What is the systematic prerequisite sequences of learning that must be mastered before approaching the subject of learning Quaternions, and then Clifford Algebra? My ultimate goal is to learn and study Maxwell’s equations in their original Quaternion form, plus other obscure scientific papers that utilize Clifford Algebra and Octonians. But before I can approach that, I need the solid mathematical background that comes before studying and understanding Quaternions and Clifford (Geometric) Algebra, beginning with a recommended syllabus that starts with pre-algebra mathematics.

I know that there is a systematic approach to learning mathematics in general. For example: Multiplication and division at the foundation, then fractions, then pre-algebra, then algebra, then geometry, then trigonometry, and so on. With each division of mathematics itself having a systematic sequence of learning the subject from its fundamentals to their ultimate advanced topics. So to state my question differently, what I want to know is this:

  1. What is the best sequence of mathematical prerequisites (the syllabus) to master before I begin studying Quaternions, and then advance to Clifford (Geometric) Algebra?

  2. Where does Quaternion and Clifford (Geometric) algebra fit in the sequence of learning algebra, and what will be their prerequisites necessary to have been already mastered before entering these subjects?

Also, can someone recommend a track of material that would properly prepare me for the topic of Quaternions and Clifford (Geometric) Algebra from a pre-algebra starting point? i.e. Completed knowledge of addition/subtraction, multiplication/division, fractions and percentages.

  • 2
    $\begingroup$ you should be comfortable with complex numbers $\endgroup$ – J. W. Tanner Feb 17 at 1:24
  • $\begingroup$ And you should be aware that learning a subject from beginning to end is an impossibility in mathematics. If I were you, I would just get the beautiful book "On Quaternions and Octonions" by Conway and Smith and then work backwards learning what you need to know to understand their exposition. $\endgroup$ – Rob Arthan Feb 17 at 1:27
  • $\begingroup$ Is it me or everyone in this question assumes that all countries have an identical and standardized notion of what is "pre-algebra", "algebra 1", "algebra 2", etc. ? $\endgroup$ – Captain Lama Feb 26 at 9:18
  • $\begingroup$ @CaptainLama It may be useful to check out MathTutorDVD's recommended course sequence at mathtutordvd.com/public/department199.cfm to follow what these terms mean. Especially if one dives into each course to see what is in the syllubus. Gook luck! $\endgroup$ – Mr. Lopez Mar 7 at 4:46
  • $\begingroup$ @Mr.Lopez Thank you for your concern, but my point was "this is an international website, please don't use American terminology as if it were universal". Or at least explicitly mention it. There might be people who are interested in this question but come from countries where courses are not organized as such. Actually that would be almost every country. $\endgroup$ – Captain Lama Mar 7 at 7:57

You learn arithmetic, basic geometry, then what is called "prealgebra", then algebra 1 and algebra 2. Once you have gotten through the chapter on complex numbers, you want to learn at least a little bit about matrices. You might get a taste in your algebra 2 class or precalculus, but the subject is called "linear algebra" and the more of it you know the better--it doesn't require calculus.

Once you have complex numbers and matrices, you are good to go. I actually would not recommend Conway; I own it and found it frustrating. I have hunted for good materials to learn quaternions and there mostly aren't any books that explain them well. But YouTube videos fill the gap; since quaternions get used in computer programming to handle rotations in three dimensions, there is some demand. Good luck!

  • $\begingroup$ Do you have any YouTube links to recommend? Ones that are more oriented to engineering physics applications rather than computer science and gaming? $\endgroup$ – Mr. Lopez Feb 17 at 3:00
  • $\begingroup$ youtube.com/watch?v=zjMuIxRvygQ This links to other videos as well. $\endgroup$ – RobertTheTutor Feb 17 at 3:57

If Science is your destination, then there is no other noble subject beyond philosophy than mathematics itself to give you the foundation you need to study and learn the sciences. It is a commendable cause.

