](https://www.physicsforums.com/insights/wp-content/uploads/2020/03/Integral-Representations-of-Some-Special-Functions.png)
A Path to Fractional Integral Representations of Some Special Functions
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
$$C_N^n:…
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SOHCAHTOA: Seemingly Simple, Conceivably Complex
What is SOHCAHTOA
SOHCAHTOA is a mnemonic acronym used in trigonometry to remember the relationships between the sides and angles of right triangles.…
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How to Find Potential Functions? A 10 Minute Introduction
Definition/Summary
Given a vector field ##\vec F(x,y,z)## that has a potential function, how do you find it?
Equations
$$\nabla \phi(x,y,z) = \vec F(x,y,z)$$…
![What is a linear equation](https://www.physicsforums.com/insights/wp-content/uploads/2019/12/what_is_a_linear_equation.png)
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…
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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,…
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How to Write a Math Proof and Their Structure
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…
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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…
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What is a Real Number? A 5 Minute Introduction
Definition of real numbers
Real numbers are a comprehensive set of numbers that encompasses all possible values on the number line. They include both…
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What is a Parabola? A 5 Minute Introduction
What is a Parabola?
A parabola is a U-shaped curve in mathematics that is defined by a specific set of points. It is a fundamental geometric shape that…
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What Is a Limit of a Function? A 5 Minute Introduction
What is a limit?
In mathematics, a limit is a fundamental concept used to describe the behavior of a function or sequence as it approaches a particular…
![What is a Tangent Line](https://www.physicsforums.com/insights/wp-content/uploads/2019/12/what_is_a_tangent_line.png)
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…
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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](https://www.physicsforums.com/insights/wp-content/uploads/2019/01/lie_algebra_structure.png)
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…
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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,…
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Learn a Simplified Synthesis of Financial Options Pricing
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…
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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…
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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…
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How to Self Study Intermediate Analysis Math
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…
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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…
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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…
![lineartransformations](https://www.physicsforums.com/insights/wp-content/uploads/2015/12/lineartransformations.png)
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…
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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…
![999](https://www.physicsforums.com/insights/wp-content/uploads/2015/08/999.png)
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]…
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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…
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Overcoming Learning Challenges 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…