Problem 63 - 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. $$z=\frac{\sqrt{2}}{2}(1+i)$$

Step-by-Step Solution

Answer

The powers of the complex number forms a square on the complex plane due to the rotation property of complex numbers.

Step 1: Calculate Powers

Firstly, calculate each of power of the complex number \(z\). 1. \(z=\frac{\sqrt{2}}{2}(1+i)\) 2. \(z^{2}=\left(\frac{\sqrt{2}}{2}(1+i)\right)^{2}=1 \) 3. \(z^{3}=\left(\frac{\sqrt{2}}{2}(1+i)\right)^{3}= \frac{\sqrt{2}}{2}(1+i) \) 4. \(z^{4}=\left(\frac{\sqrt{2}}{2}(1+i)\right)^{4}=1 \)

Step 2: Representation on Argand Diagram

Next, plot each power of z on a complex plane called an Argand diagram. This will involve placing points corresponding to each complex number on a graph, where the horizontal (x) axis represents the real part of the complex number and vertical (y) axis represents the imaginary part. The magnitude (or modulus) of the complex number is its distance from the origin.

Step 3: Describe Pattern

After graphically representing, an interesting pattern will be observed. The points for \(z, z^2, z^3, z^4\) forms a square shape on the plane. The shape is formed because the \(\frac{\sqrt{2}}{2}(1+i)\) represents a rotation of 45 degrees around the origin. Each subsequent power is simply rotating the original point by a further 45 degrees. This explains why after four rotations (i.e. \(z^4\)), we return to the starting position.

Key Concepts

Powers of Complex Numbers

When dealing with the powers of a complex number, you're essentially examining how repeated multiplication affects both the angle and magnitude of the number in the complex plane. Let's take the complex number \( z = \frac{\sqrt{2}}{2}(1+i) \). This complex number can be expressed in polar form as well, since its geometric properties reveal a rotation and scaling factor when plotted.

• The modulus, or magnitude, of \( z \) is \( \sqrt{z \cdot \overline{z}} = 1 \), meaning it lies on the unit circle in the complex plane.
• The argument (angle) of \( z \), given the form \( a + bi \), can be calculated with \( \text{arg}(z) = \tan^{-1}(1/1) = \frac{\pi}{4} \) radians or 45 degrees.

Each successive power of a complex number geometrically represents a rotation by a fixed angle. Thus, for \( z^2, z^3, \), etc., you repeatedly add the angle \( \frac{\pi}{4} \) radians to the original angle.

Argand Diagram

An Argand diagram is a graphical representation of complex numbers. It's like a coordinate plane but specifically for complex numbers.
Here:

• The horizontal axis (often called the real axis) represents the real part of any complex number.
• The vertical axis (known as the imaginary axis) stands for the imaginary part.

A complex number like \(z = a + bi\) is plotted by placing a point at coordinates \( (a, b) \). To understand the geometry of powers of a complex number, plotting them on an Argand diagram allows you to see the pattern and rotation around the origin, as seen with \(z, z^2, z^3, \text{and } z^4\). This visual approach is crucial for recognizing the recurrence and symmetry inherent to complex powers.

Complex Plane

The complex plane is an extension of the conventional two-dimensional coordinate plane used in algebra. It provides a visual method to represent and analyze complex numbers.
In the complex plane:

• The x-axis represents the real component.
• The y-axis represents the imaginary component.

Complex numbers are expressed as points or vectors in this plane. The advantage of using the complex plane is its ability to intuitively show operations like addition, subtraction, and particularly multiplication, which involve rotation and scaling. With our number \(z = \frac{\sqrt{2}}{2}(1+i)\), the plane helps visualize how powers of \(z\) map into new positions with each multiplication. This graphical relationship demonstrates key ideas in complex number theory.

Patterns in Complex Numbers

There are fascinating patterns present in the powers of complex numbers. When you calculate successive powers graphically or numerically, these patterns emerge. In our example with \(z\), the process is as follows:

• \(z\) itself represents a point on the unit circle.
• Each power of \(z\) rotates \(45^\circ\) counterclockwise on this unit circle.

This consistent rotation due to the fixed magnitude ensures that after a full 360-degree turn, the complex number returns to its starting position. This cycle results in a predictable rotation pattern that forms geometric shapes, like squares or polygons, based on the number of rotations and the specific angle. Recognizing these patterns allows more insight into the periodicity and symmetry inherent in complex number operations, and helps solve problems that involve powers or roots of complex numbers efficiently.