Tag Archives: rational trigonometry

Channel Trailer (2021) for Insights into Mathematics (YouTube)

Hi everyone, I’m Norman Wildberger, a soon-to-be retired professor of mathematics at UNSW in Sydney Australia, and I want to tell you about this channel which will introduce you to a wide variety of mathematical topics with a novel slant. The content is aimed at a very broad audience from everyday people with an interest in maths to graduate students working on a PhD in the subject. A link to this introductory video is given below, so you will be able to find quickly any of the playlists that I describe.

I believe that mathematics should be completely clear and straightforward, and that ideally a beginner should be able to navigate through one of the many branches of the subject, one step at a time, supported by lots of explicit examples and concrete computations, with the logical structure visible at all times.

That means however that I no longer buy the standard religion of “real numbers”, which are anchored in an arithmetic reliant on infinite processes. It’s not possible to add up an infinite number of things, so why do we pretend that we can?

I also don’t believe in the “hierarchies of infinite sets” that supposedly form the foundation for modern mathematics, following Cantor. It’s not possible to exhibit a “set” with an infinite number of elements, so why do we pretend that we can?

The pure mathematical community depends on these and other fancies to support a range of “theories” that appear pleasant but are not actually corresponding to reality, and “theorems” which are not logically correct. Measure theory is a good example –this is a subject in which the majority of “results” are without computational substantiation. And the Fundamental theorem of Algebra is a good example of a result which is in direct contradiction to direct experience: how do you factor x^7+x-2 into linear and quadratic factors? Answer: you can’t do this exactly — only approximately.

By removing ourselves from the seductive but false dreamings of modern pure mathematics, we open our eyes to a more computational, logical and attractive mathematics –where everything is above board, where computations actually finish in finite time, where examples can be laid out completely, and where we acknowledge the proper distinction between the exact and the only approximate. This is a pure mathematics which is closer to applied mathematics, and more likely to be able to support it. It also gives us many new insights, more precise definitions, and theorems which are actually …correct.

In this channel, we explore the beginnings of such an exciting new way of learning and doing and teaching mathematics. I present you with topics that are developed and explored in a sequence of YouTube videos, usually from rather elementary beginnings. These topics are organized in Playlists, so you can work your way through them sequentially and strengthen your understanding slowly and steadily.

The History of Maths series is great for high school teachers and anyone with a general interest in mathematics — so much of the subject makes more sense when viewed in a historical context. There is also a playlist on Ancient Mathematics and another on Old Babylonian mathematics. The latter topic is close to my heart — a paper in Historia Mathematica a few years ago with Daniel Mansfield on Plimpton 322 generated international coverage in hundreds of newspapers, including the New York Times.

Wild Trig is an introduction to Rational Trigonometry — a more general and algebraic view of trig that allows much more extensive and quicker calculation for many problems and that opens the door to many new theoretical possibilities, such as chromogeometry! This is based on my book: Divine Proportions: Rational Trigonometry to Universal Geometry.

Famous Math Problems discusses a wide range of —famous math problems, some of them with novel solutions!

Wild Lin Alg A and the follow up Wild Lin Alg B is a first year undergraduate course in Linear Algebra, from largely a geometric point of view.

The most extensive series is the MathFoundations series, which comes in parts MathFoundationsA (videos 1-79), MathFoundationsB (videos 80-149) and MathFoundationsC (videos 150-present). This series examines so many important topics in the subject. The most recent videos for example give a new treatment of the Algebra of Boole, transcending the more usual Boolean Algebra (which is not really what Boole intended) and open the door for simpler logic gate analysis by engineers.

The most elementary series is: Elementary Math (K-6) Explained which is for parents and teachers of primary school students, and will give you tools to understand the important mathematical skills and concepts their children need to learn. In this direction, there is also a course on Math Terminology for Incoming Uni Students meant for people from a non- English speaking background.

Universal Hyperbolic Geometry is a more advanced series on geometry which will give you an exciting new completely algebraic way to understand the hyperbolic geometry of Gauss, Lobachevsky and Bolyai, and which connects more naturally with relativistic physics. There are hundreds of new theorems here, many very beautiful. I will be developing this a lot more in the coming years.

There are also more advanced courses on Differential Geometry and Algebraic Topology. And there’s also a quick and more elementary course which introduces Probability and Statistics.

So this is a large amount of content that is consistently oriented towards avoiding infinite processes and arguments which are not supportable by explicit computation. It is a new kind of mathematics. If you work through some of this, your mathematical understanding will deepen, you will see connections that were invisible, and your appreciation for the logical beauty of the subject will continue to grow. Mathematics is surely the richest intellectual discipline, and I want to empower more people, young and old to experience it directly, to learn lots of fascinating things, to be challenged, and to explore on your own. For those of you aspiring to do some research on your own, there will be plenty of new directions to think about!

My understanding is very different from my fellow mathematicians. So why do I have such a unique perspective? One reason is that I have simply worked in lots of areas of mathematics.

I have done work in number theory, developing the most powerful general algorithm for solving large Diophantine equations, and unravelling the algebraic structure of Gaussian periods. I’ve done work on Pell’s equation –basically discovered the simplest explanation of why solutions are always possible.

I have worked in Lie group harmonic analysis, solving the Horn conjecture (with A. H. Dooley and J. Repka) on eigenvalues of sums of Hermitian matrices. I’ve initiated the moment map of a Lie group representation and found a geometric Fourier transform which explains *-products on coadjoint orbits of compact Lie groups. The wrapping map introduced with A. H. Dooley gives a broad explanation for the effectiveness of A. A. Kirillov’s orbit theory.

In work with D. Arnal I’ve introduced quasi-standard Young tableux, building from my geometric “diamond” construction of the irreps of SU(3), which is of considerable interest to physicists. I have also given combinatorial constructions of G2 and the simply laced Lie algebras, excluding E8.

In 2005 I wrote a book which introduces Rational Trigonometry, and then extended that to a complete rewrite of hyperbolic geometry. This gives a large scale revision of Euclidean and non-Euclidean metrical geometries. With this I have further discovered a remarkable three-fold symmetry in planar geometry called chromogeometry.

I have developed the theory of finite signed hypergroups, which are probabilistic versions of finite groups, and developed a duality theory for them, somewhat like Poyntriagin duality for abelian groups, and also applied ideas of entropy to them.

For the last five years I have been developing the Algebraic Calculus, which is a coherent approach to Calculus which avoids real numbers and infinite processes, and is correspondingly more general and often gives new insights. Videos for this can be found at the sister channel Wild Egg mathematics courses, while the course itself is on openlearning.

If you are interested in learning more about my research at the more advanced level, there is a Playlist on this channel of Math Seminars, and also a smaller one on Research Snapshots, which I hope to enlarge in the future.

I have a Vice Chancellor’s award at UNSW for teaching excellence and have been very involved in the development of online tutorials for mathematics courses there.

In summary, my aim is to put this wealth of research and teaching experience to work in framing a more fruitful path for mathematics education, and opening up a more solid approach to pure mathematics research, connected more strongly to computational reality. Come along and join me on an exciting journey to explore new and better foundations and directions for 21st century pure mathematics! Once we face the music and see things as they really are, not just how we want them to be, there is much to do.