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4.10: Powers and Roots of Complex Numbers

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Manually calculating (simplifying) a statement such as: (14 - 17i)^5 or \sqrt[4]{(3 - 2i)} in present (rectangular) form would be a very intensive process at best.

Fortunately you will learn in this lesson that there is an alternative: De Moivre's Theorem. De Moivre's Theorem is really the only practical method for finding the powers or roots of a complex number, but there is a catch...

What must be done to a complex number before De Moivre's Theorem can be utilized?

Watch This

This lesson reviews the two opposite operations involving De Moivre's Theorem: finding powers, and finding roots. The video lessons review each operation separately.

Embedded Video:

a. Raising Complex Numbers to a Power:

- James Sousa: De Moivre's Theorem

b. Finding Roots of Complex Numbers:

- James Sousa: Determining the Nth Roots of a Complex Number

Guidance

Powers of Complex Numbers

How do we raise a complex number to a power? Let’s start with an example

(-4 - 4 i ) 3 = (-4 - 4 i ) x (-4 - 4 i ) x (-4 - 4 i ) =

In rectangular form, this can get very complex. What about in r cis θ form?

(-4 - 4i) = 4\sqrt{2}\ cis \left ( \frac{5\pi}{4} \right )

So the problem becomes

4\sqrt{2} \ \mbox{cis} \left ( \frac{5\pi}{4} \right ) \cdot \ 4\sqrt{2} \ \mbox{cis}\left ( \frac{5\pi}{4} \right ) \cdot \ 4\sqrt{2} \ \mbox{cis}\left ( \frac{5\pi}{4} \right )

and using our multiplication rule from the previous section,

(-4 - 4i)^3 = (4\sqrt{2})^3 \ \mbox{cis}\left ( \frac{15\pi}{4} \right )

Notice, ( a + bi ) 3 = r 3 cis 3 θ

In words: Raise the r -value to the same degree as the complex number is raised and then multiply that by cis of the angle multiplied by the number of the degree.

Reflecting on the example above, we can identify De Moivre's Theorem :

Let z = r (cos θ + i sin θ ) be a complex number in rcisθ form. If n is a positive integer, z n is
z n = r n (cos( ) + i sin( ))

It should be clear that the polar form provides a much faster result for raising a complex number to a power than doing the problem in rectangular form.

Roots of Complex Numbers

I imagine you noticed long ago that when an new operation is presented in mathematics, the inverse operation often follows. That is generally because the inverse operation is often procedurally similar, and it makes good sense to learn both at the same time.

This is no exception:

The inverse operation of finding a power for a number is to find a root of the same number.
a) Recall from Algebra that any root can be written as x 1/ n
b) Given that the formula for De Moivre’s theorem also works for fractional powers, the same formula can be used for finding roots:
z^{1/n} = (a + bi)^{1/n} = r^{1/n} cis \left ( \frac{\theta}{n} \right )

Example A

Find the value of (1 + \sqrt{3}i)^4

r = \sqrt{(1)^2 + (\sqrt{3})^2} = 2

\mbox{tan} \ \theta_{ref} = \frac{\sqrt{3}}{1},

and θ is in the 1 st quadrant, so

\theta = \frac{\pi}{3}

Using our equation from above:

z^4 = r^4\ cis\ 4\theta

z^4 = (2)^4\ cis\ 4\frac{\pi}{3}

Expanding cis form:

z^4 = 16\left ( \mbox{cos} \left ( \frac{4\pi}{3} \right ) + i \ \mbox{sin}\left ( \frac{4\pi}{3} \right ) \right )

= 16\left ( (-0.5) - 0.866i \right )

Finally we have

z 4 = -8 - 13.856 i

Example B

Find \sqrt{1 + i}

Solution

First, rewriting in exponential form: (1 + i) ½

And now in polar form:

\sqrt{1 + i} = \left (\sqrt{2}\ cis \left (\frac{\pi}{4} \right ) \right )^{1/2}

Expanding cis form,

= \left (\sqrt{2} \left ( \mbox{cos} \left ( \frac{\pi}{4} \right ) + i \ \mbox{sin} \left ( \frac{\pi}{4} \right ) \right ) \right )^{1/2}

Using the formula:

=(2^{1/2})^{1/2} \left ( \mbox{cos} \left ( \frac{1}{2} \cdot \frac{\pi}{4} \right ) + i \ \mbox{sin} \left ( \frac{1}{2} \cdot \frac{\pi}{4} \right ) \right )

