Problem 151 - 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. Fifth roots of \(128(-1+i)\)

Step-by-Step Solution

Answer

The fifth roots of \(128(-1+i)\) in standard form are found by using the equations from De Moivre's Theorem and calculating for \(k = 0, 1, 2, 3, 4\). The roots are then represented graphically on a complex plane, and written in standard form \(a+bi\).

Step 1: Convert to Polar form

The complex number should be converted to its polar form which uses magnitude and angle instead of real and imaginary parts. The magnitude \(r\) of the complex number \(z = 128(-1+i)\) can be calculated with the formula \(r = \sqrt{Re^2 + Im^2}\). The angle \(\theta\) is determined by the arctangent formula \(\theta = \tan^{-1}(-1/1)\).

Step 2: Use De Moivre's Theorem

Using De Moivre's Theorem, we find the fifth roots by using the equation \(z_k = r^{1/n}[\cos((\theta + 2\pi k)/n)+ i \sin((\theta + 2\pi k)/n)]\) for \(k = 0, 1, 2, 3, 4\).

Step 3: Represent the roots graphically

Draw a complex plane, mark the origin (0,0), and represent each root as a point on this plane. Remember, the x-coordinate represents the real part of the root and the y-coordinate represents the imaginary part.

Step 4: Write roots in standard form

Lastly, write each of the roots in standard form \(a+bi\), where \(a\) is the real part and \(b\) is the imaginary part.

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful tool for finding roots and powers of complex numbers expressed in polar form. This theorem simplifies complex calculations by connecting trigonometry and complex algebra. To apply De Moivre's Theorem, a complex number must be in polar form, expressed as \( z = r(\cos \theta + i \sin \theta) \).

When finding the \(n\)-th root of a complex number, De Moivre's Theorem states:

• Identify the magnitude \(r\) and angle \(\theta\).
• The nth roots are given by: \[z_k = r^{1/n}\left[\cos\left(\frac{\theta + 2\pi k}{n}\right) + i\sin\left(\frac{\theta + 2\pi k}{n}\right)\right]\]
• Here, \(k = 0, 1, 2, \ldots, n-1\) will generate all distinct roots.

This means De Moivre's Theorem breaks down a complex problem into a series of manageable steps. Perfect for both powers and roots, it elegantly combines algebra with geometry. This theorem is especially useful for visualizing complex numbers, as it involves angles and magnitudes, which naturally translate into graphs.

Polar Form

Polar form is an alternate way of expressing complex numbers using a combination of a magnitude and an angle, rather than the traditional real and imaginary components. In polar form, a complex number \(z\) is represented as \(r(\cos \theta + i \sin \theta)\), where \(r\) is the modulus and \(\theta\) is the argument of \(z\).

The conversion from standard (Cartesian) form to polar form involves:

• Calculating the modulus: \(r = \sqrt{a^2 + b^2}\) for a complex number \(a+bi\).
• Finding the argument: \(\theta = \tan^{-1}(b/a)\), which is the angle made with the positive real axis.

Once in polar form, complex operations like multiplication, division, and taking roots become simpler by dealing with their magnitudes and angles. This form highlights the rotational aspect of complex numbers, making them ideal for problems involving periodic functions or rotations, like those solved using De Moivre's Theorem.

Graphical Representation of Roots

Representing complex roots graphically helps visualize their positions on the complex plane, enhancing understanding of how these roots are related to each other. This involves plotting each root as a point defined by its real and imaginary parts.

To graphically represent the roots:

• Draw the complex plane with the x-axis as the real axis and the y-axis as the imaginary axis.
• Calculate and plot each root from De Moivre's Theorem or polar form conversion, translating the polar coordinates into Cartesian by converting \((r, \theta)\) to \(x+yi\).
• Notice symmetrical patterns; for instance, fifth roots will often form a pentagon pattern around the origin since their angles are evenly distributed due to the periodic nature of sine and cosine functions.

This graphical method not only shows the real and imaginary components but also emphasizes periodicity and equidistance from origin, key features that these graphical patterns encode.

Standard Form of Complex Numbers

The standard form of a complex number, represented as \(a+bi\), combines real and imaginary numbers into a single expression. Here, \(a\) is the real part, and \(b\) is the imaginary part.

This Cartesian approach differs from polar form by explicitly expressing both halves of a complex number, providing:

• A straightforward visualization as a point on the complex plane, where the real part \(a\) is plotted on the x-axis and the imaginary part \(b\) on the y-axis.
• An immediate means to perform arithmetic operations, such as addition and subtraction, directly by combining corresponding parts.

The conversion of complex numbers from polar to standard form involves calculating the real and imaginary parts by:

• Real part: \( a = r \cos \theta \)
• Imaginary part: \( b = r \sin \theta \)

Standard form is particularly convenient for addition and subtraction, providing a straightforward representation of any complex number without involving angles or trigonometric functions.