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nyssos parent
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.
koolala
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)
ndriscoll
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:

http://blog.sigfpe.com/2006/08/geometric-algebra-for-free_30...

koolala
As this shows, ei and ej are imaginary numbers. This confusion is the biggest issue in GA to me. Typically Linear Algebra users don't use imaginary numbers either. It's all connected when doing geometry though and its pointless to draw invisible lines between these concepts.

We have no way to talk about more general types of complex / imaginary numbers besides rigid math lingo that provides no geometric intuition or grace for geometric imagination.

mkl
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
I'm glad you seem to know what I mean. I love imaginary numbers like the Square Root of -1 or the Square Root of 0. They can only be used like:

nil²=0

i²=-1 j²=-1

And combining them into complex forms like:

(i + j)

(nil + i)

What can I call this general idea of using imagination to determine new number rules and combining them together?

If I call this GA or Clifford Algebra in a math community will it trigger rigor admins to ban me from talking because I'm not using their terms?

I wish we had artistic imaginary math communities for exploring Geometry without rigor turing everything into Semantics. Geometry literally doesn't need semantics if you agree on points, lines, planes ect. Algebra to me should just be simple maps from clifford numbers to examples of intuitive geometry / physics.

bmacho
People should stop calling that set "imaginary numbers". It's net bad. That set isn't really worth it to be called anything. Maybe "pure imaginary numbers". Also "imaginary numbers" is used for the complex numbers already.
koolala
People also like colloquially saying "The Square Root of -1". Technically wrong but personally meaningful. GA is mainstream once normal people start joking about the square root of zero. We need to imagine and teach even more imaginary numbers one day.

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