# What math course is Quaternion math taught in?

Summary:: I want to learn Quaternion math. Also; I do not know what prefix my question falls under. I just learn maths from books. I don't know which parts of school they are taught in.

I was hoping that I would get to learn about that subject in my Linear Algebra text book, but I looked through the index, and the Quaternion is not mentioned there. So, what subject teaches about Quaternion math?

I am currently re-learning Calculus and Analytical Geometry, and I will be studying my Linear Algebra book right after that. Is that enough to learn the subject that teaches Quaternions?

Staff Emeritus
You might see them in a class on Abstract Algebra. Or you might not see them at all. They are not generally considered an important part of the curriculum.

• vanhees71
Stephen Tashi
If you want to make practical use of quaternions, you should study the mathematics of computer animation. Computers weren't common when I was in school, so I'm not sure what department would teach such a course. I doubt it would be taught by the math department.

In courses on abstract algebra, you will probably get a short presentation of quaternions as one of the few examples of "finite dimensional associative division algebras over the real numbers". You will study all the material needed to understand that terminology, but you won't learn how to make practical use of quaternions.

fresh_42
Mentor
What math course is Quaternion math taught in?
The way you ask, I'm tempted to answer: in a physics book. You can find some interesting answers if you google "quaternion analysis".

From a mathematical point of view, the quaternions are just an example of a division algebra, and thus treated as algebraic object, i.e. the investigation of division algebras in general, and their representation. So if you are interested in what can be done in terms of calculus, you will find better results in the realm of physics.

• vanhees71
The way you ask, I'm tempted to answer: in a physics book. You can find some interesting answers if you google "quaternion analysis".

From a mathematical point of view, the quaternions are just an example of a division algebra, and thus treated as algebraic object, i.e. the investigation of division algebras in general, and their representation. So if you are interested in what can be done in terms of calculus, you will find better results in the realm of physics.
Thank you for moving this to the correct area. I thought it was a math topic. Can you recommend a physics book?

If you want to make practical use of quaternions, you should study the mathematics of computer animation. Computers weren't common when I was in school, so I'm not sure what department would teach such a course. I doubt it would be taught by the math department.

In courses on abstract algebra, you will probably get a short presentation of quaternions as one of the few examples of "finite dimensional associative division algebras over the real numbers". You will study all the material needed to understand that terminology, but you won't learn how to make practical use of quaternions.
Thank ou for the reply. That's actually why I want to study quaternions.

jedishrfu
Mentor
Checkout 3blue1brown's youtube videos on them:

fresh_42
Mentor
Thank you for moving this to the correct area. I thought it was a math topic. Can you recommend a physics book?
Not really. I googled as quoted and found some interesting links. But they likely breached copyright rules, which is why I basically said: "look for yourself" instead of quoting them. I could imagine that it is quite difficult to perform analysis without commutativity.

I've seen expressions which looked like complex Lie algebras of differential operators. So this is not specifically about quaternions, more about Lie theory. E.g. the unit quaternions are the 3-sphere which is the Lie group ##SU(2,\mathbb{C})##. So you get immediately into Lie theory, which is also the suited theory of physical relevance.

jedishrfu
Mentor
A brief summary on Quaternions with further references:

http://stahlke.org/dan/publications/quaternion-paper.pdf

The curious thing was that Hamilton tried to steer physics in that direction but push back from other physicists notably Josiah Gibbs resulted in them taking the i,j,k notions, and creating Vector Analysis which was less cumbersome to use but lost the rotational aspect that is now prized in today's computer usage.

https://en.wikipedia.org/wiki/Josiah_Willard_Gibbs

From 1880 to 1884, Gibbs worked on developing the exterior algebra of Hermann Grassmann into a vector calculus well-suited to the needs of physicists. With this object in mind, Gibbs distinguished between the dot and cross products of two vectors and introduced the concept of dyadics. Similar work was carried out independently, and at around the same time, by the British mathematical physicist and engineer Oliver Heaviside. Gibbs sought to convince other physicists of the convenience of the vectorial approach over the quaternionic calculus of William Rowan Hamilton, which was then widely used by British scientists. This led him, in the early 1890s, to a controversy with Peter Guthrie Tait and others in the pages of Nature.

Infrared
Gold Member
Quaternions have come up in several math classes I've taken:

In topology, they give group structure on ##S^3## (view ##S^3## as the group of imaginary quaternions), and octonions give ##S^7## the structure of an H-space, so these spheres are parallelizable. In general, one can ask how many everywhere-linearly independent vector fields exist on ##S^n##, and the construction of the maximal number uses representations of Clifford algebras (which generalize quaternions).

Quaternions give a natural way of constructing a double cover ##SU(2)\to SO(3)## (Take a unit quaternion, view it as an element of ##SU(2)##. Conjugating by that quaternion gives an isometry on the vector space of imaginary quaternions; this is the map ##SU(2)\to SO(3)##).

