If we have a complex number such that z=a+biz=a+bi, then z=r(costheta+isintheta)z=r(cosθ+isinθ), where:
- r=sqrt(a^2+b^2)r=√a2+b2
- theta=tan^(-1)(b/a)θ=tan−1(ba)
For z_1=7+4iz1=7+4i :
r=sqrt(7^2+4^2)=sqrt(49+16)=sqrt(65)r=√72+42=√49+16=√65
theta=tan^(-1)(4/7)~~29.7θ=tan−1(47)≈29.7
=sqrt(65)(cos(29.7)+isin(29.7))=√65(cos(29.7)+isin(29.7))
For z_2=2+6iz2=2+6i :
r=sqrt(2^2+6^2)=sqrt(4+36)=sqrt(40)r=√22+62=√4+36=√40
theta=tan^(-1)(6/2)~~71.6θ=tan−1(62)≈71.6
=sqrt(40)(cos(71.6)+isin(71.6))=√40(cos(71.6)+isin(71.6))
For z_1+z_2z1+z2 :
z_1+z_2=sqrt(65)cos(29.7)+isqrt(65)sin(29.7)+sqrt(40)cos(71.6)+isqrt(40)sin(71.6)z1+z2=√65cos(29.7)+i√65sin(29.7)+√40cos(71.6)+i√40sin(71.6)
color(white)(z_1+z_2)=sqrt(65)cos(29.7)+sqrt(40)cos(71.6)+i(sqrt(65)sin(29.7)+sqrt(40)sin(71.6))z1+z2=√65cos(29.7)+√40cos(71.6)+i(√65sin(29.7)+√40sin(71.6))
color(white)(z_1+z_2)=8.999470954+9.995734316iz1+z2=8.999470954+9.995734316i#
color(white)(z_1+z_2)~~9+10iz1+z2≈9+10i
Proof:
7+4i+2+6i=(7+2)+i(4+6)=9+10i7+4i+2+6i=(7+2)+i(4+6)=9+10i
