Mathematics

mathanalysis1

Self-Study Analysis: A Proof-to-Manifolds Roadmap

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26 Comments
Direct answer: To self-study real analysis from a calculus background, follow this sequence: complete a proof-writing book (Velleman's How to Prove It…
complexmath

Hear the Case for Learning Complex Math

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15 Comments
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. …
complexexponents

Exponents Explained: Irrational and Imaginary Powers

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5 Comments
What Is the General Definition of Exponentiation? Exponentiation is defined for all real and imaginary exponents using the formula xy = ey ln(x), where…
Eigenvectors

What Are Eigenvectors and Eigenvalues in Math?

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37 Comments
Two important concepts in Linear Algebra are eigenvectors and eigenvalues for a linear transformation that is represented by a square matrix. Besides…
complexnumbers

Complex Number Paradoxes Explained: Why Exponent Rules Fail

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3 Comments
Complex exponentiation breaks the familiar high-school rules for powers because those rules — (x^a)^b = x^ab, (xy)^a = x^a y^a, and x^a = x^b ⇒ a =…
MillenniumPrize

Millennium Problems: Poincaré, P vs NP, Riemann

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13 Comments
IntroductionWhat this Insight Covers In this Insight, I will go over the background information for the Millennium Prize problems and briefly describe…
micro2

Peano Axioms Explained: Natural Numbers Built from Set Theory

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0 Comments
The Peano axioms are a set of three rules that define the natural numbers using only a starting element and a "successor" function. This article presents…
partialfractions

Introduction to Partial Fractions Decomposition

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3 Comments
Partial fractions decomposition is an algebraic technique that can be used to decompose (break down) a product of rational expressions into a sum…
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…
ODE2

How to Solve Nonhomogeneous Linear ODEs using Annihilators

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3 Comments
My previous Insights article, Solving Homogeneous Linear ODEs using Annihilators, discussed several examples of homogeneous differential equations, equations…
ODE1

Solving Homogeneous Linear ODEs using Annihilators

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8 Comments
In this Insights article we'll look at a limited class of ordinary differential equations -- homogeneous linear ODES with constant coefficients. Although…
lineartransformations

Learn About Matrix Representations of Linear Transformations

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19 Comments
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…
animationwalk

Fun with Self-Avoiding Walks Simulations

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16 Comments
This post is about some simulations I did of self-avoiding random walks. These are what they sound like with each step, the position of the walk moves…
999equals1

Why 1 = 0.999…: Four Simple Proofs Explained

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111 Comments
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...…
prequantumgeometry5

Higher Prequantum Geometry V: The Local Observables – Lie Theoretically

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1 Comment
 This article discusses how the previous considerations naturally follow the concepts of local observables of local field theories and of…
prequantumgeometry4

Higher Prequantum Geometry IV: The Covariant Phase Space – Transgressively

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13 Comments
  The Euler-Lagrange ##p##-gerbes discussed in the previous article are singled out as being exactly the right coherent refinement…
prequantumgeometry3

Higher Prequantum Geometry III: The Global Action Functional – Cohomologically

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9 Comments
  The previous article ended with the concept of classical locally variational field theories, of which a class of examples are…
prequantumgeometry2

Higher Prequantum Geometry II: The Principle of Extremal Action – Comonadically

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1 Comment
  The previous article motivated the importance of considering "pre-quantum field theory" in-between classical and quantum field…
prequantumgeometry

Higher Prequantum Geometry I: The Need for Prequantum Geometry

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18 Comments
 Before proceeding with a discussion of the super p-brane sigma models, whose emergence from the superpoint I discussed in the previous article,…
proofs

Proofs in Mathematics: Methods, Examples, and How to Write Them

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28 Comments
A mathematical proof is a logical argument, built only from definitions, axioms, and previously established results, that shows a statement is true in…
cardinalnumbers

Cardinal and Ordinal Numbers: An Informal Introduction

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1 Comment
Cardinal numbers measure the size of a set, including infinite sets, while ordinal numbers describe the structure of a well-ordered set. Every set can…
infinity_faq

Infinity in Mathematics: What It Really Means, Explained

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17 Comments
Infinity is not a real number but a set of distinct mathematical tools, including the extended real line, the projective real line, the Riemann sphere,…
999

Is 1 Equal to 0.999…? Rigorous Proof Explained

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101 Comments
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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7 Comments
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…
errorbars

Plus/minus What? How to Interpret Error Bars

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20 Comments
People sometimes find themselves staring at a number with a ± in it when a new physics result is presented. But what does it mean? This Insight aims to…
Distributions

Errors in Probability: Continuous and Discrete Distributions

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1 Comment
1. Classifying as discrete, continuous, or mixed These statements (or equivalents) can be found in authoritative-seeming websites:X "A random variable…
natural language errors

Frequently Made Errors in Probability: Conditionals in Natural Language

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2 Comments
  1. Turning a verbal condition into Algebra An actual thread..."A study of auto accidents has found that 40% of all fatal accidents are…
variables

Some Conceptual Difficulties in the Roles of Variables and Constants

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3 Comments
1. Variables and Constants When is a constant not a constant? When it varies.In the standard equation ##y = a x + b##, we are used to thinking of…
differentiable function 2

Nowhere Differentiable Functions: Fejér Kernel & Cesàro Sums

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2 Comments
This is Part 2 of a series of articles in which the goal is to exhibit a continuous function that is nowhere differentiable and to explore some interesting…
differentiable function

Constructing a Continuous but Nowhere Differentiable Function

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11 Comments
When studying calculus, we learn that every differentiable function is continuous, but a continuous function need not be differentiable at every point.…
abstract_algebra

Peano Axioms Explained: Natural Numbers Made Rigorous

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0 Comments
The natural numbers can be defined rigorously through the Peano axioms, a system built from a set, a successor function, and a starting element (0 or 1…
micro1

Well-Formed Formulas in Set Theory: WFFs Explained Clearly

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0 Comments
What Is a Well-Formed Formula in Set Theory? A well-formed formula (wff) in set theory is a string of symbols built strictly from the alphabet of set…