Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Answer each of the following. For the real number \(1,\) find and graph all indicated roots. Give answers in rectangular form. (a) Fourth (b) Sixth

Step-by-Step Solution

Answer

(a) 1, i, -1, -i; (b) 1, \(\frac{1}{2}+\frac{\sqrt{3}}{2}i\), \(-\frac{1}{2}+\frac{\sqrt{3}}{2}i\), -1, \(-\frac{1}{2}-\frac{\sqrt{3}}{2}i\), \(\frac{1}{2}-\frac{\sqrt{3}}{2}i\).

Step 1: Determine what root means

When asked to find the fourth root or sixth root, it means we need to find a number that, when raised to the fourth or sixth power, yields 1.

Step 2: Find fourth roots of 1

The fourth roots of 1 are all complex numbers of the form \( e^{2\pi i k/4} \) for \( k = 0, 1, 2, 3 \). These correspond to \( 1, i, -1, \text{ and } -i \), which in rectangular form are \((1, 0), (0, 1), (-1, 0), (0, -1)\).

Step 3: Find sixth roots of 1

The sixth roots of 1 are given by \( e^{2\pi i k/6} \) for \( k = 0, 1, 2, 3, 4, 5 \). These points are equivalent to \( 1, \frac{1}{2} + \frac{\sqrt{3}}{2}i, -\frac{1}{2} + \frac{\sqrt{3}}{2}i, -1, -\frac{1}{2} - \frac{\sqrt{3}}{2}i, \frac{1}{2} - \frac{\sqrt{3}}{2}i \) in rectangular form.

Step 4: Graph the fourth roots

Graph these as points on the complex plane: \((1, 0), (0, 1), (-1, 0), (0, -1)\). They will form a square centered at the origin. Each point is equidistant from the origin.

Step 5: Graph the sixth roots

Graph the points \((1, 0),\) \((\frac{1}{2}, \frac{\sqrt{3}}{2}),\) \((-\frac{1}{2}, \frac{\sqrt{3}}{2}),\) \((-1, 0),\) \((-\frac{1}{2}, -\frac{\sqrt{3}}{2}),\) \((\frac{1}{2}, -\frac{\sqrt{3}}{2})\) on the complex plane. These form a hexagon centered at the origin.

Key Concepts

Fourth Roots

When we talk about the fourth roots of a number, such as 1, we refer to the set of numbers which, when raised to the power of four, will result in that number. For the number 1, it means finding the values for which their fourth power equals 1. In complex numbers, this involves numbers along the unit circle in the complex plane. The fourth roots of 1 can be expressed using exponentials as follows:

\( e^{2\pi i k/4} \) where \( k \) can be 0, 1, 2, or 3.
These represent:

• \( k = 0 \): \( e^{0} = 1 \)
• \( k = 1 \): \( e^{\pi i/2} = i \)
• \( k = 2 \): \( e^{\pi i} = -1 \)
• \( k = 3 \): \( e^{3\pi i/2} = -i \)

In rectangular form, these roots are points on the complex plane:

• (1, 0)
• (0, 1)
• (-1, 0)
• (0, -1)

These points form a square centered at the origin, each point being equidistant from the center.

Sixth Roots

In the realm of complex numbers, finding the sixth roots of 1 involves identifying numbers that, when powered to six, equal 1. Similar to the fourth roots, the sixth roots lie on the unit circle in the complex plane. The general expression can be written as:

\( e^{2\pi i k/6} \) where \( k \) ranges from 0 to 5.

This gives us six distinct complex numbers:

• \( k = 0 \): \( e^{0} = 1 \)
• \( k = 1 \): \( e^{\pi i/3} = \frac{1}{2} + \frac{\sqrt{3}}{2}i \)
• \( k = 2 \): \( e^{2\pi i/3} = -\frac{1}{2} + \frac{\sqrt{3}}{2}i \)
• \( k = 3 \): \( e^{\pi i} = -1 \)
• \( k = 4 \): \( e^{4\pi i/3} = -\frac{1}{2} - \frac{\sqrt{3}}{2}i \)
• \( k = 5 \): \( e^{5\pi i/3} = \frac{1}{2} - \frac{\sqrt{3}}{2}i \)

The rectangular coordinates, or the positions on the complex plane, are:

• (1, 0)
• (\( \frac{1}{2}, \frac{\sqrt{3}}{2} \))
• (\( -\frac{1}{2}, \frac{\sqrt{3}}{2} \))
• (-1, 0)
• (\( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \))
• (\( \frac{1}{2}, -\frac{\sqrt{3}}{2} \))

These points form a hexagon with the center at the origin on the complex plane.

Complex Plane

The complex plane, a crucial concept in complex numbers, resembles the traditional Cartesian coordinate system but specifically represents complex numbers. Each complex number corresponds to a point in this plane which blends real and imaginary number axes.

• Real Axis: This is the horizontal axis representing the real part of complex numbers.
• Imaginary Axis: This vertical axis showcases the imaginary component.

The position of any complex number \( a + bi \) in the complex plane is denoted by the point \((a, b)\), where:

• \( a \) is the real part, lying on the real axis.
• \( b \) represents the imaginary part, lying on the imaginary axis.

Visualizing roots on the complex plane helps understand the symmetrical properties they exhibit. For instance, the fourth and sixth roots of 1 form geometric shapes (a square and a hexagon, respectively) on this plane. The complex plane thus allows a pictorial view that simplifies the relationship between different complex numbers and their roots. This graphical representation aids immensely in grasping abstract mathematical concepts.

Rectangular Form

In mathematics, especially in the study of complex numbers, the rectangular form is a method for expressing complex numbers using their separate real and imaginary parts. A complex number in rectangular form is written as \( a + bi \), where \( a \) and \( b \) are real numbers.

• Real part \( a \): This is the number located on the real axis of the complex plane.
• Imaginary part \( b \): This lies on the imaginary axis and is always multiplied by \( i \), the imaginary unit.

The rectangular form is particularly useful when adding, subtracting, or plotting complex numbers because it directly correlates to the complex plane with coordinates \((a, b)\). It allows easy computation and visualization, serving as an alternate way to the polar form where complex numbers are represented using magnitude and angle. In exercises such as finding roots, converting into this form helps quickly identify each root's exact position in the complex plane, thus easing the process of graphing them.