Advanced education and experience with mathematics
The gauge integral (Henstock–Kurzweil) The current (pure) mathematics curriculum at the university is well-established. Most of the choices made are sensible, but some important topics are still usually not taught. Some of these topics are obscure and not well known even to many professional mathematicians; others are known to specialists but for some reason are…
Introduction Integration is an incredibly useful technique taught in all calculus classes. Nevertheless, there are certain paradoxes involved with integration that are not immediately resolved. I once asked in my topology class whether anybody could resolve one such paradox; no one found the correct answer at the time. First paradox: the antiderivative of 1/x The…
Klein’s Erlangen program: groups and geometry There is a very deep link between group theory and geometry. Sadly, this link is not emphasized in most group-theory courses, even though the ideas are not difficult. The link was spelled out by Felix Klein in his Erlangen program; the goal of this article is to explain the…
There are three major areas of mathematics: geometry, analysis, and algebra. This insight gives a roadmap for learning basic abstract algebra for self-study, including the study of groups, rings, fields, and other algebraic structures. Abstract Algebra Prerequisites The requirements for self-studying abstract algebra are surprisingly low. You should be familiar with most pre-calculus mathematics and…
Ramsey theory has its origins in a very nice riddle Consider a party of 6 people. Any two of these six will either be meeting each other for the first time (in which case they are strangers) or they will know each other (in which case they are friends). Show that in this party of…
For several years I have helped students who were self-studying mathematics. I did this free, without compensation, because I found it an enriching experience. Most who sought help came from physics forums; I usually contacted them to offer assistance and, if they agreed, we proceeded. My help typically consisted of providing resources, supplying extra problems,…
If you wish to follow this guide, you should be familiar with analysis on ##\mathbb{R}## and ##\mathbb{R}^n##. See my previous insight for the list of prerequisite topics and book suggestions: https://www.physicsforums.com/insights/self-study-analysis-part-intro-analysis/ You should also be comfortable with linear algebra; see my insight on that: https://www.physicsforums.com/insights/self-study-algebra-linear-algebra/ We will now take one step further. While analysis on…
This article gives a roadmap for students to learn the basics of linear algebra. Aside from calculus, linear algebra is one of the most applicable subjects in mathematics: it is widely used in engineering, the sciences, and computer science. The best way to approach linear algebra is with a focus on vector spaces and linear…
We often see threads from new members asking whether they are cut out for mathematics, physics, engineering, or other technical fields. For some reason they become discouraged in high school and conclude they cannot succeed. I offer the following answers to common concerns. 1. My IQ Is Too Low This is a very common concern….
This insight is written for high school students who don’t feel very challenged by their high school courses or who want a taste of what university mathematics is like. You don’t have to wait until university to see exciting mathematics—you can learn it now. Of course, some topics require heavier prerequisites (for example, multiple calculus…
Introduction to Pure Geometry Geometry is one of the oldest parts of mathematics. It has been studied and advanced by the greatest minds humankind has to offer. It has been described as a subject of great beauty. How do we approach such an amazing work of art as a student? In this section, I will…
This is a sequel to my posts on self-studying mathematics. I have already given a very detailed road map on how to study high school mathematics and calculus. In this post and the next ones, I will try to give a very detailed road map on how to self-study analysis to reach a high level….
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 B We know that [itex]\sqrt{-1}=i[/itex]. But at the same time, we have [tex]i=\sqrt{-1}=(-1)^\frac{1}{2}=(-1)^\frac{2}{4}=[(-1)^2]^\frac{1}{4}=1[/tex] Example C Eulers identity tells us that [itex]e^{2\pi i}=1[/itex]. So [itex]\log(1)=2\pi i[/itex], but at…
Bloch Chapter 1.2 The Peano system in Bloch has a special element ##1\in \mathbb{N}##. The intuitive idea here is that ##\mathbb{N} = \{1,2,3,…\}##. However, we can also present much of the same material if we instead choose ##\mathbb{N} = \{0,1,2,3,…\}##. The axioms for this remain the same: A Peano system is a set ##\mathbb{N}## with…
Set-Theoretic Foundations of Mathematics It is important to realize that in standard mathematics we attempt to characterize everything in terms of sets. This means notions such as natural numbers, integers, and real and rational numbers are defined in mathematics to be certain sets. Also, the very notion of a function is defined as a set….
Introduction We often get questions here from people self-studying mathematics. One common question is: what mathematics should I study, and in what order? To answer that question I have made a list of topics a mathematician should ideally know and what prerequisites those topics require. Basic stuff Of course, we have the basic (high school)…
We often get questions here from people self-studying mathematics. One common question is: “What mathematics should I study and in what order?” To answer that, I have made a list of topics a mathematician should ideally know and their prerequisites. Calculus After high-school mathematics comes calculus. This includes the following major areas. Differentiation — Differentiation…
How to self-study mathematics? People self-study mathematics for a lot of reasons. Either out of pure interest, because they want to get ahead, or simply because they don’t want to take formal education. In this guide, I will try to provide help for those people who choose to self-study mathematics. Is it even possible…
Doubt, as odd as this may sound, can be essential to living. We all make decisions and later question whether we made the right choice; doubt helps influence our next decision when a similar choice arises. While doubt is a natural part of the human conflict in all of us, discouragement is different. Intentional discouragement…
Cardinal numbers We will now give an informal introduction to cardinal numbers. We will later formalize this by using ordinal numbers. Informally, cardinal numbers are “numbers” that measure the cardinality of a set. So for every set, we can introduce a cardinal number of this set. Let’s start with finite sets, cardinal numbers here are…
Introduction It’s 6:30 in the morning. You’ve just woken up and feel so sleepy you think, “A few more minutes can’t hurt,” and drift back under your comforter. The sounds of kids playing outside, sunlight and birds chirping at your window wake you three hours later. You realize you only have 15 minutes to study…
Introduction Understanding the behavior of infinity is one of the major accomplishments of mathematics. However, the infinite is often misunderstood and can lead to apparent paradoxes when misused or misinterpreted. This FAQ explains the role of infinity in mathematics and attempts to resolve several apparent paradoxes. 1. Infinity is not a real number! Very often,…
Yes. What 0.999… Means First, we have not addressed what 0.999… means. So it is best to first describe what the notation [tex]b_0.b_1b_2b_3…[/tex] means. The way mathematicians define this thing is [tex]b_0.b_1b_2b_3…=\sum_{n=0}^{+\infty}{\frac{b_n}{10^n}}[/tex] So, in particular, we have that [tex]0.999…=\sum_{n=1}^{+\infty}{\frac{9}{10^n}}[/tex] But all of this doesn’t make any sense until we define what the right-hand side means….
The goal of this FAQ is to clarify the concept of 0, and specifically the operations that are allowed with it. The best way to start this FAQ is to look at a bit of history. A short history of 0 Historically, there are two different uses of zero: zero as a placeholder and zero…
Using: Anderson-Feil Chapter 1.1 Is zero a natural number? This is a pretty controversial question. Many mathematicians – especially those working in foundational areas – say yes. Another good deal of mathematicians say no. It’s not really an important question, since it is essentially just a definition and it matters very little either way. I…
Hrbacek–Jech Chapter 1.2: Formal Language and WFFs Introduction Hrbacek and Jech do not go into full detail about what a property is formally. This is a part of mathematical logic, but it is important here to give a precise definition of what a property is. Alphabet of Set Theory First, we describe the alphabet of…