## Beginners Guide to Precalculus, Calculus and Infinitesimals

Introduction I am convinced students learn Calculus far too late.   In my view, there has never been a good reason for this. In the US, they go through this sequence of Pre-Algebra, Algebra 1, Geometry, Algebra 2, Precalculus, Calculus 1, and Calculus 2.  But is this required?   Recently I came across two books that turned…

## What Are Numbers?

Introduction When doing mathematics,  we usually take for granted what natural numbers, integers, and rationals are. They are pretty intuitive.   Going from rational numbers to reals is more complicated.   The easiest way at the start is probably infinite decimals.  Dedekind Cuts can be used to get a bit more fancy.  A Dedekind cut is a…

## What Are Infinitesimals – Simple Version

Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be thrown away because they are negligible. That way, when defining the derivative, for example, you do not run into 0/0, but when…

## What Are Infinitesimals – Advanced Version

Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be thrown away because they are negligible. That way, when defining the derivative, for example, you do not run into 0/0, but when…

## The LCEVC Coding Standard and 8K Television

Introduction First, for background, here is an overview of the most common video compression standard in use today, H264: https://www.maketecheasier.com/how-video-compression-works/ H265 has improved on this, but for various reasons, it was a bit of a flop. It is in use and will continue to grow, but royalty issues are a problem. To try snd address…

## Recursion in Programming and When to Use or Not to Use It

Recursion is actually quite simple. It’s a subroutine calling itself. It’s surprising but some problems that look quite hard can be trivial using recursion – but be wary – as I hope to explain there are traps you need to consider especially if working in a team environment. When I studied Computer Science the first…

## Some Useful Integrals In Quantum Field Theory

In my paper on renormalisation I mentioned what most who have studied calculations in Quantum Field Theory find, it’s rather complicated and mind-numbing. I am not sure doing these is that illuminating of anything, but every now and then you got to do, what you got to do, so I would like to go through…

## Why Renormalisation in Quantum Theory Needs a Cutoff

Introduction This is a follow on from my paper explaining renormalization. A question was raised – why exactly do we need a cut-off. There is a deep reason to do with dimensional analysis, and the power series expansion used in perturbation theory. Along the way, we will see renormalization in a more general setting, and…