I give my answer to your question for the benefit of all on being appropriately prepared to understand Quaternions and Clifford (Geometric) Algebra (which utilizes Octonions). Because of the increase in traffic to mathematical forums for queries about quaternions due to a heightened advantageous awareness of them in computer science, and an increasing popularity for applications in physics (particularly their broader inclusion within Geometric Algebra), it now necessitates a consolidated forum page to answer these many queries. Thus, for that reason, this question has been carefully considered. For my answer, I draw upon the contributions of RobertTheTutor, CyclotomicField, J. W. Tanner, Rob Arthan, along with the results of my own extensive research to learn the answer to this very same question.


I. The Proper Prerequisite Learning Sequence.

II. Recommended Sequential Learning Track Material.

III. Why Use Quaternion Based Mathematics? Benefits and Features.

IV. Historical Fun Facts About Quaternions and the Truth About Maxwell Theory.

V. James Clerk Maxwell Original Work and References.

VI. Deriving the Maxwell Source Equations In Quaternion Space.

I. The Proper Prerequisite Learning Sequence.

To be appropriately prepared for the subjects of your inquiry, the proper prerequisite you will need is as follows:

  1. Arithmetic: addition, subtraction, multiplication, division, fractions, decimals, percentages.
  2. Pre-Algebra 1
  3. Pre-Algebra 2
  4. Algebra 1
  5. Algebra 2
  6. Geometry
  7. Advanced Algebra
    • Including Matrices
    • Including Complex Numbers
  8. Trigonometry
  9. Pre-Calculus
  10. Calculus 1
  11. Linear Algebra; Basic + Advanced (very Important To Master!!!)
  12. Quaternion Algebra
    • Which includes Versor Algebra
  13. Abstract Algebra
    • Including Abstract Algebra: Group Theory
    • Including Abstract Algebra: Ring Theory
  14. Geometric (Clifford) Algebra

This sequence will not only prepare you for entering into quaternions (and octonians of Clifford Algebra), but also give you the fundamental foundation you need to advance eventually into forms of Calculus that are based upon Geometric (Clifford) Algebras.

It may be that you have already obtained a particular level of mathematics already. If that is the case, then simply choose the point where you are in your mathematical level, and proceed forward from that point.

As a side note, if you come upon anyone speaking in a video lecture online about “Versor Algebra”, know that a versor is just a unit quaternion. In other words; in mathematics, a versor is a quaternion of norm one (a unit quaternion). Thus Versor Algebra is not a separate algebra, but relates precisely to a unit in quaternion mathematics. It was something that tripped me up when I first heard the term, but now it doesn’t. I just know that I have to learn quaternions to understand their online lecture.

II. Recommended Sequential Learning Track Material.

I recommend this track of material to properly prepare you:

1.1 MathtutorDVD (MathandScience) courses. These fantastic video courses will take you by the hand, and patiently take you from arithmetic all the way through to mastery of advanced calculus if that is your desire. However, their courses from arithmetic through to engineering mathematics (which is their courses on leaner algebra) will be sufficient for your needs. This will cover all mathematics I have placed above in my proper prerequisite list from number 1 to number 11. However, instead of buying the courses separately, I recommend their package deal because by the time you buy all of these math courses separately, you may as well have bought their package deal of all courses.

1.2 If free is your thing, then you may prefer to check out the Real Not Complex website. Then after you master arithmetic through to pre-algebra, you can check out this full free Algebra Course posted to YouTube. Total length is 5:22:06. Personally, I think MathtutorDVD does a better job than these free resources. However, you can decide for yourself.

2.1 If you have mastered mathematics all the way through linear algebra, then you should be ready for the study of quaternions. Before you begin your study into quaterions, it is very important that you understand how complex numbers produce rotations in the complex plane first. If you have that down, then you can proceed to these next recommendations.

2.2 How To Visualize Quaternions, and Gain an Intuitive Grasp of Them? The best I have encountered (and learned of by RobertTheTutor, is Grant Sanderson and Ben Eater's Explorable Video Series on Visualsing Quaternions. This YouTube video is a good introduction and overview of their method of teaching: Visualizing Quaternions(4D Numbers) With Stereographic Projection. To get the full treatment, go to their website dedicated to interactive instruction on Quaternions.

