[adrian_b]

The complex numbers are 2-dimensional, but their 2 dimensions are not the 2 dimensions of a normal 2-dimensional geometric plane, they are 2 other dimensions.

The 2 dimensions of a geometric plane correspond to the 2 orthogonal translations of the plane.

The 2 dimensions of a complex number do not correspond to translations, but to scalings and rotations of the geometric plane.

The multiplication of the complex numbers corresponds to the composition of scalings and plane rotations, which are invertible operations and that is why the set of complex numbers is a commutative field, unlike the set of points of a geometric plane, which does not have such an algebraic structure.

The set of complex numbers can be viewed as a plane, but it must be kept in mind that this plane is a distinct entity from a geometric plane.

(The Cartesian product of a geometric plane with a complex plane forms a geometric algebra with 4 dimensions.)

[ogogmad]

> The 2 dimensions of a complex number do not correspond to translations, but to scalings and rotations of the geometric plane.

Similar story for other 2D number systems:

For the dual numbers, they express scalings and "Galilean" rotations (i.e. shears).

For the double numbers, they express scalings and "Minkowski" rotations (i.e. Lorentz boosts).

Unfortunately, some of the nice theory of the complex numbers doesn't generalise easily to the dual numbers or double numbers. I'm thinking specifically of complex analysis which is very, very nice, and much nicer than real analysis. But I think these planar number systems have their own intriguing character: For instance, see "screw theory" and "automatic differentiation" for two distinctive applications of the dual numbers.

[niknoble]

> The 2 dimensions of a complex number do not correspond to translations, but to scalings and rotations of the geometric plane.

I disagree. When multiplying complex numbers, they are viewed as scalings with rotation like you say. But when adding them, they are viewed as translations. If you only took one of these perspectives, then complex numbers would be very simple and not express anything interesting. Their complexity and power comes from alternating between the scaling with rotation perspective (multiplication) and the translation perspective (addition).

[boppo1]

>The 2 dimensions of a complex number do not correspond to translations, but to scalings and rotations of the geometric plane.

So, polar coordinates basically?

[morokhovets]

That's exactly what I was trying to convey, thanks.