Linked Questions

2
votes
2answers
337 views

Complex Number, Quaternions and Octonions [duplicate]

There are complex $\mathbb C$, quaternions $\mathbb H$ and octonions $\mathbb O$. Is there any higher dimensional generalization of them, such in the $\mathbb R^{16}$? Or why do we just study three ...
2
votes
0answers
91 views

Do you lose any more equational identities when you go past sedenions? [duplicate]

Every Cayley-Dickson algebra can be viewed as a $(+,-,*,0,1)$ algebra. The reals and the complexes share the same equational identities. The quaternions have a subset of the equational identities, ...
74
votes
7answers
8k views

What is lost when we move from reals to complex numbers? [duplicate]

As I know when you move to "bigger" number systems (such as from complex to quaternions) you lose some properties (e.g. moving from complex to quaternions requires loss of commutativity), but does it ...
58
votes
13answers
8k views

Why are addition and multiplication commutative, but not exponentiation?

We know that the addition and multiplication operators are both commutative, and the exponentiation operator is not. My question is why. As background there are plenty of mathematical schemes that ...
35
votes
6answers
3k views

What is the motivation for quaternions?

I know imaginary numbers solve $x^2 +1=0$, but what is the motivation for quaternions?
21
votes
2answers
3k views

Why are properties lost in the Cayley–Dickson construction?

Motivating question: What lies beyond the Sedenions? I'm aware that one can construct a hierarchy of number systems via the Cayley–Dickson process: $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H}...
28
votes
2answers
2k views

What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?

I'm starting to come around to an understanding of hypercomplex numbers, and I'm particularly fascinated by the fact that certain algebraic properties are broken as we move through each of the $2^n$ ...
2
votes
1answer
216 views

Is $\mathbb{R}^n$ a field?

Is $\mathbb{R}^n$ a field for all $n$? I suppose for n=1 and 2 the result is clear. What about higher values of $n$.
0
votes
1answer
153 views

Is there infinitely many “complex units”

As we know, $i$ = $\sqrt{-1}$, a simple complex unit. In complex space of two dimensions, you graph an axis of $a+bi$ where $i$ is your second dimension axis. Now, you also know, in three and four ...
1
vote
1answer
157 views

What properties do you lose when you extend your number set? [duplicate]

So in $\mathbb{R}$ and $\mathbb{C}$ you have both associative and commutative property, but as you extend to $\mathbb{H}$ you lose the commutative property, and $\mathbb{O}$ loses the associativity. ...