Why are quaternions more popular than tessarines despite being non-commutative?

Question

Is this simply because of marketing, hype, etc?

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

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Answer 1

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).

Answer 2

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.

Answer 3

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).