Let us write the two complex numbers in polar coordinates and let them be
#z_1=r_1(cosalpha+isinalpha)# and #z_2=r_2(cosbeta+isinbeta)#
Their multiplication leads us to
#r_1*r_2{(cosalphacosbeta-sinalphasinbeta)+(sinalphacosbeta+cosalphasinbeta)}# or
#r_1*r_2{(cos(alpha+beta)+sin(alpha+beta))#
Hence, multiplication of #z_1# and #z_2#, will be given by
#(r_1*r_2, (alpha+beta))#, so for multiplying complex numbers, take new angle as #(alpha+beta)# and modulus the product of the modulus of two numbers.
Similarly #(r_1(cosalpha+isinalpha))/(r_2(cosbeta+isinbeta))#
To simplify let us multiply numerator and denominator by denominator's conjugate #(r_2(cosbeta-isinbeta))#, then #z_1/z_2# is given by
#(r_1(cosalpha+isinalpha)*(r_2(cosbeta-isinbeta)))/(r_2(cosbeta+isinbeta)(r_2(cosbeta-isinbeta)))# which when simplified becomes
#(r_1*r_2(cosalphacosbeta+sinalphasinbeta)+i(sinalphacosbeta-cosalphasinbeta))/(r_2^2(cos^2beta+sin^2beta))# or
#(r_1/r_2)*(cos(alpha-beta)+isin(alpha-beta)# or
#z_1/z_2# is given by #(r_1/r_2, (alpha-beta))#, so for division complex number #z_1# by #z_2# , take new angle as #(alpha-beta)# and modulus the ratio #r_1/r_2# of the modulus of two numbers.
