4 ms·
That’s really misleading. The complex numbers are not a two-dimensional Euclidean space directly, but are a space of transformations (scaling & rotation) on two
by jacobolus 9y ago
That’s really misleading. The complex numbers are not a two-dimensional Euclidean space directly, but are a space of transformations (scaling & rotation) on two-dimensional Euclidean vectors, where 1 represents the identity transformation, and i represents a quarter turn anticlockwise.
In a similar way, the quaternions are the space of transformations (scaling & rotation) of three-dimensional vectors. (It’s a little more complicated because 3-dimensional rotations are not commutative, and must be combined by sandwiching, so there are 2 choices of quaternion corresponding to every scale and orientation in 3-dimensional space. For an introduction see http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf)
- eriknstr 9y ago>The complex numbers are not a two-dimensional Euclidean space directly, but are a space of transformations (scaling & rotation) on two-dimensional Euclidean vectors, where 1 represents the identity transformation, and i represents a quarter turn anticlockwise. You might be right, I don't know. Could you explain a bit more how you mean?
- hcs 9y agoI suspect the important point is that rotations cycle around (keep rotating and you come back to where you started)
- jacobolus 9y agoComplex numbers have multiplication defined on them. If you multiply two complex numbers, you get another complex number. (They compose via multiplication in exactly the same way as scaling & rotation operators on 2-dimensional vectors.) If you just have 2-dimensional vectors, there’s no obviously well-defined way to multiply two vectors and get out another vector. In other words, both 2-dimensional vectors and complex numbers are made up of 2 coordinates, but they don’t have the same mathematical structure.