Welcome to Algebraic Calculus One 2021

Hello Everyone — the Algebraic Calculus One course is now live at


and you can find a welcome video at https://www.youtube.com/watch?v=1SFI9QbRpzg

Here is the Welcome to the Course taken from it, and gives some information about it.

Introduction and Welcome

Hi everyone! Welcome to our course. I am Prof Norman Wildberger and I have been teaching at UNSW in Sydney for 30 years, from which I have recently retired, and before that I taught at the University of Toronto for 3 years and Stanford University for 2 years. I am a keen expositor of mathematics on YouTube with my channel Insights into Mathematics and the sister channel Wild Egg Maths, where the videos for this course are posted publically.

I am also the developer of Rational Trigonometry, Chromogeometry, and Universal Hyperbolic Geometry, and am currently putting together exciting series of videos on Solving Polynomial Equations and Exceptional Structures in Mathematics and Physics from Dynamics on Graphs, which you can access by becoming a member on the Wild Egg Maths YouTube channel.

Over my career I have also worked in Harmonic Analysis, Lie group representation theory, Diophantine equations, finite commutative hypergroups, Old Babylonian mathematics (you may have heard about my work with Daniel Mansfield on Plimpton 322 in the international press in 2017), combinatorics, and some mathematical physics, around quantization and star products.

I am hoping this course will open up new pathways for mathematics education and research, and that it will attract a new generation of students to the beauty and power of calculus and geometry! 

What is Calculus?

Calculus is a branch of geometry, and also a branch of physics. These two origins of the subject colour almost every part of it.

As part of geometry, calculus is concerned with the areas and tangents of curves. Given a curve in the plane, how do we define and calculate the area determined by that curve, and perhaps other curves or lines? Given a curve in the plane, how do we define and calculate the tangents to that curve at points on the curve, or perhaps near the curve? And what do we mean by a curve in the first place?

As part of physics, calculus is concerned with the motion of bodies, in particular the relationship between the position, velocity and acceleration of bodies. It is motivated by Newton’s laws, which allow us to determine the acceleration on objects from the forces that act on them. The job of calculus is then to determine from that information both velocities and positions.

Both the geometric and physical sides of calculus can be accessed through an applied, real-life point of view emphasizing approximate calculations, or through an abstract, pure point of view, focusing more on exact calculations. We are mostly interested in the theoretical development and logical structure of the subject in this course, but we will be strongly motivated by applied questions, history, and making calculations. We want our theory to support a practical, powerful calculus.

Motivating problems

Here are some fundamental problems that we would like to solve, and that the calculus helps us with.

  1. How do you explain the motion of the planets in the night sky?
  2. How do you calculate the “area” of a segment of a curve, like a circle, parabola or hyperbola? Does this question even have a precise meaning??
  3. How can you determine the trajectory of a particle if you know its starting point and subsequent velocities?
  4. What exactly is a “curve” and how can you define and determine a “tangent line” to a given one?
  5. How can you find the centre of mass of a planar lamina?
  6. What exactly is a “function”, and how can you find maxima and minima of a given one?

Early pioneers of Calculus

Calculus is usually attributed to Leibniz and Newton. These pioneers played a big role in the story. But many important aspects go back to the ancient Greeks, notably Eudoxus and Archimedes, and evidence is emerging that the Babylonians understood some aspects of it.

In modern times, the development of calculus is hard to separate from the emergence of analytic geometry in the 17th century, largely due to Fermat and Descartes. The great Bernoulli family from Switzerland made huge contributions, as did the towering figures of 18th and early 19th century mathematics: Euler, Lagrange, Laplace and Gauss, but in fact there were many others too!

A concrete computational approach

We can, and will, learn much from all of them in crafting this algebraic approach to a traditionally analytic subject. We will add modern geometric developments, notably the power of linear algebra and projective geometry. And never far from sight will be our most important modern allies: calculators and computers! Our concrete computational approach avoids the many contentious aspects of calculus that have dogged it historically.

That means we avoid any mention of completed infinite processes. We are never going to say: “and now let’s do the following infinite number of operations”. We are going to stick with algebraic operations that can be followed step by step by a computer. Limits accordingly play a very limited role in the Algebraic Calculus. We work with rational numbers. Integration comes before differentiation. Discrete situations are generally studied as motivation and prior training for continuous versions.

These are major departures from modern thinking, which considers “calculus” and “infinite processes” to be almost synonymous terms. It means we are not going to consider irrational numbers as exact numbers on the same footing as rational numbers. For us there is a world of difference between 21/7 and “\(\pi\)”. In fact we will not assume that you already know what “\(\pi\)” or “\(\sqrt2\)” or “\(e\)” already are—in fact we try to avoid them altogether. We will not work with “infinite decimals” until we find a finite, concrete, explicit way of introducing and working with “them”.

We will be very careful even about using familiar words like area, function, curve, sequence and number. Our view is that these terms need to be defined rather precisely before they can be accurately used. Until we come up with precise definitions, which are always signified by bold font, you can safely assume that we are adopting a casual, everyday, informal usage of terminology. And thinking about how to make things more precise!

Organization and thanks

The Course has 10 Chapters, which are accessible via the left navigation panel, starting with The Affine Plane. Each Chapter consists of four Modules, and each Module is always divided into four sections called Videos and Notes, Worked Problems, Homework Exercises and Links, Definitions, Notation.

You should expect to spend most of your time going through the Worked Problems and the Homework Questions carefully. The latter are graded for difficulty E (easy), M (medium), H (hard) with the occasional C (challenge or research problem). The average participant should probably expect to spend a minimum of 20 hours on each Chapter, but some of you will need to spend considerably more. That is OK — the more time you put in, the more you will get out of the course.

Comments, questions, discussion and submissions are encouraged. This is your course, so please contribute to making it exciting for everyone!

A big thanks to Dr Anna Tomskova who has worked very hard in helping me put together many of the Problems and Questions,  contributed to the nice diagrams, and worked through many of the calculations. 

This calculus course will be very different from any other one that you are likely to meet. I hope you enjoy it, and that you make it your own. And I look forward to a lot of interaction and questions and exciting developments. There’s much to learn!!

All the best,

N J Wildberger

Sydney July 2021

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