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Complex numbers along with the Quaternions are the only finite dimensional associative division algebras that properly contain the reals. The interesting thing
by yequalsx 9y ago
Complex numbers along with the Quaternions are the only finite dimensional associative division algebras that properly contain the reals. The interesting thing is that the 2d real space can be equipped with a multiplication operation that works. This isn’t always possible to do on a finite dimensional real vector space.
It happens to be true that this field called the complex numbers is a two dimensional real vector space. It took humans quite a long time to come up with 0 and negative numbers. There’s no way the jump to the complex numbers was going to be as easy and it isn’t as simple as, “numbers in 2d”. Why isn’t 3D a finite dimensional associative division algebra?
- Meegul 9y agoYou speak of a multiplication operation that "works", also stating that it's not always possible for a finite dimensional real vector space. Doesn't the definition of a real vector space require multiplication that "works" (through the scalar multiplication requirement) or are you referring to the multiplication between vectors within that space "working"?
- yequalsx 9y agoMultiplication of vectors. That’s the meaning of the algebra word in the phrase, finite dimensional associative division algebra. In R^3 you can’t define multiplication of vectors in a way that works. That is, in a way that each vector can be multiplied by itself and in which each vector has a multiplicative inverse while preserving the real vector space structure.
- Meegul 9y agoThanks! I'm still in the process of really grokking linear algebra as I only just took a class on it, so this is helpful.