Solutions To Complex Numbers (Polar Form) - Part 1

 1)1) Express each of the following end results in the form: r exp(iq):

a) (2 + 3i)(1 – 2i)

b) (1 + i) (1- i)

c) (1 + Ö-3)²

Solutions:

a) Multiplying the two factors we get:

2 + 3i -6(i²) -4(i) = 8 + 3i – 4i = 8 –i

Next convert to standard polar form, given :

z = x + iy = 8 - i

so: r = r = [8² + (-1)²]^1/2 = Ö65  =   8.1

q = arg(z) = arctan (y/x) = arctan (-1/8) = -7.1 deg.

So: z = 8.1 [cos(-7.1) + isin(-7.1)] 

     Or: z = 8.1 exp (i(-7.1))


b) (1 + i) (1- i)

Expand to get: 1 + 1i - 1i -(i)(i)  = 1 - (-1) =  1 + 1 = 2

For which arg(z) = 0  (No imaginary component, i)

And arctan (0) = 0 hence there is no polar form.

But: z =  r exp (i 0) = Ö2 (1) = Ö2

c) (1 + Ö-3)²

Expand to get:

z = (1 + Ö-3) (1 + Ö-3)

But bear in mind that: (Ö-3) = [iÖ3)] so we have:

(1 + [iÖ3]) • (1 + [i Ö3]) = 1 + 2i[Ö3)] + (-1)3 = -2 + 2i[Ö3)]

z = -2 + 3.46i

r = [(-2)² + (2(iÖ3) ²] =  [4 + 36]^1/2 = [40]^1/2 = 6.3


q = arctan (y/x) = arctan (2[Ö3)]/ -2) = arctan (-Ö3) = (-60) = -p/3
Thus, z = 6.3 exp(i(-p/3))

2) Plot the results of (b) and (c) on the same Argand Diagram and obtain the resultant.

Solution: 

We plot the results of (b) and (c) on the Argand diagram and obtain the resultant

Call z(b) = 0 + Ö2 i = 1.4i (from that solution, i.e. z =  r exp (i 0) )

And z(c) = -2 + 3.46i

The plots of the two numbers are shown in the diagram. The resultant is sketched and can be confirmed from the algebra:

z(b) + z(c)= 1.4i + (-2 +3.46i) = -2 + 4.86i

This yields a parallelogram with points at: (0, 0), (0, 2), (-2, 4.86) and (-2, 3.46).