Let  P  be a point in the  xy-plane, which corresponds to a complex number  z = x + iy.

In the polar coordinate system,

   and    ,

where  r  is the distance from the origin  0  to the point  P, and    is the angle that the ray  OP  makes with the positive direction of the  x-axis.

Therefore, a complex number  z = x + iy  can be written in the polar form as

.

The polar radius    is said to be the modulus.  The polar angle    is known as the argument (or phase) of the complex number  z.

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