For questions on the octonions, a normed division algebra over the real numbers. It is a non-associative higher-dimensional analogue in the hierarchy of real, complex, and quaternionic numbers.

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Cayley-Dickson construction: a general rule for multiplying imaginary units?

The Cayley–Dickson construction (see refs below) is a way of generating 'algebras' (in the loose sense) of increasing size over the reals, obtaining a sequence of algebras $\mathbb R = R_0 ...
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Has the Riemann Hypothesis been generalized to the Octonions and the Quaternions?

I've noticed that it uses imaginary numbers. I know that sometimes when I have too few dimensions like (-1)^n, dots show where I might expect lines due to imaginary numbers. So perhaps there is a ...
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Inner product space over generalized number systems

Apologies for the lengthy setup, but I want to make sure I am clear on how I am using the notation, and what I mean by the phrase "generalized number system". Define a generalized number system $G$ ...
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Construction of Hyper-Complex Numbers

How does one construct a hyper-complex number multiplication table? For example: Quarternions: ...
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Proof that the subalgebra generated by any two elements of $\mathbb{O}$ is isomorphic to $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$

In the wikipedia atrticle (http://en.wikipedia.org/wiki/Octonion) it is stated that "one can show that the subalgebra generated by any two elements of $\mathbb{O}$ is isomorphic to $\mathbb{R}$, ...
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Unit or non-zero octonions form an $A_\infty$-space?

If I have a Moufang loop, can it have a classifying space? I'm thinking of the unit octonions, if that's too general, so perhaps a better question is: are the unit (or non-zero) octonions an ...
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Are roots of unity in hypercomplex algebras well defined?

While playing around with cyclotomic fields, I started to wonder about taking the roots of unity in higher dimensional analogues of the complex plane. Are the roots of unity well defined in the ...
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What can you do with octonions?

How can you calculate with them and what can you actually make up from the calculations? And what is exactly meant by normed division-algebras?
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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 $2^n$ ...
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Why do we start losing algebraic properties when dealing with hypercomplex numbers? [duplicate]

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?
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Trinonions, Quaternions, Quinonions, Sextonions, Septonions, Octonions

There are quaternions and octonions and even sextonions but what about trinonions, quinonions and septonions. Are there 3, 5, and 7 dimensional algebras which could be called trinonions, quinonions ...
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Pfister's 16-Square Identity and the norm of sedenions

Consider the sequence of numbers: complex numbers $\Bbb C$, quaternions $\Bbb H$, octonions $\Bbb O$, and sedenions $\Bbb S$. The Brahmagupta-Fibonacci 2-Square identity implies that the norm of the ...
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Are H$_n$(O) (n>3) Jordan algebras?

As we know, H$_3$(O) is a 27-dimensional exceptional Jordan algebra, here O is Cayley octonion algebra.But how about n>3? I guess that when n>3, H$_n$(O) are not Jordan algebras. But I only have a ...
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Fractal derivative of complex order and beyond

Is there some precise definition of "complex (fractal) order derivative" for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would ...
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218 views

Math beyond Quaternions

Quaternions remove the commutative property and octonions eliminate the associative property can we go any higher and eliminate more properties?
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What do we lose passing from the reals to the complex numbers?

As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we ...
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A question about hypercomplex numbers: quaternions, octonions etc [duplicate]

Possible Duplicate: Why is 8 so special? First of all let me state that I am not a mathematician but I work in computer science and engineering. I was reading about hypercomplex numbers, ...
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Constructing a parallelization of the 7-sphere.

I would like to show that $S^7$, the 7-sphere, is a parallelizable manifold. Let $\mathcal{O}$ be the octonions, the normed division algebra (noncommutative, nonassociative) over $H\times H$, where ...
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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 ...
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Matrix Representation of Octonions

Since quaternions $\mathbb{H}$ have a matrix representation as elements of $\text{SU}(2,\mathbb{C})$ as the following $$ 1 \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},\quad \mathrm ...
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Why is 8 so special?

I have been reading about multi-dimensional numbers, and found out that it's been proven that Octonions are the division algebra of the largest dimension. I was wondering why despite having infinitely ...
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Octonionic Hopf fibration and $\mathbb HP^3$

Real, complex and quaternionic versions of Hopf fibration ($S^0\to S^n\to\mathbb RP^n$, $S^1\to S^{2n+1}\to\mathbb CP^n$ and $S^3\to S^{4n+3}\to\mathbb HP^n$) give rise to spherical fibrations ...
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Nonzero Octonions as a 7-sphere

While reading about Moufang loops in the book "An introduction to Quasigroups and their Representations" by Smith, I've encountered the following statement: The set $ S^7 $ of nonzero octonions of ...
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What are some real-world uses of Octonions?

... octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative. Comes from a a quote by John Baez. Clearly, the sucessor to quaterions from the Cayley-Dickson process ...