## Omissions in Mathematics Education: Gauge Integration

The current (pure) mathematics curriculum at the university is well established. Most of the choices made are sensible. But still, there are some important topics that 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…

## 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

Introduction 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. Basic stuff…

## A Guide 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…

## The Essential Guide to Self 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…

## Dealing with Doubt as a Science Student

Doubt, as odd as this may sound, can actually 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…

## 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…

## The Best Methods to Deal with Procrastination

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….

## Top 5 Misconceptions About Infinity

Introduction 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…

## 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 $$b_0.b_1b_2b_3…$$ means. The way mathematicians define this thing is $$b_0.b_1b_2b_3…=\sum_{n=0}^{+\infty}{\frac{b_n}{10^n}}$$ So, in particular, we have that $$0.999…=\sum_{n=1}^{+\infty}{\frac{9}{10^n}}$$ 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…

## Is Zero a Natural Number?

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…