Octonion

< 1 2 3 >

Numbers with eight dimensions are called octonions. You write them as

w = a + bi + cj + dk + ep + fq + gr + hs

and consists of the real numbers a, b, c, d, e, f, g and the units i, j, k, p, q, r, s. The set of octonions is indicated by the symbol O.

Explanation

When multiplying the units i, j, k, p, q, r, s the order plays a role.

1 i j k p q r s

i −1 k −j q −p −s r

j −k −1 i r s −p −q

k j −i −1 s −r q −p

p −q −r −s −1 i j k

q p −s r −i −1 −k j

r s p −q −j k −1 −i

s −r q p −k −j i −1

History

Britisch mathematician Arthur Cayley first published on octonons in 1843.

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1	i	j	k	p	q	r	s
i	−1	k	−j	q	−p	−s	r
j	−k	−1	i	r	s	−p	−q
k	j	−i	−1	s	−r	q	−p
p	−q	−r	−s	−1	i	j	k
q	p	−s	r	−i	−1	−k	j
r	s	p	−q	−j	k	−1	−i
s	−r	q	p	−k	−j	i	−1