2.3 John S. On YouTube also has a 3 part instruction on understanding Quaternions.

2.4 There is also a book I can recommend: Visualizing Quaternions by Andrew Hanson and Steve Cunningham.

2.5 When you are ready to plunge deeper into learning quaternions, Introduction to Quaternions and An Elementary Treatise On Quaternions by Philip Kelland and Peter Guthrie Tait will give you a deeper breadth of learning them from the standpoint of this mathematics's origins. These fellows were contemporaries with the inventor of quaternion mathematics.

2.6 To gain the full treatment on quaternions, what better place to turn than the author and originator of quaternion mathematics himself: Sir William Rowan Hamilton. Whom made his discovery in 1843, October 16. His three chief works on the subject are: Lectures of Quaternions, On Quaternions, and Elements of Quaternions. You can go to the embedded links to view or download them for free.

2.7 At this point, you may have follow up questions that have already been answered on this forum:

2.8 I save these last two books until later, for the simple fact that they have been frequently recommended, yet at the same time, comments about them indicate that they are better suited for someone whom has achieved junior or senior level of college mathematics. If you are a beginner in quaterion mathematics, then save these book until later, otherwise these two books are for you: Rotations, Quaternions, and Double Groups by Simon Altman, and On Quaternions and Octonians by John Horton Conway and Derek Smith.

2.9 Quaternions: Other Resources You May Want to Check Out:

3.1 If you have mastered Trigonometry, Pre-Calculus, Calculus 1, Leanear Algebra, and Quaternion Algebra, then you should now be ready for the study of Abstract Algebra on your destined goal for mastery of Geometric (Clifford) Algebra.

3.2 I recommend starting off with Abstract Algebra for Beginners by Dr. Steve Warner PhD (Mathematics). It should give you a great overview study of the course to familiarize yourself of the deeper study to come. Meanwhile at the same time, it will get you up and running with the concepts presented. You may also like instead, A Book of Abstract Algebra, Second Edition, by Charles Pinter. Both books come with 4.5 star book review ratings. However, if a single book is what you are after, and jumping right into a path for full in depth mastery of Abstract Algebra from the standpoint of a beginner, then you may like instead of either of those former books I mentioned, Abstract Algebra, Third Edition, by David Dummit and Richard Foote.

4.1 If you have mastered everything through to Abstract Algebra in the proper learning sequence list above [Roman numeral (I) in the Table of Contents], you should now be ready for the study of Geometric (Clifford) Algebra.

4.2 Geometirc (Clifford) Algebra Video Series by Prof. Alan Macdonald. Note: He also publishes two books (and has a support website for those taking his course). His course starts with linear algebra and works up from there, including all the way up to advanced Calculus based upon Geometric (Clifford) Algebra.

4.2.1 A Swift Introduction to Geometric Algebra [44:22 minute YouTube video]. I give this entry as a good visual overview of the subject that may be of interest to you meanwhile you are checking out Professor Alan Macdonald’s videos and course.

4.2.2 Geometric Algebra YouTube Playlist by Mathoma. I give this entry merely as a supplemental to Professor Alan Macdonald’s video and course, in the event that you find a different angle on the subject useful.

4.3 Geometric (Clifford) Algebra: Other Resources You May Want to Check Out:

  • Wolfram Mathword Website’s overview of Geometric (Clifford) Algebra.

  • Euclidean Space Website - Introduction and Exploration of Geometric (Clifford) Algebra.

  • Wikipedia’s Article on Geometric (Clifford) Algebra.

  • Wikipedia’s Article on the inventor of Geometric Algebra, William Kingdon Clifford.

  • In the event that you should need to follow a course for Lie Algebras on your path in Geometric (Clifford) Algebra, you may want to start with Introduction to Lie Algebras by Erdmann and Wildon (be sure to do all the exercises in the book to establish a firm foundation. Next, move on to to Humphreys' Introduction to Lie Algebras and Representation Theory (the exercises in this book are more difficult, but it gets you deeper into the subject matter). Finally, if you should like to become a heavy hitting champ in this subject, then take up the book Lie Groups Beyond an Introduction by Anthony Knapp. The prerequisite for this book, however, is that you already have a good grasp of everything before this point in the recommended learning sequence, and have knowledge of smooth manifolds (topology and differential geometry). Other than this, the book come well recommended.

