Why do we start losing algebraic properties when dealing with hypercomplex numbers?

Question

Every form of hypercomplex number I have seen (including the complex numbers) lose some important algebraic property. Why is that? Is there a pattern to what we lose?

Answer 1

The complex numbers $\mathbb{C}$ do not lose any properties that $\mathbb{R}$ has. They are both fields and hence share all the properties of fields.

Now the simple answer to your question is because they do! Unlike real numbers (and maybe arguably complex numbers) is because we create them (not that you don't create the others but you can argue the real numbers rather boldly present themselves in nature the way other 'number' structures don't). The way that we create the quaternions, $\mathbb{H}$, is such that they lose commutativity of multiplication. It comes directly from how we define multiplication of quaternions.

Similarly, how we define the octonions means that they lose both associativity and commutativity of multiplication. For both groups of hypercomplex numbers, we form an operation called multiplication in terms of real numbers and check to see that they satisfy the properties we normally have when we work with real numbers. The way we have to define the operations to construct these hypercomplex numbers for $\mathbb{H}$ and $\mathbb{O}$ causes these properties to fail.

The more complex (and more realistic reason) that they lose the properties 'numbers' normally have is that we are trading the structures they lose to get something more. That is, we want to generalize $\mathbb{R}$ to bigger and bigger sets of 'numbers' that have certain properties. We call these bigger structures algebras (because they resemble the algebra we are used to). For example, we also want to be able to tell the size of these new 'numbers' (we call this the norm of a number). In order to generalize to bigger and bigger 'numbers' with these properties, essentially something has to give and what often is lost is commutativity and associativity of multiplication.

Of course, this is only the general idea. The mechanics of where and why things break down is quite complicated in the general case (involving Lie Algebras, Clifford Algebras, and general matrix groups). However, this gives a general insight into the question. In fact, there is still much work needed to be done to understand in full the structures for these objects and why they are the way that they are.