The Argument of a Complex Number

Any complex numbercan be written in the formwhere

and

if

if

if

if

is called the argument of the complex number z and may be writtenIt is actually the angle taken clokwise from the positive real axis to the line drawn from the origin to the point z. The argument of a complex number has the property that it can increase by any multiple ofand still return the same complex number, since each increase in the argument byrotates the complex number once about the origin and returns it to the same point.

We can define the argument of a point relative to any point in the complex plane. Relative to the pointthe argument ofis writenand is the angle between the line fromto and a line drawn fromin the direction of the positive real axis.