= 2^{1/4} \left ( \mbox{cos} \left ( \frac{\pi}{8} \right ) + i \ \mbox{sin} \left ( \frac{\pi}{8} \right ) \right )

In decimal form, we get

=1.189( 0.924 + 0.383 i )

=1.099 + 0.455i

To check, we will multiply the result by itself in rectangular form:

(1.099 + 0.455i)\ \cdot \ (1.099 + 0.455\mbox{i}) = 1.099^2 + 1.099(0.455i) + 1.099(0.455i) \ + (0.455i)^2

= 1.208 + 0.500i + 0.500i + 0.208i^2

= 1.208 + i - 0.208 \ \mbox{or}

= 1 + i

Example C

Find the value of x : x^3 = \left (1 - \sqrt{3}i \right )

Solution

First we put 1 - \sqrt{3}i in polar form.

Use x = 1, \ y  = -\sqrt{3} to obtain r = 2, \ \theta = \frac{5\pi}{3}

let z = \left ( 1 - \sqrt{3}i \right ) in rectangular form

z = 2\ \mbox{cis}\ \left ( \frac{5\pi}{3} \right ) in polar form

x = \left (1 - \sqrt{3}i \right )^{1/3}

x = \left [2cis \left ( \frac{5\pi}{3} \right ) \right ]^{1/3}

Use De Moivre’s Equation to find the first solution:

x_1 = 2^{1/3} cis \left ( \frac{5\pi /3}{3} \right ) or 2^{1/3} cis \left ( \frac{5\pi}{9} \right )

Leave answer in cis form to find the remaining solutions:

n = 3 which means that the 3 solutions are \frac{2\pi}{3} radians apart or

x_2 = 2^{1/3} cis \left ( \frac{5\pi}{9} + \frac{2\pi}{3} \right ) and x_3 = 2^{1/3} cis \left ( \frac{5\pi}{9} + \frac{2\pi}{3} + \frac{2\pi}{3} \right )

Note : It is not necessary to add \frac{2\pi}{3} again. Adding \frac{2\pi}{3} three times equals 2 π . That would result in rotating around a full circle and to start where it all began- that is the first solution.

The three solutions are:

x_1 = 2^{1/3} cis \left (\frac{5\pi}{9} \right )

x_2 = 2^{1/3} cis \left (\frac{11\pi}{9} \right )

x_3 = 2^{1/3} cis \left (\frac{17\pi}{9} \right )

Each of these solutions, when graphed will be \frac{2\pi}{3} apart.

Check any one of these solutions to see if the results are confirmed.

Checking the second solution:

x_2 = 2^{1/3} cis \left (\frac{11\pi}{9} \right )

= 1.260 \left [ \mbox{cos} \left (\frac{11\pi}{9} \right ) + i \ \mbox{sin} \left (\frac{11\pi}{9} \right ) \right ]

= 1.260[-0.766 - 0.643i]

= -0.965 - 0.810i

Does (-0.965 – 0.810 i ) 3 or (-0.965 – 0.810 i ) (-0.965 – 0.810 i ) (-0.965 – 0.810 i )

= \left (1-\sqrt{3}i\right )?

Concept question follow-up

A complex number operation written in rectangular form, such as: (13 - 4i)^3 must be converted to polar form to utilize De Moivre's Theorem.

Vocabulary

De Moivre's Theorem is the only practical manual method of identifying the powers or roots of complex numbers.

Guided Practice

1) What are the two square roots of i ?

2) Calculate \sqrt[4]{(1 + 0i)} What are the four fourth roots of 1?

3) Calculate (\sqrt3 + i)^7

Solutions

1) Let z = \sqrt{0 + i}

r = 1, \ \theta = \pi/2 or z = \left [1 \times \ cis \frac{\pi}{2} \right ]^{1/2} Utilizing De Moivre’s Theorem:
z_1 = \left [1 \times \ cis \frac{\pi}{4} \right ] or z_2 = \left [1 \times \ cis \frac{5\pi}{4} \right ]
z_1 = 1\left ( \mbox{cos}\frac{\pi}{4} + i \ \mbox{sin}\frac{\pi}{4} \right ) or z_2 = 1 \left (\mbox{cos} \frac{5\pi}{4} + i \ \mbox{sin} \frac{5\pi}{4} \right )
z_1 = 0.707 + 0.707i or z_2 = -0.707 - 0.707i

Check for z 1 solution: (0.707 + 0.707 i ) 2 = i?

