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.

Question

Complex Arithmetic in Rectangular Form
Given that [latex]Z_1 = 5 + j5 ext{ and } Z_2 = 3 - j4[/latex], reduce [latex]Z_1 + Z_2 , Z_1 - Z_2 , Z_1Z_2 , ext{ and } Z_1 /Z_2[/latex] to rectangular form.

Accepted Answer

For the sum, we have

[latex]Z_1+Z_2=(5+5j)+(3-j4)=8+j1[/latex]

Notice that we add (algebraically) real part to real part and imaginary part to imaginary part.

The difference is

[latex]Z_1-Z_2=(5+5j)-(3-j4)=2+j9[/latex]

In this case, we subtract each part of [latex]Z_2[/latex] from the corresponding part of [latex]Z_1[/latex] .

For the product, we get

[latex]\begin{matrix} Z_1Z_2&=&(5+j5)(3-j4) \ &=&15-j20+j15-j^220 \ &=&15-j20+j15+20 \ &=&35-j5 \end{matrix} [/latex]

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

[latex]\frac{Z_1}{Z_2}=\frac{5+j5}{3-j4} [/latex]

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

[latex]\begin{matrix} \frac{Z_1}{Z_2} &=&\frac{5+j5}{3-j4} imes \frac{Z_2^\ast }{Z_2^\ast } \ &=&\frac{5+j5}{3-j4} imes \frac{3+j4}{3+j4} \ &=&\frac{15+j20+j15+j^220}{9+j12-j12-j^216} \ &=& \frac{15+j20+j15-20}{9+j12-j12+16} \ &=&\frac{-5+j35}{25} \ &=& -0.2+j1.4 \end{matrix} [/latex]

Question A.1: Complex Arithmetic in Rectangular Form Given that Z1 = 5 + j...

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