Imaginary numbers aren't a field, so there's no such thing. Clifford algebras over the complex numbers work fine, but it's usually not what the people talking about "geometric algebra" are doing.
How is complex / imaginary numbers not what they are doing? Numbers that square to -1, 0, and 1 are the bread and butter of the GA I know. Exploring different combinations of types of imaginary numbers and their products and space describing algebras. (including naturally quaternions, duel quaternions, i-rotation, nilpotent, ect)
Typically GA people are working with real algebras, meaning the coefficients are real, and things like a square root of -1 appear as some object in the algebra (like a 2-blade). But you could also have a Clifford algebra with coefficients in e.g. the complex numbers or fields of finite characteristic.
In fact using different coefficient rings is one way to write a compact recursive definition of real Clifford algebras:
You seem to be calling complex numbers imaginary numbers, but they're not the same thing. Imaginary numbers are a subset of the complex numbers consisting of the imaginary axis without 0, e.g. i, 2i, -3.1i. Complex numbers also include the real numbers and all combinations of real and imaginary.
koolala|1 year ago
ndriscoll|1 year ago
In fact using different coefficient rings is one way to write a compact recursive definition of real Clifford algebras:
http://blog.sigfpe.com/2006/08/geometric-algebra-for-free_30...
mkl|1 year ago