  • If electrical engineering will be your application of Geometric (Clifford) Algebra, then you may like to check out the book: Geometric Algebra for Electrical Engineers: Multivector Electromagnetism, by Peeter Joot. However, this book is for graduate level. So if you are just a beginner, or at intermediate level, then this text is not for you.

  • If physics will be your application of Geometric (Clifford) Algebra, and you want a good coverage of what Geometric Algebra is capable of doing when applied in modern physics, then you will want to follow the mathematical presentation given by Professor Anthony Lasenby of Cambridge University here. The total length of this YouTube video is 2:05:15 hours in length [Title: GAME2020 3. Professor Anthony Lasenby. A new language for physics]. After watching his presentation, you may be inspired to obtain his two textbooks on Geometric Algebra: Geometric Algebra for Physicists, and Space-Time Algebra Second Edition. However, both these Geometric Algebra textbooks presume an already existing understanding of concepts in physics such as quantum mechanics, relativity (special and general), and several other higher undergraduate physics topics. So that would be your prerequisite before approaching these textbooks by Professor Anthony Lasenby.

III. Why Use Quaternion Based Mathematics? Benefits and Features.

  • Native to problems consisting of four dimensional space and easily handles 3D and 4D problems.

  • Bypasses the common problem of 3D rotational locking in Euler’s angles (known as Gimbal Lock). It is often an inevitably incurred problem when using vector calculus [If that is the case, I can only imagine what errors could happen when trying to describe particles and field rotational variables in electromagnetism (just saying)].

  • Integrates easily into 3D programming routines for multi-physics problems.

  • Requires less computational power to implement than vector calculus in 3D and multi-physics simulation situations. Although not popularly known outside of game developers, 3D gaming graphics engines implement multi-physics simulations to accurately display user interactivity with physical events in the simulated environment. A majority of 3D games utilize Quaternions in their 3D physics engines and not Vector Calculus.

  • In comparison to rotation matrices, Quaternions use less computer memory, compose faster, and are naturally suited for efficient interpolation of rotations.

  • Meanwhile vector calculus often fails at interpolate orientations, quaternions excel exceedingly at them.

  • It helps to ground the practicing physicist utilizing them in describing the sciences in mathematical reality.

  • Finally, check out this YouTube video, Quaternions and 3D Rotation, Explained Interactively by Grant Sanderson and Ben Eater.

IV. Historical Fun Facts About Quaternions and the Truth About Maxwell Theory.

Oliver Heavside and his side-kick Gibbs back in the day called Quaternions, “pure evil”, and “the work of the devil”… no joke! True story! I reference Grant Sanderson and Ben Eater’s YouTube video. The reason why vector calculus won the day back in the late 19th century over Quaternions was not because vector calculus was a superior way to describe events in 3D and 4D space, but rather because Quaternions generally are not readily intuitive, seem contrary to convention of normal thinking (seem convoluted), and thus were difficult to learn and grasp. However, with the advent of modern instructional methods that can add interactive video graphics to the learning methodology (such as provided at Ben Eater’s website, this is no longer the case. Quaternions are now increasingly accessible to the person desiring to learn them and whom have completed at least tenth grade high school mathematics courses in linear algebra (in reference to the United States). With Quaternions and Octonians (used in Clifford/Geometric Algebra), it is very possible and easy to describe and simulate most aspects of physics (from the standpoint of computational power of computers) without venturing into vector calculus at all.

“Somehow quaternions are a fundamental building block of the Universe” - Sir William Rowan Hamilton, originator and developer of Quaternion Algebra.

From my investigation and what I have learned, I believe retrospectively, that James Clerk Maxwell was ultimately correct to use Quaternions in his equations to describe his electromagnetic theory, and Oliver Heavside was wrong to mutilate them with his truncated vector calculus translation of them. Has anyone ever heard the term: "lost in translation"?

