The Polar Form of a Complex Number

Figure 7.3 Polar form of complex numbers

This latter form is usually abbreviated to Z=r∠θ, and is called the polar form of a complex number.

r is called the modulus (or magnitude of Z) and is written as mod Z or |Z|. r is determined from Pythagoras’s theorem on triangle OAZ:

The modulus is represented on the Argand diagram by the distance OZ. θ is called the argument (or amplitude) of Z and is written as arg Z. θ is also deduced from triangle OAZ: arg Z=θ=tan^-1y/x.

For example, the cartesian complex number (3+j4) is equal to r∠θ in polar form, where  and,

Hence, 

Similarly, (–3+j4) is shown in Figure 7.3(b),

where, 

and, 

Hence, (–3+j4)=5∠126.87