Question

Compute each of the following complex numbers, giving your answers in both rectangular and exponential forms. Sketch each complex number, on individual pairs of axes, and indicate on each plot the real part, imaginary part, magnitude, and phase in radians.(a) q = [(e - jπ)/(π - je)]^(2/9)(b) r = abcdf, wherea = √3(1 + j) + (1- j) d = 1 + j√3b = √3 + j f = jc = 1+ j

Answer 1

The complex numbers computed are:

A)
`q=0.8752+j0.4838=1e^(-j0.5049)`

B)
`r=-8-j8√(3) =16e^{j\pi (4)/(3)}`

The sketches are attached to this answer

Explanation:

To compute these complex numbers you have to remember these rules:

`Z=a+jb=(a^2+b^2)^{(1)/(2)}e^{jtan^(-1)(b/a)}` (a)

`Z=|z|e^(j\alpha)=|z|cos(\alpha)+j|z|sin(\alpha)` (b)

Also for multiplication, division, and powers, if W and U are complex numbers and k is a real number:

`{W}\cdot{U}=e^(j\alpha)=We^(j(\alpha+\beta))` (1)

`(W)/(U)=(|W|e^(j\alpha))/(|U|e^(j\beta))=(|W|)/(|U|)e^(j(\alpha-\beta))` (2)

`W^(k)=|w|^(k)e^(j(\alpha\cdot k))` (3)

With these rules we will do the followings steps:

for A:

1) We solve first the divition, writing the 2 complex numbers exponential form (equation (a)).

2) With the rule (2) we solve the division.

3) with rule (3) we solve the power.

For B:

1)We write the numbers a, b, c, d, and f in exponential form (equation (a)).

2) We use the rule (1) for the product.