Complex numbers in the form a+bia+bi can be represented as:
z=r(costheta+isintheta)z=r(cosθ+isinθ)
Where:
bbr=sqrt(a^2+b^2)r=√a2+b2
bbtheta=arctan(b/a)
We now put our complex numbers in this form:
Let:
z_1=5+4i and z_2=6+i
For z_1
r=sqrt((5)^2+(4)^2)=sqrt(41)
theta=arctan(4/5)=38.66
z_1=sqrt(41)(cos(38.66)+isin(38.66))
For z_2
r=sqrt((6)^2+(1)^1)=sqrt(37)
theta=arctan(1/6)=9.46
z_2=sqrt(37)(cos(9.46)+isin(9.46))
Addition is the same as for that of complex numbers in the form a+bi
:.
a_1+b_1i +a_2+a_2i=(a_1+a_1) +(b_1+b_2)i
z_1+z_2
sqrt(41)(cos(38.66)+sqrt(37)(cos(9.46)=11.00002714
isqrt(41)(sin(38.66)+isqrt(37)sin(9.46)=4.999773551i
Rounding to 2 .d.p.
11.00+5.00i=11+5i
Notice the result is exactly the same as adding in rectangular form.
(5+4i) +(6+i)=(5+6)+(6+1)i=11+5i
