What is the motivation for normed division algebras? The famous Hurwitz theorem states that the only normed division algebras are  $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$. What is some good pedagogical motivation why we should think about normed division algebras at all?
 A: Much of this can be gleaned from the introduction section of John Baez's survey article on the octonions. Hamilton wanted a 3D number system analogous to the complex numbers, in that the length of a number was multiplicative and hence multiplication by unit norm numbers had the effect of rotating space. Hamilton used sum-of-squares identities to eventually realize he would need a fourth dimension, and hence quaternions. This is why composition algebras are connected to number theory and binary quadratic forms. In particular, we have


*

*Two-squares identity (Brahmagupta-Fibonacci) connected to complex numbers $\mathbb{C}$: $$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2.$$

*Four-squares identity (Euler) connected to the quaternions $\mathbb{H}$,

*Eight-squares identity (Degen) connected to the octonions $\mathbb{O}$,

*Sixteen-squares identity (Pfister), using rational expressions instead of bilinear.


Of note are Hurwitz's theorem on composition algebras and Pfister's theorem on sums of squares.
Hamilton  then spent his life writing, lecturing and expanding on the usages of quaternions in geometry and physics. They even became useful in Maxwell's equations of electromagnetism. IIRC, they became standard curriculum in Irish academia. But not everybody was a fan.
At the time, there was a great debate between vectors and quaternions. The vectors had the dot product and cross product as distinct binary operations, while the quaternions incorporated them both into its multiplication operation. Many mathematicians came out on one side or the other; in particular Gibbs and Heaviside for vectors, Hamilton and Graves for quaternions. Other algebras came to light too, of note Grassman's algebras (aka exterior algebras) and Clifford's algebras (aka geometric algebras), which are related to modern-day spinors.
Quaternions may be used to understand 3D and 4D rotations. The unit quaternions with the operation of multiplication form a symmetry group $\mathrm{Sp}(1)$ (stands for "symplectic," same meaning as "complex," a term Weyl used in The Classical Groups since "complex" was already in use). Every quaternion has a real and imaginary part, or equivalently a scalar and a vector part. Conjugating a vector by an arbitrary unit quaternion yields an arbitrary 3D rotation, and multiplying a quaternion on the left and right by a pair of arbitrary unit quaternions yields an arbitrary 4D rotation.
The whole time Hamilton was trying to create a 3D number system he was in correspondence with Graves. When Hamilton managed to create the quaternions, Graves then created the octonions, although Cayley beat him to publication (also some guy called Degen had discovered them even earlier). Anyway, octonions never caught on at the time, and as we know today vectors eventually usurped quaternions. Lately though octonions have been reborn after it was noticed the octonions play a surprisingly central role in a web of connected exceptional structures in mathematics:


*

*As we've seen, the bilinear sum-of-squares identities arise from composition algebras.

*The number of pointwise orthogonal vector fields on a sphere is maximized when it is the unit sphere in one of the four composition algebras $\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$.

*As per Adams' theorem, the only Hopf fibrations between three spheres are the fiber bundles $\mathbb{S}^{n-1}\to\mathbb{S}^{2n-1}\to\mathbb{S}^n$, with fiber $\mathbb{S}^{n-1}$ the unit sphere in a composition algebra $\mathbb{K}$, total space $\mathbb{S}^{2n-1}$ the unit sphere in  $\mathbb{K}^2$, and base space $\mathbb{S}^n$ the projective line $\mathbb{KP}^1$ ($n=\dim\mathbb{K}$).

*Projective planes are built from composition algebras. For $\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H}$ there are projective planes of any dimension, $\mathbb{KP}^k$. But for $\mathbb{O}$ there is only $\mathbb{OP}^k$ for $k=0,1,2$.

*When quantum mechanics was first being developed, some physicists wanted to develop a barebones set of axioms for the algebra of observables. One candidate was formally real Jordan algebras, which are all direct sums of simple such algebras, which are in turn classified using composition algebras, the exceptional Albert algebra coming from $\mathbb{OP}^2$.

*Over half of the accidental isomorphisms between low-dimensional spin groups and classical matrix Lie groups come from constructions built from composition algebras.


