Complex Numbers (III)

Fig. 1: This finds the resultant of two complex numbers (A, B from earlier example). Note how it can be done graphically - as well as algebraically!

From the diagram (see previous blog):

A= z1 = -2 + 2i

B = z2 = -2 -3i


To get the resultant: z1 + z2 = z3:

A + B = z1 + z2 =[ (-2 + 2i) + (-2 – 3i)] = -4 –i

BUT – this could also have been obtained by completing the parallelogram and finding the resultant of A + B, graphically (See. Fig. 1).

In any case:

z3 = x3 + iy3 = -4 –i

and: arg(z3) = arctan(y3/x3) = arctan(-1/-4) = arctan(0.25)

so theta = arg(z3) = 14 deg.

The value of r3= [x3^2 + y3^2]^1/2 = [(-4)^2 + (-1)^2]^1/2 = [17]^1/2 = 4.1

So: z3 = 4.1 cos(14) + isin(14) = 4.1 cis(14)

What about now combining this resultant z3 with that from C (see original diagram) which we call z4, to obtain z5?

As we saw: C = z4 = 4 + 3i

Thence: z5 = [z3 + z4] = [(-4 – i) + (4 + 3i)] = 0 + 2i

WHERE would this resultant be and what is arg(z5)?

We will obtain this in the final instalment on basic complex number operations, but now go on to division:

We want to divide: z1 = -2 + 2i by z2 = -2 - 3i

As we saw from before, these may be written:

z1 = 1.414 (cos(-45) + isin(-45)) = 1.414 cis(-45)

and

z2 = 3.6(cos(56.3) + isin(56.3) = 3.6 cis(56.3)


In all such cases of complex division we require that the z, r in the denominator not be zero.

Thus:

(z1/z2) = (r1 cis(theta1)/ r2 cis(theta2)) = (r1/ r2) cis (theta1 – theta2)

Now: r1/ r2 = (1.414/ 3.6) = 0.39

And we saw previously:

(theta1 – theta2) = arg(z1) – arg(z2) = (-45) – (56.3) = -101.3

Thus, the basic procedure for division entails dividing the lengths (r’s) and subtracting the angles (theta).

So:

(z1/ z2) = 0.39 (cos (-101.3) + isin(-101.3))

= 0.39((-0.195) + i(-0,98)) = -0.07 + 0.38i

For next time: I invite readers to divide:

(1 + i) by [3]^1/2 – i

Until then!