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What properties do we lose as we go from real numbers to quaternions, then to octonions? Do any new properties arise, or do calculations just become more "path dependant"?

Some loony with a calculator
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  • What exactly is a "property"? Arguably being a $4$-dimensional vector space over $\Bbb{R}$ is a "new property" of the quaternions! – user771918 Apr 24 '20 at 23:46
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    You can google Cayley-Dickinson construction to see some properties these algebras lose as you double the dimension. If not, there are infinitely many interesting properties that distinguish them, eg topological. – Bcpicao Apr 24 '20 at 23:48
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    In my opinion, you should add the word "quaternions" or "real division algebras" or something similar to yous title. Otherwise people get confused as "number system" has no inherited meaning. For example $p$-adic numbers, surreal numbers and many more falls under to the same name. At the same time "properties" also a very vague term. They could be algebraic, geometric, topological or analytic or something else. Please be specific in the problem statement. – Bumblebee Apr 24 '20 at 23:52
  • I meant like, starting with multiplication, division, integration etc., how do those change as you add more dimensions, and if there are any specific (yet general) relationships that begin to arise as well. I'm not talking about any abstractions like vector spaces, just "simple" manipulations like (on the topic of vectors) not being able to multiply vectors in any order you choose. – Some loony with a calculator Apr 24 '20 at 23:57
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    You can have a vector space structure (addition +scalar multiplication+some compatibilities) on any $\mathbb{R}^n.$ But only $n=1, 2$ has field structures (addition+multiplication+some nice properties) which give rise to complex numbers. From the algebraic prospective quaternions has this nice characterization which is bit more complicated. Measure theory (integration) works nicely in any dimension. – Bumblebee Apr 25 '20 at 00:01
  • Also the trivial analogue of fundamental theorem of algebra dose not holds for quaternions. But I don't know how it works with other hyper-complex numbers in general. – Bumblebee Apr 25 '20 at 00:25
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    There are several questions about this topic on this site; e.g. see here for what happens beyond the octonions. – Noah Schweber Apr 25 '20 at 01:19

2 Answers2

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From reals to complex you lose order. From complex to quaternions you lose commutativity. From quaternions to octonions you lose associativity. From octonions to ..?

I had written this as a comment, then I followed Noah Schweber's link, which essentially says this plus more. And pregunton's answer is meaty.
What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?

EDIT: Bcpicao's comment led me to
https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction
a fabulous link.

dcromley
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The most important property (IMHO) that $\mathbb{R}$ has that its extensions don't have is that it is a totally ordered field with least upper bound property. You don't have an order (that works nicely with the field) in these extensions.

Physical Mathematics
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