z1 = Ö2(cos(-45) + isin(-45)) = Ö2 cis(-45)

by

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:

(z

_{1}/z_{2}) = (r_{1}cis(q_{1})/ r_{2 }cis(q_{2})) = (r_{1}/ r_{2}) cis (q_{1}– q_{2})
Now: (r1/ r2) = (1.414/ 3.6) = 0.39

And we saw previously:

(q

_{1}– q

_{2}) = arg(z

_{1}) – arg(z

_{2}) = (-45) – (56.3) = -101.3

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

_{1}– q_{2}).So:

(z

_{1}/ z

_{2}) = 0.39 (cos (-101.3) + isin(-101.3))

= 0.39((-0.195) + i(-0.98)) = -0.07 + 0.38i

What about? (1 + i) ¸ Ö3 – i

The first order of business is to get dividend and divisor each into polar form, specifically as a (cis) function:

Then (1 + i) = z

_{1}= x1 + iy1, so arg(z

_{1}) = arctan (y1/x1)

Further:

arctan (y1/x1) = arctan (1/1) = arctan (1) so q

_{1}= 45 deg

What about r1?

r1= [1

^{2}+ 1

^{2}]

^{1/2}= Ö2 = 1.4

so z

_{1}= 1.4 [cos (45) + isin(45)] = 1.4 cis(45)

Now: z

_{2}= Ö3 – i

So arg(z

_{2}) = arctan(y

_{2}/x

_{2}) = arctan(-1/ Ö3) so q

_{2}= (-30 deg)

And for r

_{2}: r

_{2}= [(Ö3)

^{2}+ (-1)

^{2}]

^{1/2}= Ö4 = 2

Then: z

_{2}= 2[cos(-30) +isin(-30)] = 2cis(-30)

We divide: (z

_{1}/z

_{2})

Which means dividing the r’s first:

r

_{1}/r

_{2}= Ö2/ 2

Then subtract angles: [(q

_{1}– q

_{2}) ] = {(45 deg) – (-30 deg)} = 75 degrees

So the end result of the division is:

(z

_{1}/z

_{2}) = Ö2/ 2 cis(75) = Ö2/ 2 {cos(75) + isin(75)}

= 0.707{cos(75) + isin(75)}

Since cos(75) = 0.258 and sin(75) =0.966, we have:

(z

_{1}/z

_{2}) = 0.707[(0.258) + i(0.966)] = 0.183 + 0.683i

Another very convenient way to express complex numbers is in the exponential form.

Thus, we can write:

*cos(q) + isin(q) = r exp (iq )*Thus, the previous numbers we divided (z

_{1 }and z

_{2}) may be expressed:

z

_{1}= Ö2 [cos (45) + isin(45)] = Ö2 exp (i p/4)

z2 = 2[cos(-30) +isin(-30)] = 2 exp(i (-p/6))

Problems for the Math Maven:

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}2) Plot the results of (b) and (c) on the same Argand diagram and obtain the resultant. Check algebraically!

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