In particular, the spin groups $\mathrm{Spin}(3)=\mathrm{Sp}(1)$ and $\mathrm{Spin}(4)=\mathrm{Sp}(1)^2$ correspond to the fact that quaternions yield 3D and 4D rotations (with the slightest of redundancy). Letting $\mathrm{SU}(2,\mathbb{K})$ act on $\mathbb{K}^2$ (suitably interpreted), hence act on $\mathbb{KP}^1\simeq\mathbb{S}^n\subset\mathbb{R}^{n+1}$ yields another handful of spin groups $\mathrm{Spin}(n+1)=\mathrm{SU}(2,\mathbb{K})$ (see my explanation here), and more generally $\mathrm{SL}(2,\mathbb{K})$ acts by similarity transformations on $\mathfrak{h}_2(\mathbb{K})$ ($2\times 2$ hermitian matrices over $\mathbb{K}$), a mask being worn by some Lorentzian (in particular Minkowski) spacetimes, yielding $\mathrm{Spin}(n+1,1)=\mathrm{SL}(2,\mathbb{K})$.


*

*John Baez at least has made numerological insinuations about octonions causing various dimensional restrictions in string theory.

*The simple lie algebras (infinitessimal linearizations of continuous symmetries) come in a handful of infinite families $\mathfrak{so}(n)$, $\mathfrak{su}(n)$, $\mathfrak{sp}(n)$ (notice the connection to composition algebras), with a finite handful of exceptional simple lie algberas $\mathfrak{g}_2$, $\mathfrak{f}_4$, $\mathfrak{e}_6$, $\mathfrak{e}_7$, $\mathfrak{e}_8$. These are the infinitessimal symmetries of the octonions and the projective planes over $\mathbb{K}\otimes\mathbb{O}$ respectively, generalized by the Freudenthal-Tits magic square.

*Simple lie algebras are classified according to Dynkin diagrams, the most symmetric of which is $\mathrm{D}_4$ having $S_3$ type symmetry. This can be explained by $\mathrm{Spin}(8)$s realization as the isotopies $(\alpha,\beta,\gamma)$ of $\mathbb{O}$. The outer symmetries are generated by cyclical shifts and negating the first two coordinates. These symmetries also correspond to permuting the vector representation and positive and negative spinor representations of $\mathrm{Spin}(8)$. Keyword: Triality. Somehow all of this can be depicted by a Feynman diagram relating to gauge bosons and fermions.

*As I understand it, octonions are related to the $E_8$ lattice and the Leech lattice. How this works and how this fits into lattices and sphere packing etc. I am not familiar with.


Octonions also cause the topology of rotations. More specifically, there is a form of Bott periodicity which states $\pi_{i+8}(\mathrm{O}(\infty))=\pi_i(\mathrm{O}(\infty))$ (where $\mathrm{O}(\infty)=\bigcup_{n\ge1}\mathrm{O}(n)$ is the stable orthogonal group), the period $8$ connecting back to the $8$-dimensional octonions. Furthermore, in dimensions $0$-$7$, the only nontrivial $\pi_i(\mathrm{O}(\infty))$s have $i=0,1,3,7$, each generated by $\mathbb{S}^{n-1}\to\mathrm{O}(n)$ defined by using unit elements of $\mathbb{K}$ for left multiplication maps.
All of this is in pure math and theoretical physics, though. If you want real-life application, Wikipedia says the quaternions have had a revival

... primarily due to their utility in describing spatial rotations. The representations of rotations by quaternions are more compact and quicker to compute than the representations by matrices. In addition, unlike Euler angles they are not susceptible to gimbal lock. For this reason, quaternions are used in computer graphics, computer vision, robotics, control theory, signal processing, attitude control, physics, bioinformatics, molecular dynamics, computer simulations, and orbital mechanics. For example, it is common for the attitude control systems of spacecraft to be commanded in terms of quaternions.

As for pedagogical motivation, not much can be said at an elementary level besides the original historical motivations of sum-of-squares identities and quadratic forms in number theory, and generalizing the 2D number system of the complex plane to get rotations in higher dimensions.
However, there are at least two natural concepts more or less equivalent to composition algebras that just might count as motivation:


*

*The dot product is an examples of a normed duality. That is, it is a bilinear map $V\times V\to\mathbb{R}$ on a normed real vector space $V$ with the property that $|\langle a,b\rangle|\le\|a\|\|b\|$ identically, and for all $b$ there is a $a$ that makes the $\le$ an $=$. Then it's natural to consider if there is a triality $V\times V\times V\to\mathbb{R}$ with the same property for $|\langle a,b,c\rangle|\le\|a\|\|b\|\|c\|$.  One may dualize between this and the condition that $V\times V\to V$ defines a composition algebra.

