For a complex number z=a+biz=a+bi it can be represented as z=r(costheta+isintheta)z=r(cosθ+isinθ) where r=sqrt(a^2+b^2)r=√a2+b2 and theta=tan^-1(b/a)θ=tan−1(ba)
(2+i)/(14+9i)=(sqrt(2^2+1^2)(cos(tan^-1(1/2))+isin(tan^-1(1/2))))/(sqrt(14^2+9^2)(cos(tan^-1(9/14))+isin(tan^-1(9/14))))~~(sqrt5(cos(0.46)+isin(0.46)))/(sqrt277(cos(0.57)+isin(0.57)))2+i14+9i=√22+12(cos(tan−1(12))+isin(tan−1(12)))√142+92(cos(tan−1(914))+isin(tan−1(914)))≈√5(cos(0.46)+isin(0.46))√277(cos(0.57)+isin(0.57))
Given z_1=r_1(costheta_1+isintheta_1)z1=r1(cosθ1+isinθ1) and z_2=r_2(costheta_2+isintheta_2)z2=r2(cosθ2+isinθ2), z_1/z_2=r_1/r_2(cos(theta_1-theta_2)+isin(theta_1-theta_2))z1z2=r1r2(cos(θ1−θ2)+isin(θ1−θ2))
z_1/z_2=sqrt5/sqrt277(cos(0.46-0.57)+isin(0.46-0.57))=sqrt1385/277(cos(-0.11)+isin(-0.11))~~sqrt1385/277(0.99-0.11i)~~0.134-0.015iz1z2=√5√277(cos(0.46−0.57)+isin(0.46−0.57))=√1385277(cos(−0.11)+isin(−0.11))≈√1385277(0.99−0.11i)≈0.134−0.015i
Proof:
(2+i)/(14+9i)*(14-9i)/(14-9i)=(28-4i+9)/(14^2+9^2)=(37-4i)/277~~0.134-0.014i2+i14+9i⋅14−9i14−9i=28−4i+9142+92=37−4i277≈0.134−0.014i
