Problem 142 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use the theorem on page 590 to find the indicated roots of the complex number, (b) represent each of the roots graphically, and (c) write each of the roots in standard form. Fourth roots of \(625 i\)

Step-by-Step Solution

Answer

The fourth roots of \(625i\) are \(5 (\cos (\frac{\pi}{8}) + i \sin (\frac{\pi}{8}))\), \(5 (\cos (\frac{5\pi}{8}) + i \sin (\frac{5\pi}{8}))\), \(5 (\cos (\frac{9\pi}{8}) + i \sin (\frac{9\pi}{8}))\), \(5 (\cos (\frac{13\pi}{8}) + i \sin (\frac{13\pi}{8}))\). They are represented graphically in the complex plane, and their standard forms can be computed using the given identities.

Step 1: Identify the complex number and convert it to polar form

Identify the complex number \(625i\) from the problem statement and understanding that it can be written in polar form as \(r (\cos\theta + i \sin\theta)\), where \(r=|625i|=625\) is the modulus of the complex number and \(\theta = \arg(625i) = \frac{\pi}{2}\) is the argument.

Step 2: Apply De Moivre's Theorem

De Moivre's theorem states that \( (r (\cos\theta + i \sin\theta))^n = r^n (\cos(n\theta) + i \sin(n\theta)) \). For n fourth roots, we can say n=4. So \( z^4 = 625 (\cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2})) \). Therefore, the roots are: \( z_k = 625^{1/4} (\cos( \frac{\frac{\pi}{2} + 2k\pi}{4} ) + i \sin( \frac{\frac{\pi}{2} + 2k\pi}{4} )) \) for \( k = 0, 1, 2, 3 \).

Step 3: Compute the roots

Substitute the values of k into the equation from step 2 to get the fourth roots of \(625i\). The roots will be: \[ z_0 = 5 (\cos (\frac{\pi}{8}) + i \sin (\frac{\pi}{8})) \]\[ z_1 = 5 (\cos (\frac{5\pi}{8}) + i \sin (\frac{5\pi}{8})) \]\[ z_2 = 5 (\cos (\frac{9\pi}{8}) + i \sin (\frac{9\pi}{8})) \]\[ z_3 = 5 (\cos (\frac{13\pi}{8}) + i \sin (\frac{13\pi}{8})) \]

Step 4: Represent roots graphically

Plot each of the roots on the complex plane. Each root corresponds to a point in the plane, with the real part of the root as the x-coordinate and the imaginary part as the y-coordinate. Remember to label each root.

Step 5: Write roots in standard form

Express each root in standard form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. To do this, use the equation \(\cos\theta = \cos(-\theta)\) and \(\sin(-\theta) = -\sin\theta\).

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in the realm of complex numbers. It connects complex numbers in polar form with their powers and roots.
According to the theorem, if you have a complex number in the polar form \[ z = r (\cos\theta + i\sin\theta) \] and raise it to the power of \( n \), it becomes:\[ z^n = r^n (\cos(n\theta) + i\sin(n\theta)) \].
This principle is crucial when you're finding the roots of complex numbers.Here's what's useful about De Moivre’s Theorem:

• It simplifies the computation of roots and powers of complex numbers.
• It connects trigonometry (through sine and cosine) and exponential growth.
• It provides a geometric perspective when representing complex numbers on the plane.

By implementing this theorem, you can break down problems involving complex numbers into more manageable parts, like determining their roots.

Polar Form

The Polar Form of a complex number is a way to express the number in terms of its magnitude and angle.
In polar form, a complex number \( z \) is expressed as:\[ z = r (\cos\theta + i\sin\theta) \]where:

• \( r \) is the modulus of the complex number, calculated as \( r = |z| \).
• \( \theta \) is the argument, or the angle measured from the positive real axis to the line representing the complex number.

Polar Form captures the essence of the number's position on the complex plane:

• The modulus \( r \) gives the distance from the origin.
• The argument \( \theta \) specifies the direction from the origin.

This form allows for easier manipulation when multiplying or taking roots. Most importantly, it simplifies many calculations due to its alignment with Euler's formula.

Complex Plane

The complex plane is a two-dimensional plane where complex numbers are graphically represented.
It’s also known as the Argand plane after Jean-Robert Argand, who introduced it. In this plane:

• The horizontal axis (x-axis) represents the real part of the complex number.
• The vertical axis (y-axis) displays the imaginary part.

Each complex number corresponds to a point on this plane. Understanding the complex plane helps in:

• Visualizing complex numbers and their operations, like addition or multiplication.
• Seeing the geometric aspect of operations such as rotations and scaling.

When working through problems, converting numbers to a form that allows for plotting on the complex plane can be enormously helpful. Each root or power calculated using polar coordinates has a clear representation within the plane, showing its magnitude and direction.

Roots of Complex Numbers

Finding the roots of complex numbers can initially seem daunting. However, using the polar form and De Moivre's theorem makes it a seamless task.
The nth roots of a complex number \( z \) are given by:\[ z_k = r^{1/n} (\cos(\frac{\theta + 2k\pi}{n}) + i\sin(\frac{\theta + 2k\pi}{n})) \]where \( k = 0, 1, 2, ..., n-1 \).Here's how to approach finding roots:

• Determine the modulus \( r \) and argument \( \theta \) of the original complex number.
• Divide the argument by the number of roots you are finding \( n \).
• Calculate \( n \) roots by adjusting the argument by adding multiples of \( 2\pi \) divided by \( n \).

Each root represents a distinct point on the complex plane. They are often symmetrically distributed around the origin. Once you have these roots, you can convert them to standard form to find their real and imaginary parts. This method not only gives you the numerical roots but provides a full graphical representation on the complex plane.