Entries by micromass

Interview with a Physicist: David Hestenes

For those who don’t know the great David Hestenes, he is the inventor of the geometric algebra formalism of physics. Here we go! 1) What is the best application of geometric algebra in theoretical physics that you can think of? In other words, what application shows the power and elegance of geometric algebra best? The […]

Some Misconceptions on Indefinite Integrals

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 […]

Groups and Geometry

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 […]

How to Self-Study Algebra. Part II: 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 algebra. This includes the study of groups, rings and fields and many other structures. Prerequisites The requirements for self-studying abstract algebra are surprisingly low. Basically, you should be acquainted with […]

Friends, Strangers, 7825 and Computers

This insight will be about Ramsey theory. 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 eachother for the first time (in which case they are strangers), or they will know eachother (in which case they are friends). Show […]

Problems with Self-Studying

For several years already I have been trying to help people who have been self-studying math. I did this completely freely with no compensation for me. I saw this as a very enriching experience. Most of the people I helped came from physicsforums. I usually contacted them and told them I might be able to […]

How to Self-study Analysis. Part II: Intermediate Analysis

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 […]

How to Self-study Algebra: Linear Algebra

In this insight I will give a roadmap to learn the basics of linear algebra. 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 on vector […]

Am I Cut Out for Mathematics or Sciences?

We often get threads by 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 question. My IQ is too low. This is a very common one. […]

Things Which Can Go Wrong with Complex Numbers

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 […]

Axioms for the Natural Numbers

** 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 […]

A Forward 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 numbers, integers, 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 done […]

How to Self-Study Basic High School Mathematics

  How to self-study basic high school mathematics We often get questions here from people self-studying mathematics. One of those questions is what of mathematics should I study and in what order. So in order to answer those questions I have decided to make a list of topics a mathematician should ideally know and what […]

How to Self-Study Calculus

  We often get questions here from people self-studying mathematics. One of those questions is what of mathematics should I study and in what order. So in order to answer those questions I have decided to make a list of topics a mathematician should ideally know and what prerequisities the topics have.. Calculus After high […]

How to Study Mathematics

  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 a formal education. In this guide, I will try to provide help for those people who chose to self-study mathematics. Is it even […]

Informal Introduction to Cardinal Numbers

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 […]

Questions About Infinity

Understanding the behaviour 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. Infinity is not a […]

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] 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 really make any sense until we define what the right-hand side means. […]

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 at a bit of history A short history of 0 Historically, there are two different uses of zero: zero as a placeholder and […]

Abstract Algebra: The Natural Numbers

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 […]

What is a Property Formally

** Hrbacek-Jech Chapter 1.2 Hrbacek and Jech do not go into full detail into 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 […]