Quaternions also came up in my representation theory/abstract algebra classes because of this double cover (and for many other reasons I'm sure). This map lets us classify the finite subgroups of ##SU(2),## for example.

There are definitely other situations in which quaternions came up that I'm not thinking of right now.

• DartomicTech
Quaternions have come up in several math classes I've taken:

In topology, they give group structure on ##S^3## (view ##S^3## as the group of imaginary quaternions), and octonions give ##S^7## the structure of an H-space, so these spheres are parallelizable. In general, one can ask how many everywhere-linearly independent vector fields exist on ##S^n##, and the construction of the maximal number uses representations of Clifford algebras (which generalize quaternions).

Quaternions give a natural way of constructing a double cover ##SU(2)\to SO(3)## (Take a unit quaternion, view it as an element of ##SU(2)##. Conjugating by that quaternion gives an isometry on the vector space of imaginary quaternions; this is the map ##SU(2)\to SO(3)##).

Quaternions also came up in my representation theory/abstract algebra classes because of this double cover (and for many other reasons I'm sure). This map lets us classify the finite subgroups of ##SU(2),## for example.

There are definitely other situations in which quaternions came up that I'm not thinking of right now.
Could you recommend a Topology text book?

A brief summary on Quaternions with further references:

http://stahlke.org/dan/publications/quaternion-paper.pdf

The curious thing was that Hamilton tried to steer physics in that direction but push back from other physicists notably Josiah Gibbs resulted in them taking the i,j,k notions, and creating Vector Analysis which was less cumbersome to use but lost the rotational aspect that is now prized in today's computer usage.

https://en.wikipedia.org/wiki/Josiah_Willard_Gibbs
Thank you for the links to the videos and the paper.

• jedishrfu
Infrared
Gold Member
• DartomicTech
I first got exposed to them in a one-semester course in abstract algebra as an undergraduate, but they were mostly presented as a curiosity. When I took graduate mechanics in physics out of the classic Goldstein, Classical Mechanics, we learned of quaternions as expressing the rotation /orientation of an object. In addition I have met quaternions in the study of rotations in some aerospace engineering courses. They do not experience singularities in their description the way Euler angles do. All told, to really get to know them on a practical level, physics and aerospace fields are the best treatment. If you want to know them at a theoretical level, perhaps math is the way to go. With quaternions you can prove, the product of two numbers which is the sum of 4 squares is also a number which is the sum of 4 squares. An interesting result related to Waring's problems

• jasonRF, vanhees71 and DartomicTech
I first got exposed to them in a one-semester course in abstract algebra as an undergraduate, but they were mostly presented as a curiosity. When I took graduate mechanics in physics out of the classic Goldstein, Classical Mechanics, we learned of quaternions as expressing the rotation /orientation of an object. In addition I have met quaternions in the study of rotations in some aerospace engineering courses. They do not experience singularities in their description the way Euler angles do. All told, to really get to know them on a practical level, physics and aerospace fields are the best treatment. If you want to know them at a theoretical level, perhaps math is the way to go. With quaternions you can prove, the product of two numbers which is the sum of 4 squares is also a number which is the sum of 4 squares. An interesting result related to Waring's problems
Thanks for the physics book recommendation.

Schwichtenberg's Physics from Symmetry has a nice intro section on Quaternions (very introductory and not very rigorous) and it is a physics book.

• DartomicTech, vanhees71, weirdoguy and 1 other person
mathwonk
Homework Helper
2020 Award
Quaternions are treated in the book Basic Algebra by Jacobson, and briefly in the books of Hungerford, Mike Artin, and Dummit and Foote. Their application to the expression of all integers as sums of 4 squares is in Herstein, Topics in Algebra, I believe.

I myself enjoyed the preliminary section of Maxwell's classic Treatise on electricity and magnetism vol. 1, where he praises Hamilton's quaternions and uses them in the discussion of vector calculus. For this reason, minus signs occur in his discussion which result in his use of the concept of "convergence" in place of the usual "divergence" in vector calculus. (quaternions can be expressed in the form a +bi +cj +dk, where a,b,c,d are real and i^2 = j^2 = k^2 = -1.)

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• DartomicTech and jasonRF
vanhees71
Gold Member
There is some reason, why since Heaviside and Gibbs physicists use tensor calculus rather than quaternions...

• jasonRF
jasonRF
Gold Member
I recommend googling

rotation matrix quaternion site:.edu
or
spacecraft attitude quaternion site:.edu

or some variation on those. You should find a number of pdf files posted by professors describing their use to represent rotations. I learned about them in grad school from “spacecraft attitude dynamics and control” by Wertz, but do not recommend purchasing that giant, expensive book just for that.

jason

• DartomicTech and vanhees71