Problem 147 - 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 \(1.\)

Step-by-Step Solution

Answer

The fifth roots of 1 are \(1, e^{\frac{2πi}{5}}, e^{\frac{4πi}{5}}, e^{\frac{6πi}{5}}, e^{\frac{8πi}{5}}\). They can be represented as points on the unit circle in the complex plane forming a regular pentagon. In standard form, these roots are calculated as \(cos(\frac{2nπ}{5}) + isin(\frac{2nπ}{5})\), for \(n = 0, 1, 2, 3, 4\).

Step 1: Apply De Moivre's theorem

We start by expressing the number 1 in exponential form. 1 can be written as \(1 = e^{2nπi}\) for integer values of n because \(e^{2nπi} = cos(2nπ) + isin(2nπ)\) which always equals 1 for any integer n. The fifth roots of 1 are given by \(e^{\frac{2nπi}{5}}\) for \(n = 0, 1, 2, 3, 4\).

Step 2: Compute the roots

On using these values of n, we get these fifth roots; n = 0: \(e^{0} = 1\), n = 1: \(e^{\frac{2πi}{5}}\), n = 2: \(e^{\frac{4πi}{5}}\), n = 3: \(e^{\frac{6πi}{5}}\), n = 4: \(e^{\frac{8πi}{5}}\). These are the fifth roots of 1 in exponential form.

Step 3: Graphical Representation

We can plot these points on the unit circle of the complex plane. Since all our roots have magnitude 1, they will all lie on a circle of radius 1 centered at the origin. They will be equidistant from each other, forming a pentagon.

Step 4: Roots in Standard Form

Given roots in exponential form can be expressed in standard form \(a+bi\). For example, for n=1, \(e^{\frac{2πi}{5}} = cos(\frac{2π}{5}) + isin(\frac{2π}{5})\). This can be computed directly for each root, n.

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in mathematics, particularly when dealing with complex numbers. It allows us to raise complex numbers to a power or find their roots. This is done using trigonometric forms. The theorem states: if we have a complex number, say in polar form \[ z = r \cdot (\cos \theta + i \sin \theta) \]then raising it to the power of \( n \) is simple:\[ z^n = r^n \cdot \left(\cos(n\theta) + i \sin(n\theta)\right) \]This expression provides a straightforward way to find powers and roots of a complex number.

• Express the complex number in polar form.
• Apply the theorem to find powers or roots.

It's especially useful for complex roots of unity, that is, roots of numbers like 1.

Complex Roots

When we solve equations involving powers, such as finding the fifth root of 1, we often look at complex roots. These are the solutions involving complex numbers.The key to finding these roots lies in representing the original number in exponential form. For instance, a complex number, 1, can be written as\[ 1 = e^{2n\pi i} \]This is because the exponential form represents a complete circle in the complex plane.

• Every integer \( n \) will yield 1 in this form.

To find the fifth roots, you divide the angle by 5:\[ \text{Fifth Root: } e^{\frac{2n\pi i}{5}} \]for \( n = 0, 1, 2, 3, \) and \( 4 \). Each value of \( n \) gives us one of the distinct roots.

Unit Circle

The unit circle plays a fundamental role in understanding complex numbers, particularly when visualizing their roots. It is a circle with a radius of 1 centered at the origin in the complex plane.

• The circle represents all numbers with a magnitude (absolute value) of 1.

For complex roots of unity, each root lies on this unit circle. This circle helps to visually represent the position and angle of each root. For example, the fifth roots of unity form a pentagon on this circle as each root is evenly spaced at a certain angle from the others. This is a geometric interpretation of why these roots are equidistant and can be visualized easily on the unit circle.

Standard Form in Complex Numbers

The standard form for expressing complex numbers is of the style \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.Converting from exponential or polar form is straightforward using: \[ r \cdot (\cos \theta + i \sin \theta) = a + bi \]This equivalency allows for seamless transition between forms.

• Calculate \( a = r \cdot \cos \theta \)
• Calculate \( b = r \cdot \sin \theta \)

Remember, for roots of unity on the unit circle, \( r \) is always 1. Thus, \( \cos \theta \) is the real part and \( \sin \theta \) is the imaginary part of the standard form.This makes it easy to convert roots like \( e^{\frac{2\pi i}{5}} \) into its standard form by finding the respective cosine and sine values.