0.500 + 0.500 i + 0.500 i + 0.500 i 2 = 0.500 + i + 0.500(-1) or i

2) Let z = 1 or z = 1 + 0 i Then the problem becomes find z 1/4 = (1 + 0 i ) 1/4

Since r = 1 \ \theta = 0, \ z^{1/4} = [1 \times cis \ 0]^{1/4} with z_1 = 1^{1/4} \left (\mbox{cos}\ \frac{0}{4} + i \ \mbox{sin}\ \frac{0}{4} \right ) or 1(1 + 0) or 1

That root is not a surprise. Now use De Moivre’s to find the other roots:

z_2 = 1^{1/4} \left [\mbox{cos}\left (0 + \frac{\pi}{2}\right ) + i \ \mbox{sin}\left (0 + \frac{\pi}{2} \right ) \right ] Since there are 4 roots, dividing 2π by 4 yields 0.5π
or 0 + i or just i z_3 = 1^{1/4} \left [\mbox{cos}\left (0 + \frac{2\pi}{2}\right ) + i \ \mbox{sin}\left (0 + \frac{2\pi}{2} \right ) \right ] which yields z 3 = -1
Finally z_4 = 1^{1/4} \left [\mbox{cos}\left (0 + \frac{3\pi}{2}\right ) + i \ \mbox{sin}\left (0 + \frac{3\pi}{2} \right ) \right ] or z_4 = -i

The four fourth roots of 1 are 1, i, -1 and -i

3) To calculate (\sqrt3 + i)^7 start by converting to rcis form First find r Recall r = \sqrt{\sqrt{3}^2 + 1^2}

r = \sqrt{3 + 1}
r = 2

If cos \theta = \frac{\sqrt3}{2} and sin \theta = \frac{1}{2} then \theta = 30^o and is in quadrant I. Now that we have trigonometric form, the rest is easy:

(\sqrt{3} + i)^7 = [2 (cos 30^o + i sin 30^o)]^7 ..... Write the original problem in r cis form
2^7[ (cos (7 \cdot 30^o) + i sin(7 \cdot 30^o)] ..... De Moivre's Theorem
128 [-\frac{\sqrt3}{2} + \frac{-1}{2}i] ..... Simplify
(\sqrt{3} + i)^7 = -64\sqrt3 - 64i ..... Simplify again

\therefore (\sqrt{3} + i)^7 = -64\sqrt3 - 64i

Practice

Perform indicated operation on these complex numbers:

  1. Divide: \frac{2 + 3i}{1 - i}
  2. Multiply: (-6 - i)(-6 + i)
  3. Multiply: \left (\frac{\sqrt{3}}{2} - \frac{1}{2} i \right )^2
  4. Find the product using polar form: (2 + 2i)(\sqrt{3} - i)
  5. Multiply: 2(\mbox{cos} \ 40^\circ + i \ \mbox{sin} \ 40^\circ) \bullet 4(\mbox{cos} \ 20^\circ + i \ \mbox{sin} \ 20^\circ)
  6. Multiply: 2 \left (\mbox{cos} \ \frac{\pi}{8} + i \ \mbox{sin} \ \frac{\pi}{8} \right ) \bullet 2 \left (\mbox{cos} \ \frac{\pi}{10} + i \ \mbox{sin} \ \frac{\pi}{10} \right )
  7. Divide: 2(\mbox{cos} \ 80^\circ + i \ \mbox{sin} \ 80^\circ) \div 6(\mbox{cos} \ 200^\circ + i \ \mbox{sin} \ 200^\circ)
  8. Divide: 3\ \mbox{cis}(130^\circ) \div 4\ \mbox{cis}(270^\circ)

Use De Moivre’s Theorem:

  1. [3(\mbox{cos} \ 80^\circ + i \ \mbox{sin} \ 80^\circ)]^3
  2. \left [\sqrt{2} \left (\mbox{cos}\ \frac{5\pi}{16} + i \ \mbox{sin} \ \frac{5\pi}{16} \right ) \right ]^4
  3. \left (\sqrt{3} - i \right )^6
  4. Identify the 3 complex cube roots of 1 + i
  5. Identify the 4 complex fourth roots of -16i
  6. Identify the five complex fifth roots of i

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Date Created:

Nov 01, 2012

Last Modified:

Aug 04, 2014

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