Problem 64 - Precalculus with Limits

Question

Graphing Powers of a Complex Number In Exercises 63 and 64 , represent the powers \(z, z^{2}, z^{3},\) and \(z^{4}\) graphically. Describe the pattern. $$z=\frac{1}{2}(1+\sqrt{3} i)$$

Step-by-Step Solution

Answer

The powers of the complex number \(z = \frac{1}{2}(1 + \sqrt{3} i)\), when represented graphically, are located at the vertices of a regular hexagon. The pattern is that for \(z = \cos(\theta) + i\sin(\theta)\) the powers of \(z\) will produce a regular polygon with n-vertices, rotating counterclockwise as the power increases.

Step 1: Convert the complex number into Polar Form

It is easier to deal with powers of a complex number when the number is in polar form. The given complex number \(z=\frac{1}{2}(1+\sqrt{3} i)\) can be written in polar form using the formula \(z= r (\cos(\theta) + i \sin(\theta))\), where \(r = \sqrt{\text{real part}^2 + \text{imaginary part}^2}\) and \(\theta = \arctan(\text{Imaginary Part}/\text{Real Part})\). For the given number, r becomes \(\frac{1}{2} \sqrt{1^2 + (\sqrt{3})^2} = 1\) and \(\theta = \arctan (\sqrt{3}) = \frac{\pi}{3}\). So polar form is \(z = \cos (\frac{\pi}{3}) + i \sin (\frac{\pi}{3})\)

Step 2: Compute the powers of z using De Moivre's Theorem

De Moivre's Theorem can be used to simply the multiplication of complex numbers in polar form. The theorem states that \((\cos(\theta) + i \sin(\theta))^n = \cos(n\theta) + i \sin(n\theta)\). This means that to find the powers of z (or any complex number in polar form) you just multiply the angle by the power. Using this theorem the powers become: \(z^2 = \cos(2*\frac{\pi}{3}) + i \sin(2*\frac{\pi}{3}) = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\), \(z^3 = \cos(3*\frac{\pi}{3}) + i \sin(3*\frac{\pi}{3}) = -1\), and \(z^4 = \cos(4*\frac{\pi}{3}) + i \sin(4*\frac{\pi}{3}) = -\frac{1}{2} - i\frac{\sqrt{3}}{2}\)

Step 3: Represent the powers graphically

The complex numbers can be represented graphically on the complex plane with x-axis as the real part and y-axis as the imaginary part. Plot the points \(z=(\frac{1}{2}, \frac{\sqrt{3}}{2})\), \(z^2=(-\frac{1}{2}, \frac{\sqrt{3}}{2})\), \(z^3=(-1, 0)\), and \(z^4=(-\frac{1}{2}, -\frac{\sqrt{3}}{2})\)

Step 4: Describe the pattern

Looking at the graph, we can see that these points are all located at the vertices of a regular hexagon (a six-sided polygon with all sides equal). The powers get plotted along the vertices of the hexagon with subsequent vertices visited as the power increases. Thus, the pattern appears to be that for \(z = \cos(\theta) + i\sin(\theta)\), the powers of \(z\) will produce a regular polygon with n-vertices, rotating counterclockwise as the power increases.

Key Concepts

Polar Form

Complex numbers can be a bit tricky to handle when it comes to multiplication and power calculations. Luckily, they can be expressed neatly in something called polar form. This is especially useful for dealing with powers of a complex number, such as those in our exercise.
You see, every complex number can be transformed into the polar format: \[ z = r (\cos(\theta) + i \sin(\theta)) \] Here, \( r \) is the magnitude (or modulus) of the complex number, and \( \theta \) is the angle (or argument) with the positive x-axis in the complex plane.

• To find \( r \), use the formula \[ r = \sqrt{\text{real part}^2 + \text{imaginary part}^2} \]
• The angle \( \theta \) is derived using \( \arctan(\text{Imaginary Part}/\text{Real Part}) \).

In our example, converting the complex number \( z = \frac{1}{2}(1+\sqrt{3} i) \) gives us \( r = 1 \) and \( \theta = \frac{\pi}{3} \), thus transforming it into polar form as \( \cos \left( \frac{\pi}{3} \right) + i \sin \left( \frac{\pi}{3} \right) \).
Polar form simplifies many operations, especially raising numbers to powers, as it neatly separates the magnitude from the angle.

De Moivre's Theorem

When you need to compute powers of complex numbers, De Moivre's Theorem is a valuable ally. This theorem elegantly handles the multiplication of complex numbers in polar form.
It states that for a complex number in polar form: \[ (\cos(\theta) + i \sin(\theta))^n = \cos(n\theta) + i \sin(n\theta) \] Practically, this means you only have to multiply the angle \( \theta \) by the power \( n \) to find the angle of the resultant complex number.

• For \( z^2 \), multiply \( \theta \) by 2 resulting in \( 2 \times \frac{\pi}{3} \).
• For \( z^3 \), multiply \( \theta \) by 3 resulting in \( 3 \times \frac{\pi}{3} \).
• For \( z^4 \), multiply \( \theta \) by 4 resulting in \( 4 \times \frac{\pi}{3} \).

In our example, the corresponding results in rectangular form are \( z^2 = -\frac{1}{2} + i \frac{\sqrt{3}}{2} \), \( z^3 = -1 \), and \( z^4 = -\frac{1}{2} - i \frac{\sqrt{3}}{2} \).
So, thanks to De Moivre's Theorem, computing these powers becomes as simple as multiplying angles.

Complex Plane Graphing

Once you have calculated the powers using De Moivre's Theorem, plotting them on the complex plane provides an intuitive visual understanding. The complex plane is much like the Cartesian plane, except here every point corresponds to a complex number.
- The horizontal axis (real axis) represents the real part - The vertical axis (imaginary axis) represents the imaginary part For \[ z = \frac{1}{2}(1+\sqrt{3} i) \] and its powers, you get different complex number coordinates:

• \( z = (\frac{1}{2}, \frac{\sqrt{3}}{2}) \)
• \( z^2 = (-\frac{1}{2}, \frac{\sqrt{3}}{2}) \)
• \( z^3 = (-1, 0) \)
• \( z^4 = (-\frac{1}{2}, -\frac{\sqrt{3}}{2}) \)

These points can be plotted on the plane to visualize how they relate to each other. Graphically representing these powers can show interesting patterns, simply by observing their arrangement on the complex plane.

Regular Polygon Patterns

By plotting the powers of a complex number on the complex plane, distinctive polygon patterns emerge. For complex numbers with modest magnitudes (like our example \( z \)), these points often form the vertices of regular polygons as they grow in power.
In our exercise, the points can be seen arranging themselves into a hexagon on the complex plane. This is because the polar form representation, once powered, repeats every full circle (or \( 2\pi \) radians) leading to symmetrical distributions.

• The powers \( z, z^2, z^3, \) and \( z^4 \)
form consecutive vertices of a hexagon (a regular 6-sided polygon).

These regular polygon formations occur because each additional power amounts to an equal rotation in the plane. Thus, each power incrementally rotates the number, plotting another vertex of the polygon.
This beautiful symmetry in the pattern showcases a wonderful aspect of complex numbers: connect the dots between each power, and the pattern resembles the elegant contours of a regular polygon, illustrating the harmony hidden within the complex numbers themselves.