If you want to understand the geometric basis for the square root of –1, start with http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.

If you want to understand the geometric basis for the square root of –1, start with http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf A complex number is most intuitively thought of as a quotient of Euclidean planar vectors, i.e. a quantity z = v / u with which a Euclidean vector can be multiplied to scale and rotate it into another vector: zu = (v / u)u = v(u \ u) = v. A unit bivector i then represents the quotient of two perpendicular vectors of the same magnitude. It naturally has the orientation of the plane spanned by the two vectors. Multiplying i by any vector in that plane serves to rotate it by a quarter turn. Multiplying a vector in the plane by i twice rotates by a half-turn.