Is this simply because of marketing, hype, etc?

The bicomplex numbers (especially tessarines) look just great being commutative and all.

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Images source:https://citeseerx.ist.psu.edu/viewdoc/download?doi=

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    $\begingroup$ Quaternions are useful. In particular for representation of rotations of 3-space. $\endgroup$ Feb 13, 2021 at 16:45
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    $\begingroup$ Thanks. Besides, I have written, some years ago an answer on MathStackExchange giving as well different correspondances of bi-complex numbers, quaternions, etc. with matrices here $\endgroup$ Feb 13, 2021 at 20:22
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    $\begingroup$ Maybe, it is a little exaggerate to say such definitive things like "a huge advantage of tessarines over all other algebras and total uselessness of quaternions"... $\endgroup$ Feb 13, 2021 at 22:29
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    $\begingroup$ @Anixx If a study shows the 'total uselessness' of quaternions, then that study is clearly flawed, as illustrated by the many uses of quaternions. $\endgroup$
    – user13928
    Feb 15, 2021 at 12:53
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    $\begingroup$ Deleting my previous comment and saying it differently: Being non-commutative is not a Bad Thing. There's lots of ways to represent the orientation of a rigid body in 3-Space, but what what makes quaternions attractive is that if we have the quaternions representing a sequence of rotations, we can multiply them together to get a single rotation that puts the object into the same, final orientation. Performing rotations in sequence is not a commutative operation. Being "numbers like" might make the math easier, but being reality like is what makes it actually useful. $\endgroup$ Feb 16, 2021 at 17:29

3 Answers 3


Commutativity is over-rated: in fact, it holds back bicomplex numbers:

  • It prevents your number system characterising non-commuting operations, e.g. rotations in $3$-dimensional space, Hamilton's original focus.
  • Since $0=i^2-j^2=(i-j)(i+j)$ in bicomplex numbers, you have zero divisors, so it's not a normed division algebra; no convenient conjugates, no general definition of $a/b$ even with $b\ne0$, which hinders the description of invertible processes - again, such as rotations. (Apart from conjugation, these are also issues with "hyperbolic" or split-complex numbers, viz. $0=1-j^2=(1-j)(1+j)$.) In this respect, your number system runs into difficulties even octonions don't face, though sedenions do. I can't offer a comprehensive overview of how this has stymied historical use, but uses of split-complex numbers have never become widespread (even when they arguably deserved to be, e.g. you can use split-complex numbers in relativity, but ironically complex numbers have proved more popular for this).
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    $\begingroup$ @Anixx That's just a made-up rule. You might as well say complex numbers aren't numbers because you want numbers to be totally ordered. "Numbers" are whatever we need. $\endgroup$
    – J.G.
    Feb 13, 2021 at 23:13
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    $\begingroup$ @Anixx Do you mean because fewer rearrangements are legal? That never upset group theorists. For many applications, that division is undefined is much worse than some equations no longer being true. $\endgroup$
    – J.G.
    Feb 13, 2021 at 23:17
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    $\begingroup$ @Anixx, the most important attribute a mathematical system can have is that it is useful. Things like "stylistic elegance" take a distant second place. $\endgroup$
    – Mark
    Feb 13, 2021 at 23:59
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    $\begingroup$ @Anixx As far as I can tell, people don't call the quaternions numbers, generally. They call them quaternions, or the quaternion algebra (compare matrix and matrix algebra). I think that no occurrence of "number" in this answer refers to the quaternions, although the one in the first bullet point is debatable I suppose. $\endgroup$
    – benrg
    Feb 14, 2021 at 2:03
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    $\begingroup$ @Anixx You asked an HSM historical question, why have bicomplex numbers been used less than quaternions? The meaning of the word "number" is irrelevant to that question. Mathematicians, physicists etc. use whichever mathematical objects they've found use for. My answer focused on the historical reasons invertible non-commutative objects (which, yes, have a matrix representation) were more helpful than non-invertible commutative objects (which, yes, have a matrix representation). $\endgroup$
    – J.G.
    Feb 14, 2021 at 7:29

Your description "total uselessness of quaternions" in a comment above is poorly chosen, and reflects more on your interests than on the real state of knowledge of mathematics. The Hamilton quaternions are the simplest nontrivial example of a quaternion algebra, which has turned out to be a really important concept in mathematics.

