Problem 91 - Precalculus with Limits

Question

Finding the \(n\) th Roots of a Complex Number \(\operatorname{In}\) Exercises \(81-96,(\) a) use the formula on page 446 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 \( 1, e^{2\pi i/5}, e^{4\pi i/5}, e^{-4\pi i/5}, e^{-2\pi i/5} \) in rectangular form.

Step 1: Convert to polar form and Identify the modulus and argument

The complex number 1 can be represented in polar form as \( r (cos(\theta) + i sin(\theta)) \) where \( r = 1 \) and \( \theta = 0 \) because the argument of a real number is zero and its modulus is the absolute value of the number.

Step 2: Use De Moivre's theorem to find the n-th roots

According to De Moivre's Theorem, an n-th root of a complex number in polar form is given by \( r^{1/n} [ cos( (\theta + 2k\pi) / n ) + i sin( (\theta + 2k\pi) / n ) ] \) where \( k = 0, 1, ... , n-1 \). As we have identified \( r = 1 \) and \( \theta = 0 \), the fifth roots are calculated by inserting these values to the theorem for \( k = 0, 1, 2, 3, 4 \).

Step 3: Convert back to rectangular form

Although the roots are now found, they are still in polar form. Convert these back to rectangular form by recognizing that \( e^{i\theta} = cos(\theta) + isin(\theta) \).

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in mathematics used to find the roots of complex numbers. It connects complex numbers and trigonometry in a beautiful way. The theorem states that if you have a complex number in polar form:

• \( r (\cos(\theta) + i \sin(\theta)) \)

then its \( n \)-th root is given by:

• \[ r^{1/n} [ \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) ] \]

where \( k = 0, 1, 2, \ldots, n-1 \).

In simple terms, De Moivre's Theorem helps us break down complex numbers into their fundamental roots by using angles and rotation. This theorem is especially useful when dealing with complex numbers expressed in polar form. The concept revolves around manipulating the angle \( \theta \), and finding each root by adding multiples of \( 2\pi \) (a full circle in radians) and dividing by \( n \), the root you're looking for. Through this process, you're essentially finding all the equally spaced points on a circle in the complex plane, which are the \( n \)-th roots.

Polar Form

Understanding the polar form of complex numbers is essential for using De Moivre's Theorem. Complex numbers can be expressed in two main forms: rectangular (or Cartesian) and polar. In polar form, a complex number is represented as:

• \( r (\cos(\theta) + i \sin(\theta)) \) or equivalently \( re^{i\theta} \)

where:

• \( r \) is the modulus (or magnitude) of the complex number, representing the distance from the origin in the complex plane.
• \( \theta \) is the argument (or angle) of the complex number, measured in radians from the positive x-axis.

Polar form offers a more intuitive understanding of complex numbers, especially when dealing with rotation and angles, which are inherent properties of complex numbers.

When you convert a complex number to polar form, you're essentially interpreting complex values as vectors. This interpretation makes multiplying and raising powers of complex numbers much simpler, as it mainly concerns adding angles and multiplying magnitudes. In practice, the polar form is invaluable for simplifying multiplication and division, as well as for finding powers and roots of complex numbers.

Rectangular Form

The rectangular form of a complex number is the most straightforward and is often the first way students encounter complex numbers. In rectangular (or Cartesian) form, a complex number is expressed as:

• \( a + bi \)

Here:

• \( a \) is the real part of the complex number.
• \( b \) is the imaginary part, and \( i \) is the imaginary unit, where \( i^2 = -1 \).

This form is useful for simple arithmetic operations like addition and subtraction, as these operations are performed component-wise.

While the rectangular form simplifies basic arithmetic, it can be cumbersome for multiplication or finding powers and roots. For these tasks, converting to polar form provides an elegant solution as complex rules of exponential growth mirror those of trigonometric circle rotations. After calculations in polar form, you typically convert your result back to the rectangular form to represent it in the standard, more familiar format. This interchange between forms allows you to leverage the strengths of each method, be it simplicity or geometric interpretation.