Advanced education and experience with mathematics
The current (pure) mathematics curriculum at the university is well-established. Most of the choices made are sensible. But still, some important topics are usually not taught. Some of these topics are very obscure and not even very well known to many professional mathematicians, others are known to experts but for some reason are not touched…
Integration is an incredibly useful technique taught in all calculus classes. Nevertheless, there are certain paradoxes involved with integration that are not easily solved. At least, I asked in my topology class whether anybody could resolve the paradox, and nobody found the correct answer. The first paradox arises in the following integral: [tex]\int \frac{1}{x} dx…
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 not so difficult. The link between groups and geometry was detailed by Klein in his Erlangen program. It is the goal of this insight to give…
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 algebra for self-study. This includes the study of groups, ring fields, and many other structures. Abstract Algebra Prerequisites The requirements for self-studying abstract algebra are surprisingly low. You should be…
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 (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 already I have been trying to help students who have been self-studying math. I did this completely free with no compensation for me. I saw this as a very enriching experience. Most of the people I helped came from physics forums. I usually contacted them and told them I might be able…
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 know what kind of topics you need to know and for suggestions of books: https://www.physicsforums.com/insights/self-study-analysis-part-intro-analysis/ Also in many parts, you should be comfortable with linear algebra, see my…
In this insight, I will give a roadmap to learn the basics of linear algebra for students. Aside from calculus, linear algebra is one of the most applicable subjects of all of mathematics. It is used a lot in engineering, sciences, computer sciences, etc. The right way to see linear algebra is with a focus…
We often get threads from new members asking whether they are cut out for mathematics/physics/engineering/whatever. For some reason, they have become discouraged in high school and don’t think they can make it. I am using this insight to provide an answer to those questions. 1. My IQ is too low This is a very common…
This insight is written for high school students who don’t feel very challenged by their high school courses. Or maybe those high school students who want to have a taste of what university math is like. The good news is that you don’t have to wait until university to see exciting mathematics, you can learn…
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 attempt to give you…
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…
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 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. Why is this…
Introduction We often get questions here from people self-studying mathematics. One of those questions is what mathematics should I study and in what order. So to answer those questions, I have decided to make a list of topics a mathematician should ideally know and what prerequisites the topics have. Basic stuff Of course, we…
We often get questions here from people self-studying mathematics. One of those questions is what mathematics should I study and in what order. So to answer those questions, I have decided to make a list of topics a mathematician should ideally know and what prerequisites the topics have. Calculus After high school stuff comes…
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 our living. We all make decisions and later have questions on whether we made the right choice or not so doubt will help influence our next decision when it comes to the same question or choice. While doubt is a natural part of the…
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…
It’s 6:30 in the morning. You’ve just woken up and you feel so sleepy you think to yourself “A few more minutes can’t hurt.” And so you drift on to sleep under your warm comforter. The sounds of kids playing outside, mixed with the sunlight and birds chirping at your window wakes you up three hours later….
Introduction Understanding the behavior of infinity is one of the major accomplishments of mathematics. Sadly, the infinite is often misunderstood and could lead to various paradoxes when used or interpreted the wrong way. This FAQ attempts to explain the role of infinity in mathematics and tries to resolve a few apparent paradoxes. 1. Infinity is…
Yes. First, we have not addressed what 0.999… means. So it’s best first to describe what on earth 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. There is…
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 at a bit of history A short history of 0 Historically, there are two different uses of zero: zero as a placeholder and…
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 Hrbacek and Jech do not go into full detail about what a property is formally. This is a part of mathematical logic, but it seems to be important here to give a full definition of what a property is. First, we describe the alphabet of set theory. An alphabet is merely the…