Fundamentals Of Hypercomplex Numbers | UCLA Extension

Dr. Michael Miller, a retired researcher at RAND, has been teaching upper level undergraduate/graduate level math courses for fun at UCLA Extension for over 50 years. This winter, he’ll be introducing hypercomplex numbers to those interested in abstract math: Fundamentals Of Hypercomplex Numbers.

His courses are thorough and rigorous, but geared toward lifelong learners and beginners in abstract mathematics to allow people better entry points into higher level mathematics. His classes are interesting and relatively informal, and most students who take one usually stay on for future courses. The vast majority of students in the class (from 16-90+ years old) take his classes for fun and regular exposure to mathematical thought, though there is an option to take it for a grade if you like. There are generally no prerequisites for his classes, and he makes an effort to meet the students at their current level of sophistication. Some background in calculus and linear algebra will be useful going into this particular topic.

If you’re in the Los Angeles area (there are regular commuters joining from as far out as Irvine, Ventura County and even Riverside) and interested in joining a group of dedicated hobbyist and professional mathematicians, engineers, physicists, and others from all walks of life (I’ve seen actors, directors, doctors, artists, poets, retirees, and even house-husbands in his classes), his class starts on January 6th at UCLA on Tuesday nights from 7-10PM.

If you’re unsure of what you’re getting into, I recommend visiting on the first class to consider joining us for the Winter quarter. Sadly, this is an in-person course. There isn’t an option to take this remotely or via streaming, and he doesn’t typically record his lectures. I hope to see all the Southern California math fans next month!

Course Description

A survey of those systems of numbers that can be constructed by adding “imaginary units” to the real numbers. The simplest and most familiar example is the two-dimensional system of complex numbers. Much less familiar, but equally fascinating, are the systems of quaternions and Cayley numbers, of dimensions four and eight, respectively. These “algebras” still enable meaningful notions of addition, multiplication, and division, but only at a price: the loss of commutativity and (in the case of Cayley numbers) associativity. Things get even more bizarre when sedenions (dimension 16) and trigintaduonions (dimension 32) are brought into play. The latter part of the course is devoted to the theorems of Hurwitz and Frobenius on the existence of suitably behaved division algebras over the real numbers.

The course should appeal to those seeking a better understanding of the arithmetical underpinnings of our number system.

Prerequisites: advanced calculus and linear algebra

January 6 – March 17
Tuesday 7:00PM – 10:00PM PT
REG# 407060
Fee: $450.00

Recommended textbook: TBD

Register here:
If you’ve never joined the class before (Dr. Miller has been teaching these for 53 years and some of us have been with him for nearly that long; I’m starting into my 20th year personally), I’ve written up some tips and hints.