The imaginary operator i rotates the vector OP into OP", and – on a second application – rotates OP" into OP'.
An imaginary operator, also called an i-operator or j-operator, is the part of a complex number that defines the magnitude of the part of the complex number at right angles to the real number part. It may be understood as follows: consider a vector OP where O is the origin of a set of Cartesian coordinates and P is the point (a, b) – see diagram. Multiplying OP by -1 will produce a vector OP'; that is, it will rotate OP through 180° such that OP' = -OP. Now consider a number i (called the imaginary operator) such that multiplying OP by i produces OP", a vector perpendicular to OP. Multiplying OP" by i will produce OP'.