POWERS AND ROOTS OF COMPLEX NUMBERS

10. POWERS AND ROOTS OF COMPLEX NUMBERS

Using the rules (9.6) for multiplication and division of complex numbers, we have (10.1)

z n = (re iθ ) n =r n e inθ

for any integral n. In words, to obtain the nth power of a complex number, we take the nth power of the modulus and multiply the angle by n. The case r = 1 is of particular interest. Then (10.1) becomes DeMoivre’s theorem:

(10.2) (e iθ ) n = (cos θ + i sin θ) n = cos nθ + i sin nθ.

You can use this equation to find the formulas for sin 2θ, cos 2θ, sin 3θ, etc. (Prob- lems 27 and 28).

The nth root of z, z 1/n , means a complex number whose nth power is z. From (10.1) you can see that this is

z 1/n = (re iθ ) 1/n =r 1/n e iθ/n = n

θ r cos + i sin

This formula must be used with care (see Examples 2 to 4 below). Some examples will show how useful these formulas are.

Example 1.

[cos(π/10) + i sin(π/10)] 25 = (e iπ/10 ) 25 =e 2πi e iπ/2 = 1 · i = i.

Section 10 Powers and Roots of Complex Numbers

Example 2. Find the cube roots of 8. We know that 2 is a cube root of 8, but there are also two complex cube roots of 8; let us see why. Plot the complex number 8 (that is, x = 8, y = 0) in the complex plane; the polar coordinates of the point are r = 8,

and θ = 0, or 360 ◦ , 720 , 1080 , etc. (We can use either degrees or radians here; read the end of Section 3.) Now by equation (10.3), z 1/3 =r 1/3 e iθ/3 ; that is, to

find the polar coordinates of the cube root of a number re iθ , we find the cube root √ of r and divide the angle by 3. Then the polar coordinates of 3 8 are

(10.4) ◦ r = 2, θ=0 , 360 /3, 720 /3, 1080 /3 · · ·

) and the point (2, 360 ) are the same. The points in (10.4) are all on a circle of radius 2 ◦ ◦ ◦

We plot these points in Figure 10.1. Observe that the point (2, 0 ◦

and are equally spaced 360 /3 = 120 apart. Starting with θ = 0, if we add 120 repeatedly, we just repeat the three angles shown. Thus, there are exactly three cube roots for any number z, always on a circle of radius 3 spaced 120 ◦ apart.

Now to find the values of

8 in rectangular

form, we can read them from Figure 10.1, or we can calculate them from z = r(cos θ + i sin θ) with

r = 2 and θ = 0, 120 ◦ = 2π/3, 240 = 4π/3. We can also use a computer to solve the equation

z 3 = 8. By any of these methods we find

Example 3. Find and plot all values of −64. From Figure 10.2 (or by visualizing a plot of −64), we see that the polar coordinates of −64 are r = 64, θ = π + 2kπ

√ 4 (where k = 0, 1, 2, · · · ). Then since z

Figure 10.2 Figure 10.3

=r e , the polar coordinates of −64 are

4 4 4 4 4 4 4 4 We plot these points in Figure 10.3. Observe that they are all on a circle of √ radius 2

2, equally spaced 2π/4 = π/2 apart. Starting with θ = π/4, we add π/2

66 Complex Numbers Chapter 2 √ 4

repeatedly, and find exactly 4 fourth roots. We can read the values of −64 in rectangular form from Figure 10.3:

√ 4 −64 = ±2 ± 2i (all four combinations of ± signs)

or we can calculate them as in Example 2, or we can solve the equation z 4 = −64 by computer.

Example 4.

−8i. The polar coordinates of −8i are r = 8, √ 6 θ = 270 + 360 k = 3π/2 + 2πk. Then the polar coordinates of −8i are

Find and plot all values of 6

In Figure 10.4, we sketch a circle of radius

2. On

it we plot the point at 45 ◦ and then plot the rest of the 6 equally spaced points 60 ◦ apart. To find the roots in rectangular coordinates, we need to find all the values of r(cos θ + i sin θ) with r and θ given by (10.5). We can do this one root at a time or more simply by using a computer to solve the equation

z 6 = −8i. We find (see Problem 33) √

Figure 10.4 ± {1 + i, 1.366 − 0.366i, 0.366 − 1.366i}. √ n

Summary In each of the preceding examples, our steps in finding re iθ were: (a) Find the polar coordinates of the roots: Take the nth root of r and divide

θ + 2kπ by n.

(b) Make a sketch: Draw a circle of radius n r, plot the root with angle θ/n, and then plot the rest of the n roots around the circle equally spaced 2π/n apart. Note that we have now essentially solved the problem. From the sketch you can see the approximate rectangular coordinates of the roots and check your answers in (c). Since this sketch is quick and easy to do, it is worthwhile even if you use a computer to do part (c).

(c) Find the x + iy coordinates of the roots by one of the methods in the examples. If you are using a computer, you may want to make a computer plot of the roots which should be a perfected copy of your sketch in (b).

PROBLEMS, SECTION 10

Follow steps (a), (b), (c) above to find all the values of the indicated roots. √ 3 √ 3 √ 4

Section 11 The Exponential and Trigonometric Functions

27. Using the fact that a complex equation is really two real equations, find the double angle formulas (for sin 2θ, cos 2θ) by using equation (10.2).

28. As in Problem 27, find the formulas for sin 3θ and cos 3θ.

29. Show that the center of mass of three identical particles situated at the points

z 1 ,z 2 ,z 3 is (z 1 +z 2 +z 3 )/3.

30. Show that the sum of the three cube roots of 8 is zero.

31. Show that the sum of the n nth roots of any complex number is zero.

32. The three cube roots of +1 are often called 1, ω, and ω 2 . Show that this is reasonable, that is, show that the cube roots of +1 are +1 and two other numbers, each of which is the square of the other.

33. Verify the results given for the roots in Example 4. You can find the exact √ values in terms of

3 by using trigonometric addition formulas or more easily by using a computer to solve z 6 = −8i. (You still may have to do a little work by hand to put the computer’s solution into the given form.)