Modulus and Argument

Thinking in terms of the Argand diagram we can specify the position of the complex number z = x + jy on the plane by giving the polar coordinates of the point (x, y).

Figure 10.3: The modulus - argument representation of z.

The polar coordinate r is the distance from O to P and is called the modulus of the complex number z and written as | z|.

r = | z| = =

The polar coordinate is called an argument of z. If we take in the range - < then we call it the (principal) argument of z and we denote it by arg(z). Note that any argument of z differs from arg(z) by an integer multiple of 2 (working in radians) or of 360^o (working in degrees)10.2.

Since x = r cos and y = r sin we can write z in terms of its modulus and argument as

This is called writing z in polar form or modulus - argument form. Any non-zero complex number can be written in this form. The point 0 is a slightly special case, it has r = 0 but the angle is not defined.