It is useful to think of the Hamilton quaternions $\mathbf H$ as being analogous to the ring of $2 \times 2$ real matrices ${\rm M}_2(\mathbf R)$. At first these might seem quite different: one is a division ring and the other is not. But here are analogies between them: both are 4-dimensional with center the real numbers and in both of them there is no 2-sided ideal other than $(0)$ and the whole ring. A ring that is 4-dimensional over a subfield $K$, has $K$ as its center, and has no 2-sided ideals other than $(0)$ and the whole ring is called a quaternion algebra over $K$. The matrix ring ${\rm M}_2(K)$ is a quaternion algebra over $K$, and it is the only kind that is not a division ring.

Sometimes ${\rm M}_2(K)$ is the only quaternion algebra over $K$, e.g., when $K = \mathbf C$ and when $K$ is a finite field. There are two quaternion algebras over $\mathbf R$: ${\rm M}_2(\mathbf R)$ and $\mathbf H$. Similarly, there are two quaternion algebras over the $p$-adic numbers for each prime $p$. For $p = 2$ the two examples are ${\rm M}_2(\mathbf Q_2)$ and the Hamilton quaternions with $\mathbf Q_2$-coefficients, but for odd primes $p$ you need to do something different to get the "nontrivial" example because the Hamilton quaternions with $\mathbf Q_p$ coefficients is isomorphic to ${\rm M}_2(\mathbf Q_p)$ rather than being a division ring.

Things get really interesting for fields like $K = \mathbf Q$, because there are infinitely many different quaternion algebras over $\mathbf Q$. The ring ${\rm M}_2(\mathbf Q)$ and the rational Hamilton quaternions are merely two examples out of a huge list of rational quaternion algebras. These "new" rational quaternion algebras show up in various places in mathematics: (supersingular) elliptic curves, Brauer groups, Shimura curves, and hyperbolic geometry in 2 and 3 dimensions. Take a look at John Voight's book in order to appreciate that you should not dismiss quaternions as being a "totally useless" concept. If you look at Ribet's paper that showed Fermat's Last Theorem is a consequence of modularity of elliptic curves over $\mathbf Q$, you'll see a role there for rational quaternion algebras that are division rings and are not the rational Hamilton quaternions.

  • $\begingroup$ The 2x2 real matrices are called split-quaternions and they unfortunately are not in the table in question. Still, they are both non-commutative and have zero-divisors, so not quite good at having numerical properties. $\endgroup$
    – Anixx
    Apr 1, 2022 at 6:32

Hamilton expected that the quaternions would be of physical interest. In this, he was right. But he was too early. He had discovered them in 1843, it was almost a century later, in 1928, when Dirac discovered his equation involving the Pauli matrices, that it was seen that the quaternions were naturally implicated in quantum field theory. (Here, the Pauli matrices are a representation of the quaternions on the space of 2 complex dimensional space).

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    $\begingroup$ Maxwell actually based his equations on quaterions. The whole vector calculus was in quaternions before Gibbs and Heaviside. There are other ways how to represent 3D rotations and Pauli matrices too (e.g. geometric algebra). $\endgroup$ Feb 14, 2021 at 15:15
  • $\begingroup$ @Vladimir F: I'd forgotten that Maxwell had used quaternions. Coincidentally, I was just looking at geometric algebra - aka Clifford alfebras - before just coming onto the site now. $\endgroup$ Feb 14, 2021 at 16:29
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    $\begingroup$ The quaternions may have helped inspire the algebra of Dirac gamma matrices, but the two are different. The crucial difference is that $\gamma^0$ anticommutes with all of $\gamma^1$ through $\gamma^3$, whereas in the quaternions $1$ commutes with all of $i$, $j$, and $k$. For the Pauli matrics, we are just back to 3D rotations. $\endgroup$
    – Lars H
    Feb 15, 2021 at 12:22
  • $\begingroup$ @Lars H: Thanks for pointing this out. I'll edit my post when I find out a bit more. I think I should be thinking about Clifford algebras rather than quaternions here. $\endgroup$ Feb 15, 2021 at 13:19

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