Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-22: Find the indicated roots, and represent them geometrically. The five fifth roots of \(1+i\)

Step-by-Step Solution

Answer

The five fifth roots of \(1+i\) are evenly spaced on a circle in the complex plane.

Step 1: Express the Complex Number in Polar Form

The complex number given is \(1+i\). Start by converting it to polar form. The magnitude \(r\) is calculated as \(r = \sqrt{1^2 + 1^2} = \sqrt{2}\). The argument \(\theta\) is found using \(\tan^{-1}(\frac{1}{1}) = \frac{\pi}{4}\). Thus, we express \(1+i\) as \(\sqrt{2}(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4})\).

Step 2: Find the Fifth Roots Using De Moivre’s Theorem

According to De Moivre’s Theorem, the \(n\)th root of a complex number \(r(\cos\theta + i\sin\theta)\) is given by \(r^{1/n}(\cos(\frac{\theta + 2k\pi}{n}) + i\sin(\frac{\theta + 2k\pi}{n}))\) for \(k = 0, 1, ..., n-1\). For the fifth roots, set \(n = 5\) and \(r = \sqrt{2}\) and solve: \(\sqrt[10]{2}(\cos(\frac{\pi + 2k\pi}{20}) + i\sin(\frac{\pi + 2k\pi}{20}))\).

Step 3: Calculate All Five Roots

Compute each root for \(k = 0, 1, 2, 3, 4\):- For \(k = 0\), the root is \(\sqrt[10]{2}(\cos\frac{\pi}{20} + i\sin\frac{\pi}{20})\).- For \(k = 1\), the root is \(\sqrt[10]{2}(\cos\frac{9\pi}{20} + i\sin\frac{9\pi}{20})\).- For \(k = 2\), the root is \(\sqrt[10]{2}(\cos\frac{17\pi}{20} + i\sin\frac{17\pi}{20})\).- For \(k = 3\), the root is \(\sqrt[10]{2}(\cos\frac{25\pi}{20} + i\sin\frac{25\pi}{20})\).- For \(k = 4\), the root is \(\sqrt[10]{2}(\cos\frac{33\pi}{20} + i\sin\frac{33\pi}{20})\).

Step 4: Represent Roots Geometrically

These roots can be represented on the complex plane as points evenly spaced around a circle centered at the origin with radius \(\sqrt[10]{2}\). Each point corresponds to one of the complex numbers calculated in Step 3 and forms the vertices of a regular pentagon when connected.

Key Concepts

Polar Form of Complex Numbers

Complex numbers can be expressed in different forms, one of which is the polar form. This is especially useful when dealing with multiplication, division, and finding roots of complex numbers. To convert a complex number to polar form, we rely on two key elements:

• The **magnitude** (or modulus), denoted by \(r\), which is the distance from the origin to the point on the complex plane.
• The **argument**, denoted by \(\theta\), which is the angle the line from the origin to the point makes with the positive x-axis.

For instance, consider the number \(1+i\). We calculate the magnitude as \(r = \sqrt{1^2 + 1^2} = \sqrt{2}\), and find the argument to be \(\theta = \tan^{-1}(\frac{1}{1}) = \frac{\pi}{4}\). Thus, the polar form of \(1+i\) is represented as \(\sqrt{2}(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4})\). This alternative representation makes it much easier to manipulate complex numbers in various calculations.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool to simplify the process of raising complex numbers to a power or extracting roots. According to this theorem, if you have a complex number in polar form, \(r(\cos\theta + i\sin\theta)\), its nth power or root is simplified to:

• **nth Power**: \(r^n(\cos(n\theta) + i\sin(n\theta))\)
• **nth Root**: \(r^{1/n}(\cos(\frac{\theta + 2k\pi}{n}) + i\sin(\frac{\theta + 2k\pi}{n}))\), for \(k = 0, 1, \ldots, n-1\)

For example, to find the fifth roots of the complex number \(1+i\), we utilize this theorem with \(n = 5\) and \(r = \sqrt{2}\). Substituting these into the nth root formula gives us roots based on different values of \(k\), specifically from 0 to 4. Each of these calculations will provide us with one of the fifth roots of \(1+i\). This method considerably simplifies what would otherwise be complex calculations.

Complex Plane Representation

Representing complex numbers on the complex plane helps us visualize operations in a geometric space. The complex plane consists of a real and an imaginary axis, similar to the x and y axes on a Cartesian plane. Each complex number corresponds to a point in this two-dimensional space.

• The **real part** of the complex number determines its position along the x-axis.
• The **imaginary part** determines its position along the y-axis.

When we calculate the roots of a complex number, as we did with the fifth roots of \(1+i\), each root can be mapped onto the complex plane.
These roots are points that, when connected, form symmetrical shapes—in this case, a regular pentagon surrounding the origin, as they are evenly spaced around the circle with radius equal to the modulus of each root, \(\sqrt[10]{2}\). This geometric representation makes it easier to understand the distribution and relationship between the roots visually.