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Chapter 2: Problem 24

Given \(z=(1+i \sqrt{3})^{100}\), then \([\operatorname{RE}(z) / \operatorname{IM}(z)]\) equals a. \(2^{100}\) b. \(2^{50} \quad\) c. \(\frac{1}{\sqrt{3}}\) d. \(\sqrt{3}\)

Short Answer

Expert verified
The answer is c: \( \frac{1}{\sqrt{3}} \).

Step by step solution

01

Recognize the Binomial Form

The given expression is \( z = (1 + i\sqrt{3})^{100} \). We identify that \( 1 + i\sqrt{3} \) can be expressed in binomial form as \( a + bi \), where \( a = 1 \) and \( b = \sqrt{3} \).
02

Convert to Polar Form

Convert the complex number \( 1 + i\sqrt{3} \) to polar form. Calculate the modulus: \( r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2 \). Find the argument: \( \theta = \tan^{-1}(\sqrt{3}/1) = \pi/3 \). Now, express it as \( z = 2 (\cos(\pi/3) + i\sin(\pi/3)) \).
03

Apply De Moivre's Theorem

Apply De Moivre's theorem to compute \( z^{100} = [2 (\cos(\pi/3) + i\sin(\pi/3))]^{100} = 2^{100}(\cos(100\pi/3) + i\sin(100\pi/3)) \). Calculate \( 100\pi/3 \mod 2\pi \) to simplify the expression.
04

Calculate the Argument Simplification

Simplify \( 100\pi/3 \mod 2\pi \). This is done by dividing \( 100\pi/3 \) by \( 2\pi \) and finding the remainder: \( (100/3) \mod 2 = 1.67 \mod 1 = 0.67 \), thus the angle \( \theta = 0.67\) which approximately equals \(2\pi / 3\).
05

Compute Real and Imaginary Components

Now calculate the real and imaginary parts using the angle \( 2\pi/3 \): \[ \operatorname{RE}(z) = 2^{100} \cos(2\pi/3) = -2^{100}/2 \] \[ \operatorname{IM}(z) = 2^{100} \sin(2\pi/3) = 2^{100} \sqrt{3}/2 \].
06

Find the Ratio of Real to Imaginary Parts

Finally, compute the ratio \( \frac{\operatorname{RE}(z)}{\operatorname{IM}(z)} = \frac{-2^{100}/2}{2^{100} \sqrt{3}/2} = -\frac{1}{\sqrt{3}} \).
07

Determine Answer Choice

The value matches answer choice c: \( \frac{1}{\sqrt{3}} \). Therefore, the correct answer is option c.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

De Moivre's Theorem
De Moivre's Theorem is a powerful tool in complex number analysis, especially for raising complex numbers to integer powers. It connects the polar form of complex numbers with exponential growth. This theorem states that for a complex number in polar form, given as
  • \( z = r(\cos \theta + i\sin \theta) \)
if you want to compute \( z^n \) where n is an integer, the expression becomes
  • \( z^n = r^n(\cos (n\theta) + i \sin(n\theta)) \).
The beauty of De Moivre's Theorem lies in its simplicity and ease of use as shown in how it transforms the problem of finding \( (1+i\sqrt{3})^{100} \) into a more manageable expression.
Polar Form
To understand De Moivre's Theorem, one must first be comfortable with converting complex numbers into polar form. Instead of expressing a complex number \( a + bi \) in its standard form, polar form expresses it in the format of
  • \( r(\cos \theta + i \sin \theta) \)
where \( r \) is the modulus of the complex number, calculated as
  • \( r = \sqrt{a^2 + b^2} \)
and \( \theta \), the argument, is obtained using the arctangent function:
  • \( \theta = \tan^{-1}(b/a) \).
In our example, \( 1+i\sqrt{3} \) was converted to polar form, resulting in \( 2(\cos(\pi/3) + i\sin(\pi/3)) \).Understanding this conversion is key as it simplifies the computation of complex numbers raised to any power.
Real and Imaginary Parts
The real and imaginary components of a complex number play a crucial role in many mathematical operations. For a complex number expressed as \( z = x + yi \),
  • The real part, \( \operatorname{RE}(z) \), is \( x \).
  • The imaginary part, \( \operatorname{IM}(z) \), is \( y \).
Once we apply De Moivre's Theorem and simplify using the polar form, we can calculate these parts for any power of a complex number. In the case of \( (1+i\sqrt{3})^{100} \), we determined:
  • \( \operatorname{RE}(z) = 2^{100} \cos(2\pi/3) = -2^{100}/2 \)
  • \( \operatorname{IM}(z) = 2^{100} \sin(2\pi/3) = 2^{100} \sqrt{3}/2 \)
Understanding these components helps perform operations like addition, multiplication, or in this case, verifying steps in a computation.
Binomial Form
Complex numbers in binomial form are denoted as \( a + bi \), where \( a \) and \( b \) are real numbers, \( a \) represents the real part, and \( bi \) is the imaginary part. This representation is useful for straightforward arithmetic operations involving complex numbers. In the context of our problem:
  • \( 1 + i\sqrt{3} \) is in binomial form with \( a = 1 \) and \( b = \sqrt{3} \).
Converting such a number to polar form allows us to utilize De Moivre's Theorem for finding powers, as binomial form can be cumbersome for such operations. Recognizing and correctly applying both forms is crucial when working with complex numbers in higher mathematical contexts.

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Most popular questions from this chapter

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