Powers of Complex Numbers & Finding Principal Values

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  • 0:04 Complex & Principal Values
  • 0:32 Example 1
  • 5:04 Example 2
  • 6:49 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we look at powers of complex numbers and how to express results with principal values. Using two examples and a step-by-step approach, we show how this is done.

Complex & Principal Values

Complex numbers can be expressed in both rectangular form -- Z ' = a + bi -- and in polar form -- Z = re. The radius r and the angle θ may be determined from the a and the b of the rectangular form. For multiplying, dividing, and raising a complex number to a power, the polar form is preferred. In this lesson we will work two examples showing how to raise a complex number to a power.

Example 1

Find Z4 for Z = -1/√3 - i

Comparing to Z = a + bi, we see a = -1/√3 and b = -1. This complex number is in rectangular form.

Step 1: Convert to polar form (if necessary).

To convert to polar form, we need r and θ. Using the equation for r:


Before finding θ let's figure out which quadrant we're in. For both a and b negative, we are in the third quadrant. The angle θ is referenced to the horizontal positive real axis, but the angle α is the angle in the right triangle formed by the lengths of a and b.

The angle alpha

The tangent of α is the opposite side over the adjacent side; thus, tan α = |b| / |a|. Note: a length can't be negative, so we use absolute value signs to keep the numbers positive. Using the inverse tangent, tan-1, we can solve for α:


We can get to the same location by rotating clockwise with respect to the real axis. In this case, θ is negative.

The angles theta and alpha

Thus, θ = -180o + α = -180o + 60o = -120o.

From Z = re-iθ we get Z = (2/√3)e-i120o .

The argument of Z, abbreviated arg Z, is the angle θ. Since -π < θ ≤ π, the value of the angle satisfies the principal value requirement. To maintain unique arguments, the convention is to express angle θ between -π and π where π is 180o. Since -π and π are the same angle, the left-most value for θ is strictly less than π while the right-most value includes θ = π. When the arg Z is the principal value, we use the designation Arg Z.

The radius r = 1.15 is slightly greater than 1 and the angle θ = -120o. In the complex plane, the point locating the complex number Z is just outside the unit circle (the unit circle is a circle centered at the origin with radius r = 1):

Z in the complex plane

Step 2: Raise to the power n.

For n is 4, Zn is Z4:


The radius r gets raised to the 4th power, and the angle θ gets multiplied by 4. Now, 480o is greater than 360o, meaning the point has rotated fully around the circle back to where it started. To find the equivalent angle less than a full circle, keep subtracting 360o from 480o until the angle is less than a full circle 360o. In this example, we only have to subtract once.

-(480o - 360o) = -120o.

The radius r has grown from 1.15 to 16/9 = 1.78.

Plotting Z and Z4 in the same complex plane:

Z and Z^4

Step 3: Change to principal value (if necessary).

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