[technical stuff] An algebra is just a vector space V with a bilinear operation * :VxV->V. [/technical stuff] There are plenty of interesting and intuitive a

[technical stuff] An algebra is just a vector space V with a bilinear operation * :VxV->V. [/technical stuff] There are plenty of interesting and intuitive algebras other than the stuff you get from the Cayley-Dickson constructions. Some interesting algebras: * you can have some fun making the imaginary unit i square to 0 instead of -1. The resulting algebra is called the dual numbers, and has some surprising properties. You can gain a bit of understanding of it by using analogies with the complex numbers. * You can make i square to +1 without being equal to +1 or -1. Again you can gain some understanding using analogies with the complex numbers. It's perhaps not the most useful examples because it's isomorphic to R \oplus R. * Another family of examples, a very important one actually, is the algebra of 2x2 or 3x3 or nxn matrices over the real numbers or the complex numbers. The first two examples are useful for understanding the general concept of a quotient ring. They're hardly exhaustive, but they are easy to picture. Now I can return to your claim about CD: What makes the Cayley-Dickson family significant is that it produces all the division algebras over the real numbers, which are the algebras for which division by nonzero elements is always possible. [edit] Needed to escape an asterisk. Third time lucky. Corrected grammar.