Problem 22 - Precalculus

Question

Find the result of each expression using De Moivre's theorem. Write the answer in rectangular form. $$(1-i)^{4}$$

Step-by-Step Solution

Answer

The result is \(-4 + 0i\).

Step 1: Convert Rectangular Form to Polar Form

Convert the complex number \(1-i\) to polar form. To do this, find the magnitude \(r\) and the argument \(\theta\). The magnitude is given by \[ r = \sqrt{1^2 + (-1)^2} = \sqrt{2}. \] The argument is \[ \theta = \tan^{-1}\left(\frac{-1}{1}\right) = -\frac{\pi}{4}. \] Thus, \( 1-i \) in polar form is \( \sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4})). \)

Step 2: Apply De Moivre's Theorem

De Moivre's Theorem states that \[ (r(\cos(\theta) + i\sin(\theta)))^n = r^n(\cos(n\theta) + i\sin(n\theta)). \]Apply this theorem to \((1-i)^4\) with \(r = \sqrt{2}\), \(\theta = -\frac{\pi}{4}\), and \(n = 4\):\[ (\sqrt{2})^4 \left( \cos\left(4\left(-\frac{\pi}{4}\right)\right) + i\sin\left(4\left(-\frac{\pi}{4}\right)\right) \right). \] This simplifies to \[ 4 \left( \cos(-\pi) + i\sin(-\pi) \right). \]

Step 3: Simplify Trigonometric Functions

Evaluate the trigonometric functions for the angle \(-\pi\):\[ \cos(-\pi) = -1, \] \[ \sin(-\pi) = 0. \]Thus, \[ 4(-1 + 0i) = -4. \]

Step 4: Write in Rectangular Form

The expression \( (1-i)^4 \) simplifies to \(-4\), which is already in rectangular form as \(-4 + 0i.\)

Key Concepts

Polar Form

Polar form is a way to represent complex numbers that provides insights into its magnitude and direction on the complex plane. Unlike the usual rectangular form, which uses real and imaginary parts, polar form uses the length of the vector and the angle it makes with the positive real axis. This can be particularly useful for multiplying or raising complex numbers to powers, like in De Moivre's Theorem.

To convert a complex number like \(1-i\) into its polar form, we start by finding its magnitude \(r\), which is the distance from the origin to the point \((1, -1)\) on the complex plane. This is calculated using the formula:

• \[ r = \sqrt{a^2 + b^2} \]

where \(a\) is the real part, and \(b\) is the imaginary part. For \(1-i\), \(a = 1\) and \(b = -1\), giving us \(r = \sqrt{2}\).

Next, we need the argument \(\theta\), the angle between the positive real axis and the line connecting the origin to the point. It can be computed with the arctangent function:

• \[ \theta = \tan^{-1} \left( \frac{b}{a} \right) \]

For \(1-i\), this gives \(\theta = -\frac{\pi}{4}\).

Thus, \(1-i\) in polar form is expressed as:

• \[ \sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4})) \]

representing the number's magnitude and direction.

Rectangular Form

Rectangular form, also known as Cartesian form, is the standard way of representing complex numbers using their real and imaginary components. A complex number in rectangular form is written as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

In situations like this exercise, you often need to convert between polar and rectangular forms. After using De Moivre's Theorem, you end up with a result in polar form, which sometimes needs conversion back to the rectangular format for clarity or final answers.

Consider our computed result of \((1-i)^4\). Initially, De Moivre's theorem gives us

• \[ 4(\cos(-\pi) + i\sin(-\pi)) \]

This simplifies trigonometric terms as mentioned earlier:

• \[ \cos(-\pi) = -1 \]
• \[ \sin(-\pi) = 0 \]

Thus, the polar expression quickly converts back into rectangular form as \(-4 + 0i\), or simply \(-4\).

This shows that rectangular form is particularly helpful in visualizing the concrete values of the real and imaginary parts or performing straightforward arithmetic operations.

Complex Numbers

Complex numbers, fundamental in many mathematical areas, consist of a real part and an imaginary part. They are expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, satisfying \(i^2 = -1\).

In mathematical operations involving powers and roots of complex numbers, such as this exercise, the polar form and De Moivre's Theorem come into play. Complex numbers can be seamlessly translated between rectangular form and polar form, aiding in various calculations and applications.

In this task, we harnessed the concept of complex numbers to first change \(1-i\) into polar form, and then apply De Moivre's Theorem. This theorem was crucial for efficiently calculating the power:

• \[ (r(\cos \theta + i \sin \theta))^n = r^n(\cos(n \theta) + i \sin(n \theta)) \]

By converting the complex number to polar form, raising it to the fourth power, and interpreting the result back in rectangular form, we get a straightforward solution.

This exemplifies how the clarity and ease afforded by complex numbers in polar form simplify operations that would be otherwise cumbersome in rectangular form. Thus, complex numbers become a versatile tool for algebra, trigonometry, and calculus, bridging various mathematical concepts together.