*The 3D cross product can be generalized to an antisymmetric bilinear operation on $\mathbb{R}^d$: it outputs a vector which is orthogonal to the inputs and whose size is the volume of the parallogram generated by the inputs. These cross products correspond to multiplication of pure imaginaries in composition algebras.


(In fact cross products can be generalized to arbitrary arity and then classified. There are three main families, which I would name degenerate, chiral, and complex, plus with two exceptions both coming from octonions.)
Beyond this, it takes someone enamored by the mysticism of the interconnectedness of exceptional structures in pure math and theoretical physics to be sufficiently motivated to learn the more highfalutin, (deliberate pun warning) high-octane stuff.
A: I think the motivation in pretty clear : the first step was to understand that complex numbers can represent points in the euclidian plane. That was a highly non-trivial and very deep idea. I think everyone will agree that this idea brings a lot to our understanding of the geometry of the plane (and conversely, of course, the geometrical point of view brings a lot to the algebra over complex numbers...).
Thus it made perfect sense that people (and of course most notably Hamilton) wondered whether you could use this wonderful method in higher dimension : can you represent the euclidian space of dimension $n$ by some sort of algebra with the euclidian norm compatible with the operations ?
Hamilton realized that this was not possible in dimension $3$, and that in dimension $4$ this was only possible if you dropped the commutativity. That was surely a bit disappointing, but the rewards were great : quaternions bring a lot to the study of the geometry of $\mathbb{R}^4$ (and ironically also $\mathbb{R}^3$ using the subspace of pure quaternions), for instance exceptional isomorphisms for orthogonal and unitary groups.
Then it's quite obvious that people would look for further generalizations, and this lead to the octonions and the theorem of Hurwitz (which can be adequately generalized to any field).
A: I think that the sympler motivation is that in a division algebra we can always solve an equation of the form $AX+B=0$ using the existence of opposite and inverse elements.
If the algenbra is also associative and commutative we can solve also more complex equations and, in the case of $\mathbb{C}$ we have the wonderful result that any polinomial equation of degree $n$ has always $n$ solution.
This last result fails for a non commutative algebra as $\mathbb{H}$ where equations as $AX^2+BX+C=0$ and$XAX+XB+C=$ does not have ( in general) the same solutions, and the situation becomes more difficult for the octonions, where also associativity is not valid.
The fact that the Hurwitz theorem limits the possible normed algebras of this type is important because the existence of a norm is the standard way to introduce a distance, so that from the algebra emerges naturally a vector space that is also a metric space.
A: A normed division algebra is related to a "squares identity"; there are notworthy only 4 of them, as was proven by A. Hurwitz in 1898: the 1-square identity (trivial case of the real numbers which form a normed division algebra); the 2-squares identity (discovered by Diophantus, which is the base of the complex numbers, also forming a normed division algebra); the 4-squares identity (discovered by Leonhard Euler, and forming the base of the quaternion number system, also forming a normed division algebra); and the 8-squares identity (discovered by Ferdinand Degen, and forming the base of the octonion number system, also forming a normed division algebra). As was proven by A. Hurwitz and later authors, there are no more squares identities which give rise to a normed division algebra.
Now, these squares identities state that the product of two sums of 1, 2, 4, or 8 squares is again a sum of 1, 2, 4, or 8 squares. A sum of such squares, on the other hand, is the squared length of a vector in 1, 2, 4, or 8 dimensional space. A physical vector in 1, 2, 4, or 8 dimensional space can thus be transformed by a unity transformation in the same space, which lets the length of the vector unchanged. This is immediately obvious by considering that the squared lenght of the vector is a sum of 1, 2, 4, or 8 squares, which can be written as a product of two sums of 1, 2, 4, or 8 squares, one of them being of unit value. 
This means that in the non-trivial spaces with 2, 4, or 8 dimensions, we can have transformations which let some intrinsic scalar quantity of the system unchanged. Applied to physics, we can identify said transformations with physical movement, and said intrinsic scalar quantity with the total energy.
See here to also see some of my publications
