Question A.1: Complex Arithmetic in Rectangular Form Given that Z1 = 5 + j...
Complex Arithmetic in Rectangular Form
Given that Z1 = 5 + j5 and Z2 = 3 – j4, reduce Z1 + Z2, Z1 – Z2, Z1Z2, and Z1/Z2 to rectangular form.
For the sum, we have
Z_1+Z_2=(5+j5)+(3-j4)=8+j1
Notice that we add (algebraically) real part to real part and imaginary part to imaginary part.
The difference is
Z_1-Z_2=(5+j5)-(3-j4)=2+j9
In this case, we subtract each part of Z2 from the corresponding part of Z1.
For the product, we get
Z_1Z_2=(5+j5)(3-j4)=15-j20+j15-j^2 20=15-j20+j15+20=35-j5
Notice that we expanded the product in the usual way for binomial expressions. Then, we used the fact that j2 = 1.
To divide the numbers, we obtain
\frac{Z_1}{Z_2}=\frac{5+j5}{3-j4}
We can reduce this expression to rectangular form by multiplying the numerator and denominator by the complex conjugate of the denominator. This causes the denominator of the fraction to become pure real. Then, we divide each part of the numerator by the denominator. Thus, we find that
\frac{Z_1}{Z_2}=\frac{5+j5}{3-j4}\times \frac{Z^*_2}{Z^*_2}=\frac{5+j5}{3-j4}\times \frac{3+j4}{3+j4}=\frac{15+j20+j15+j^2 20}{9+j12-j12-j^2 16}=\frac{15+j20+j15-20}{9+j12-j12+16}=\frac{-5+j35}{25}=-0.2+j1.4