Problem 66 - Precalculus with Limits

Question

In Exercises 65 and 66, represent the powers \(z\), \(z^2\), \(z^3\), and \(z^4\) graphically. Describe the pattern. \(z\ =\ \dfrac{\sqrt{1}}{2}(1\ +\ \sqrt{3}i)\)

Step-by-Step Solution

Answer

The pattern for the powers is that each power 'z' rotates through an angle of 60° in the complex plane. As a result, 'z' is \(e^{i60}\), 'z^2' is \(e^{i120}\), 'z^3' is \(e^{i180}\), and 'z^4' is \(e^{i240}\).

Step 1 - Calculating Power Values

Firstly, need to compute four power values for 'z'. Since \(z = \frac{\sqrt{1}}{2}(1 + \sqrt{3}i)\), can use the Polars form of z: \(z = re^(iΘ)\) where r = modulus of z, and Θ = argument of z. Modulus r = 1 (as \(\sqrt{2^2 + (-1^2)} = 1\)) and Θ = 60° (as atan(1/1)). So, \(z = e^{i60}\).\n\nNow,\n\n'z' = \(e^{i60}\),\n\n'z^2' = \(e^{i120}\),\n\n'z^3' = \(e^{i180}\),\n\n'z^4' = \(e^{i240}\)

Step 2 - Graphical Representation of Power Values

In the next step, represent these powers of 'z' graphically in the complex plane. Can use the polar plot to visualize these values. Because Argument Θ increases by 60° each time. Graphically, it's equivalent to rotating through an angle of 60° in the complex plane counter-clockwise.

Step 3 - Describe the Pattern

Looking at the values and the graphical representation, it can be seen that there is a shift of 60° angle for each power. This means, multiplying by z rotates a point through an angle of 60° in the complex plane. Hence, such rotation represents the pattern for the powers of 'z'.

Key Concepts

Understanding Complex Numbers

In mathematics, complex numbers represent a significant expansion from the one-dimensional number line of real numbers to a two-dimensional plane. They consist of two parts: a real part and an imaginary part, and are typically written as a + bi, where a is the real component and bi is the imaginary component with i being the imaginary unit satisfying i^2 = -1.

Complex numbers can be added, subtracted, multiplied, and divided, following specific rules based on combining their real and imaginary parts. They're essential in various fields including engineering, physics, and applied mathematics because they make many equations easier to solve. Complex numbers also play a crucial role in understanding oscillations, waves, and other phenomena that involve periodicity.

Polar Form Representation

The polar form of a complex number is another way to represent complex numbers, which is quite useful when dealing with complex powers and roots. While the standard form is described as a + bi, the polar form employs a radial distance (r, the modulus) and an angle (θ, the argument) to describe the position of the complex number in the two-dimensional plane. The polar form is written as r(cos θ + i sin θ) or re^{iθ}.

• r is found by calculating the square root of the sum of squares of the real and imaginary parts.
• θ can be determined by the function known as the arctangent, which compares the imaginary and real parts of the number.

Using Euler's formula e^{iθ} = cos θ + i sin θ, the expression simplifies complex number calculations, especially when multiplying or taking powers of complex numbers. Polar form makes it clear that multiplying a complex number by itself corresponds to increasing the angle, which is key for understanding the graphical representation of complex powers.

Complex Plane Rotations

Rotations in the complex plane are a geometric interpretation of multiplying complex numbers, particularly in their polar forms. A rotation is performed by altering the angle θ of the complex number while keeping the modulus r constant. For the powers of a complex number, such as with our given number z, each power corresponds to the rotation in the complex plane by a certain angle.

With each successive power of z, the angle of the original complex number is increased by its own argument, which for the exercise given was 60 degrees. Therefore, raising z to the second power (z^2) rotates it by 60 degrees again, totaling a 120-degree rotation from the positive real axis. This pattern continues with further powers, leading to the understanding that a complex number raised to a power results in the rotation of that number in the complex plane by multiples of its original argument. The result is a sequence of points on the complex plane that show a clear rotational pattern, a concept that is key in fields such as phasor analysis in electrical engineering.