The imaginary number 3 + 4i is represented on the complex plane as (3, 4). It could also be written as 5(cos 53o + isin 53o). Cos 53o + isin 53o can be abbreviated as cis53o where cis stands for cos + isin. This conversion is done the same as converting regular Cartesian coordinates to polar coordinates.

It is easy to add or subtract complex numbers, but multiplying and dividing them is more complicated.

For example (3 + 4i) x (8 – 6i) = (3)(8) + (3)(-6i) + (4i)(8) + (4i)(-6i)

= 24 – 18i + 32i + 24

= 48 + 14i

This was the same as multiplication of binomials. So how would it look in polar form? First, convert the standard form to polar form. 3 + 4i = 5(cos 53o + isin 53o).

For example 5(cos 53o + isin 53o) x 10(cos -37o + isin -37o) =50[(cos 53o)(cos -37o) + (cos 53o)(isin -37o) + (isin 53o)(cos -37o) + (isin 53o)(isin -37o)

Rearranging and remembering i2 = -1 =50[(cos 53o)(cos -37o) – (sin 53o)(sin -37o)] + i[(cos 53o)(sin -37o) + (sin 53o)(cos -37o)]

Then using double angle formulas =50[cos (53o + -37o) + isin(53o + -37o)]

Giving this as the answer in polar form =50[cos 16o + isin16o]

Which is the same as above in standard form. = 48 + 14i

DEMOIVRE’S THEOREM

Abraham DeMoivre developed a theorem to with complex numbers raised to a power. It is stated thusly:

If z = a + bi is any complex number with trig form r (cos + isin) and n is any positive integer, then the nth power z is given by

zn = rn (cos n + isin n).

Here is an example.

Find [2(cos 45o+ isin 45o)]6. Write the result in standard form.

[2(cos 45o + isin 45o)]6 =26[cos 6(45o) + isin 6(45o)]

=26[cos(270o) + isin(270o)]

=64[0 + -1i]

= -64i

Now work these problems and turn in the solutions to your teacher.

Write 2 – 7i in polar coordinates on the complex plane.

Find (1 + i)8 using DeMoivre’s Theorem.

I. Write 2 – 7i in polar coordinates on the complex plane.

II. Find ( 1 + i) ^8using DeMoivre’s theorem.