Original Maxwell equations were written using quaternions and potentials. Quaternion is a combination of vector and scalar part. Electric and magnetic fields was defined as difference in potential. There was two kinds of potentials - electric and magnetic. Today's "Maxwell equations" are actually Heaviside equations, which are a truncated limited edition of the original Maxwell theory.

Meanwhile there is much that can be practically implemented with Heaveside's equations (like in any translation), there is much to be mined and extrapolated upon in the original and authentic theory of Maxwell and his equations described in QUATERNIONS... NOT in Vector Calculus (as is often now currently taught in Electrical Engineering Schools).

One may find it relevant to see a comparison study between original Maxwell’s Equations and Heavside’s Equations made by André Waser in his essay, On the Notation of Maxwell's Field Equations. Back up link here.

V. James Clerk Maxwell Original Work and References.

Since Maxwell’s original quaternion equations were one of the goals of your question, you may find these original references helpful if you do not have them already. Once one has Quaternions mastered, one would do well to pay a visit upon Maxwell’s original work done in Quaternions, and thoroughly examine them. One may even discover their ingenious mathematical beauty and ditch Oliver Heavside’s equations for the original Maxwell’s Equations. Maxwell’s original unadulterated work is available here:

A Dynamical Theory Of The Electromagnetic Field - 1865. Maxwell's 1865 paper describing his 20 Equations in 20 Unknowns - Predecessor to the 1873 Treatise. Proceedings of the Royal Society Of London Vol XIII.

A Treatise on Electricity and Magnetism, Volume 1 – 1873.

A Treatise on Electricity and Magnetism, Volume 2 – 1873.

Five of Maxwell's Papers by James Clerk Maxwell (On Project Gutenberg Website). These papers relates to light being integrated into the electromagnetic field.

On Physical Lines of Force, J.C. Maxwell, 1861. Back up link here.

There is much more than these reference works mentioned above. The full accumulation of his life’s work can be found in: The scientific letters and papers of James Clerk Maxwell published by Cambridge University Press.

PLEASE NOTE: The above reference mentioned is Volume 1 and 2 of a total 3 volume set of his complete collection containing all of his works and writings. This single set online containing volume 1 and 2 is 1064 pages long. Its free online, but registration is required to view it online. Google Books also has all three volumes online, but as I reviewed them, I saw that several hundred pages were censored out (although listed they cannot be read or accessed). I find it very interesting also that the hard back edition of volume two is no longer available in print, but has been substituted with a paperback edition. However, to get the full three volume set in print, I recommend looking up ISBN-13: 978-0521757942, or ISBN-10: 0521757940 (but this edition too appears to be missing several hundred pages when compared to the original hardback three volume editions by the same publisher – Cambridge University Press… very strange).

VI. Deriving the Maxwell Source Equations In Quaternion Space.

I recommend looking into these references if deriving the Maxwell source equations in Quaternion Space is your goal:

M. M. Acevedo, J. López-Bonilla, et al: Quaternions, Maxwell Equations and Lorentz Transformations.

Vic Chrisitianto and Florentin Smarandache, et al, A Derivation of Maxwell Equations in Quaternion Space.

Doug Sweetser’s YouTube Channel here apparently derives the Maxwell theory in quaternions, and addresses several applications of Quaternions to modern field equations (i.e. Relativity, Gravity, Quantum Mechanics, et al). However, this particular video series may be of greater relevance to your question:

  1. GEM Blog 28: Deriving How to Derive Field Equations. (1/5)

  2. GEM Blog 29: Deriving the Maxwell Source Equations Using Quaternions. (2/5)

  3. GEM Blog 30: Deriving the Maxwell Homogeneous Equations Using Quaternions. (3/5)

  4. GEM Blog 31: Deriving the Hypercomplex Gravity Field Equations. (4/5)

  5. GEM Blog 32: Deriving the GEM Field Equations. (5/5)

  6. Using Quaternions for EM and GEM, Manifestly Covariant Form.


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