Transformations on the Argand Diagram

Conjugation

\(z_1\mapsto z_1^*\): reflect in the real axis.

Negation

\(z_1\mapsto-z_1\): rotate anticlockwise by \(\pi\), or reflect in the origin.

Addition and subtraction

Adding \(z_2=x_2+y_2i\) translates by \((x_2,y_2)\); subtracting \(z_2\) translates by \((-x_2,-y_2)\).

Multiplication by i

\(z_1=x_1+y_1i\mapsto iz_1=-y_1+x_1i\): rotate anticlockwise by \(\dfrac{\pi}{2}\).