Why do we have real numbers with one term, complex numbers with two terms, quaternions with four terms, octonions and so on, and nothing in between?
In: 9
First, there is not really a “and so on”. It depends on what you allow, but it ends as soon as you add some common requirements. We often encounter:
– the reals ℝ, an (actually, *the* archimedean complete) _ordered_ field, i.e. we have ≤ and it is compatible in various way with + and ·,
– the complex numbers ℂ, an (algebraically closed, complete) _commutative_ division algebra, i.e. a·b = b·a holds,
– the quaternions ℍ, an _associative_ division algebra, i.e. a·(b·c) = (a·b)·c holds,
– the octonions 𝕆, an _alternative_ division algebra, i.e. a·(a·b) = (a·a)·b and a·(b·b) = (a·b)·b hold
– the sedenions 𝕊, which are not even a division algebra, i.e. for some x there is no y with x·y = 1; they at least still satisfy a·(a·a) = (a·a)·a and (a·a)·(a·a) = ((a·a)·a)·a.
The cursive words are the most important aspect here: each of them is also true for the ones above them, but not below. At each step, something is lost.
With 𝕊, almost none of the typically used properties are left, we barely(!) can define powers x^n without having to write down in which order we multiply them. Most people hence stop the list there, only very few people seriously work with them.
Generally, classification theorems exist for algebras over the reals, which say that the dimensions 1, 2, 4, 8 are the only ones where algebras with the above properties exist. However, depending on what exactly you require, it is not always true that the above list contains all such algebras; only the dimensions are certain powers of 2!
For convenience, I will now write “ℝ-algebra” for what is typically called a “unitary finite dimensional algebra over ℝ”: a set A with addition + and multiplication · containing (and compatible with) ℝ, satisfying:
– distributivity (that is: a·(b+c) = a·b+a·c and (a+b)·c = a·c+b·c) for any a,b,c in A,
– commutativity of addition (a+b = b+a),
– centrality of ℝ (s·a = a·s for all a in A and s in ℝ, in other words, multiplication of two numbers is commutative if at least one factor is in ℝ)
– negativity (a+(-1)·a = 0, noting that -1 as a real number is in A),
– addition&multiplication turn A into a real vector space of finite dimension dim(A) (there is a finite list a1, a2, …, an of elements of A such that every element of A can be written in the form s1·a1 + s2·a2 + … + sn·an for suitable s1, s2, …, sn in ℝ).
If you then require that A satisfies some sane other properties, e.g. allowing division, or associativity or commutativity, then one can show that only very few examples exist. The exact version depends on what you want, there are many different results.
Lets maybe dive a bit into one such result by Hurwitz on “composition algebras”, where the standard absolute value |·| on ℝ extends linearly to the algebra and furthermore satisfies |x·y| = |x|·|y|. Then such algebras exist only in dimensions 1, 2, 4 and 8. Some examples are
– ℝ,
– ℂ, with |a+bi|² = a²+b²,
– ℍ, with |a+bi+cj+dk|² = a²+b²+c²+d²,
– 𝕆 with the same, but 8 terms.
– ℝ², with |(a,b)| = a²-b²,
– and many more.
Note that the last one is not on the list above, and there are similar examples of dimensions 4 and 8.
But the really interesting part about this theorem is that |x·y| = |x|·|y| on ℂ corresponds, with x = a+bi and y = c+di, to the formula (a²+b²)·(c²+d²) = (ac-bd)²+(ad+bc)², hence a product of two sums of two squares is again a sum of two squares! And we can verify this formula directly, never even talking about complex numbers, or even composition algebras; it actually holds in any ring! Similarly, we get such formulas for sums of 4 or 8 squares by invoking ℍ or 𝕆, they are just a bit longer.
Yet, almost magically, there cannot be such a formula for other numbers of variables by the theorem!
Lastly, a quick statement on the proof behind such results:
– Proving that ℝ is the only archimedean ordered complete field is typically done in the first year of a calculus course and not difficult, just a bit technical.
– The only ℝ-algebra (except ℝ itself) that also happens to be a field is ℂ, as follows from the fundamental theorem of algebra (i.e. that every non-constant polynomial with complex coefficients has a root within ℂ); this famous result is shown similarly early when studying mathematics.
– There is a very vast theory of “Brauer groups” that classifies, for any field K, all the K-algebras that also happen to be division rings (similar to fields, but without requiring commutativity). For K=ℝ, the only examples are ℝ, ℂ and ℍ. This is already pretty advanced, typically a Master’s course, and often done when also dealing with “Group/Galois cohomology”.
– Almost all proofs where the answer is either 1, 2, 4, 8 or exactly the list ℝ, ℂ, ℍ, 𝕆 are seriously involved and usually done by invoking advanced methods from algebraic topology such as K-theory or higher homotopy groups. Even sketching anything here would go way beyond this already long post.
Edit: forgot to mention:
Those results on composition / division algebras also have ramifications on very different looking things that are quite interesting in their own rights, such as:
– “Sphere eversions” (turning a sphere inside-out without creasing or tearing it) exists for spheres of dimensions 0, 2 and 6 within 1, 3 and 7 dimensional space, respectively. There are some nice videos of the 2-sphere being everted in 3-space such as https://www.youtube.com/watch?v=OI-To1eUtuU&t=1131s, and I would recommend the entire video as worth your time.
– A “cross product” exists only in dimensions 0, 1, 3 and 7. The [3-dimensional one](https://en.wikipedia.org/wiki/Cross_product) is widely known among multiple fields and is related to quaternions, but the [7-dimensional cross product](https://en.wikipedia.org/wiki/Seven-dimensional_cross_product) coming from octonions is something even many mathematics professors don’t know.
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