Is it possible to perform basic operations on complex numbers in polar form?

1 Answer
Jan 6, 2015

Yes, of course.

Polar form is very convenient to multiply complex numbers.
Assume we have two complex numbers in polar form:
#z_1=r_1[cos(phi_1)+i*sin(phi_1)]#
#z_2=r_2[cos(phi_2)+i*sin(phi_2)]#
Then their product is
#z_1*z_2=r_1[cos(phi_1)+i*sin(phi_1)]*r_2[cos(phi_2)+i*sin(phi_2)]#
Performing multiplication on the right, replacing #i^2# with #-1# and using trigonometric formulas for cosine and sine of a sum of two angles, we obtain
#z_1*z_2=r_1r_2[cos(phi_1+phi_2)+i*sin(phi_1+phi_2)]#
The above is a polar representation of a product of two complex numbers represented in polar form.

Raising to any real power is also very convenient in polar form as this operation is an extension of multiplication:
#{r[cos(phi)+i*sin(phi)]}^t=r^t[cos(t*phi)+i*sin(t*phi)]#

Addition of complex numbers is much more convenient in canonical form #z=a+i*b#. That's why, to add two complex numbers in polar form, we can convert polar to canonical, add and then convert the result back to polar form.
The first step (getting a sum in canonical form) results is
#z_1+z_2=[r_1cos(phi_1)+r_2cos(phi_2)]+i*[r_1sin(phi_1)+r_2sin(phi_2)]#

Converting this to a polar form can be performed according to general rule of obtaining modulus (absolute value) and argument (phase) of a complex number represented as #z=a+i*b# where
#a=r_1cos(phi_1)+r_2cos(phi_2)# and
#b=r_1sin(phi_1)+r_2sin(phi_2)#

This general rule states that
#z=r[cos(phi)+i*sin(phi)]# where
#r=sqrt(a^2+b^2)# and
angle #phi# (usually, in radians) is defined by its trigonometric functions
#sin(phi)=b/r#,
#cos(phi)=a/r#
(it's not defined only if both #a=0# and #b=0#).
Alternatively, we can use these equations to define angle #phi#:
If #a!=0#, #tan(phi)=b/a#. Or, if #b!=0#, #cot(phi)=a/b#.