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A comment on A Visual, Intuitive Guide to Imaginary Numbers (2007). Parent comment

Re:A Visual, Intuitive Guide to Imaginary Numbers (2007) by yiyus on Friday August 10, 2018 @06:54PM (#17737285)
You want geometric algebra. In 2D GA, you would have two unitary vectors, i and j, such that i * i = 1 and j * j = 1. The (non-commutative) product of between them (or their division) would be a bivector: i * j = ij = - j * i. You rotate 90 degrees the i and j vectors using the bivector.

The good thing about GA is that the same concept can be easily extended to 3D (quaternions), and in fact to 4D and nD.

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Re:A Visual, Intuitive Guide to Imaginary Numbers (2007) by westoncb on Friday August 10, 2018 @08:13PM (#17737575)
That does sound pretty close to what I had in mind. Are i and j from your example associated with orthogonal axes in a plane?
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Re:A Visual, Intuitive Guide to Imaginary Numbers (2007) by yiyus on Saturday August 11, 2018 @05:55AM (#17739076)
Yes, they are. i would be the unitary vector in the horizontal direction, x, and j in the vertical one, y.

In fact, geometric algebras are very general, and the one I briefly sketched is not the only possible interpretation. If you are interested, you can find many good introductions online, directed at different audiences. You can also search for the term "Clifford algebras" if you are interested in a more formal approach.

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