Agnijo's Mathematical Treasure Chest banner
Home > Abstract Algebra


Octonions are an extension of the quaternions. They were discovered by Arthur Cayley and also independently by John T. Graves, There are 8 main octonions: 1, i, j, k, {l}, m, n, o. (I put the letter l in braces to prevent confusion with the number 1 with this font. It is normally written without braces) Each octonion is the sum of a multiple of these. This makes adding or subtracting octonions trivial: you just need to add or subtract each of the parts together. For example, (1 + 3i + 4{l}) + (j + 2k - 3l) = 1 + 3i +j + 2k + {l}. Multiplication is MUCH harder. You will need to memorise a large table before trying to multiply octonions. Here is that table.

X 1 i j k {l} m n o
1 1 i j k {l} m n o
i i -1 k -j m -{l} -o n
j j -k -1 i n o -{l} -m
k k j -i -1 o -n m -{l}
{l} {l} -m -n -o -1 i j k
m m {l} -o n -i -1 -k j
n n o {l} -m -j k -1 -i
o o -n m {l} -k -j i -1

Octonions are NOT commutative. For example, ij = k but ji = -k. This is also true of quaternions. However, unlike quaternions, octonions are not even associative. Thus (ij)m = km = -n but i(jm) = io = n. You can also have octonion conjugates. If an octonion X is x1 + x2i + x3j + x4k + x5{l} + x6m + x7n + x8o then the conjugate X* is x1 - x2i - x3j - x4k - x5{l} - x6m - x7n - x8o There are many other things, such as the modulus. ||X||2 = x12 + x22 + x32 + x42 + x52 + x62 + x72 + x82. Also, the reciprocal X-1 is X*/||X||2. There are actually two distinct ways of dividing one octonion by another. Using left-division X/Y = Y-1X and using right-division X/Y = XY-1. Like quaternions, these give different answers. Also, if X and Y have no real part, you can have (XY - YX)/2. This is very useful as you can get cross-products in 7-dimensional space. This is an extension of the cross-product in ordinary 3-dimensional space, which is defined in exactly the same way using quaternions. There are many more discoveries about octonions.

Related entries

   • Complex Matters

   • Colour Octonions

   • Quaternions