## OctonionsOctonions 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.
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 x _{1} + x_{2}i
+ x_{3}j + x_{4}k + x_{5}{l} + x_{6}m +
x_{7}n + x_{8}o then the conjugate X* is x_{1} -
x_{2}i - x_{3}j - x_{4}k - x_{5}{l} - x_{6}m
- x_{7}n - x_{8}o There are many other things, such as the
modulus. ||X||^{2} = x_{1}^{2} + x_{2}^{2}
+ x_{3}^{2} + x_{4}^{2} + x_{5}^{2}
+ x_{6}^{2} + x_{7}^{2} + x_{8}^{2}.
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^{-1}X 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Home • Contact |