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Octonions
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