Problem 51 - Algebra and Trigonometry

Question

In Exercises \(45-52,\) find the quotient \(\frac{z_{1}}{z_{2}}\) of the complex numbers. Leave answers in polar form. In Exercises \(49-50,\) express the argument as an angle between \(0^{\circ}\) and \(360^{\circ}\). $$ \begin{aligned} &z_{1}=2+2 i\\\ &z_{2}=1+i \end{aligned} $$

Step-by-Step Solution

Answer

\(\frac{z_{1}}{z_{2}} = 2 + 0i\)

Step 1: Convert to Polar Form

The first step is to convert the given complex numbers from rectangular form \(a + bi\) to polar form \( r(\cos \theta + i \sin \theta)\). For \(z_{1} = 2 + 2i\), the magnitude \(r_{1}\) is \( \sqrt{2^2 + 2^2} = 2\sqrt{2} \) and the argument \(\theta_{1}\) can be found using the formula \(\tan^{-1}(b/a) = \tan^{-1}(2/2) = 45^\circ\). Similarly, for \(z_{2} = 1 + i\), the magnitude \(r_{2}\) is \(\sqrt{1^2 + 1^2} = \sqrt{2}\) and the argument \(\theta_{2}\) is \(\tan^{-1}(1) = 45^\circ\). Thus, \(z_{1}\) and \(z_{2}\) in polar form are \(2\sqrt{2}(\cos 45^\circ + i \sin 45^\circ)\) and \(\sqrt{2}(\cos 45^\circ + i \sin 45^\circ)\) respectively.

Step 2: Find the Quotient

The quotient \(\frac{z_{1}}{z_{2}}\) is found by dividing the magnitudes and subtracting the arguments. This will give: \(\frac{r_{1}}{r_{2}} = \frac{2\sqrt{2}}{\sqrt{2}} = 2\) \(\theta_{1}-\theta_{2} = 45^\circ - 45^\circ = 0^{\circ}\) Thus, the quotient \(\frac{z_{1}}{z_{2}}\) in polar form is \(2(\cos 0^\circ + i \sin 0^\circ)\).

Key Concepts

Polar Form

Polar form is a way of representing complex numbers, characterized by a combination of a magnitude (or modulus) and an angle (or argument). Instead of the usual rectangular form, which is written as \(a + bi\) (where \(a\) is the real part, and \(b\) is the imaginary part), polar form expresses complex numbers as \(r(\cos \theta + i \sin \theta)\). Here, \(r\) is the magnitude and \(\theta\) is the argument.

This form is particularly useful in many mathematical operations, such as multiplication, division, and taking powers of complex numbers because it simplifies the calculations. In polar coordinates, multiplication and division operations involve simple arithmetic with magnitudes and arguments rather than more complex algebraic manipulation.

To convert a complex number from rectangular to polar form, you need the following steps:

• Calculate the magnitude as \(r = \sqrt{a^2 + b^2}\).
• Find the argument using \(\theta = \tan^{-1}(b/a)\).

Quotient of Complex Numbers

The quotient of complex numbers in polar form becomes much simpler and more intuitive to compute than in rectangular form. When dividing two complex numbers, \(z_1\) and \(z_2\), represented in polar form, the result is obtained by dividing their magnitudes and subtracting their arguments.

Consider two complex numbers \(z_1 = r_1 (\cos \theta_1 + i \sin \theta_1)\) and \(z_2 = r_2 (\cos \theta_2 + i \sin \theta_2)\). The quotient \(\frac{z_1}{z_2}\) results in a new complex number:

• The magnitude is \(\frac{r_1}{r_2}\).
  
• The argument is \(\theta_1 - \theta_2\).
  

This simplifies to \(\frac{r_1}{r_2} (\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2))\). Thus, polar form offers elegance and efficiency, especially when working with division.

Magnitude and Argument

The magnitude and argument are crucial components in polar representation of complex numbers. Together, they define a complex number's position within the complex plane.

• Magnitude: This is the distance from the origin to the point representing the complex number in the complex plane. It is computed using the formula \(r = \sqrt{a^2 + b^2}\), where \(a\) and \(b\) are the real and imaginary parts, respectively.
  
• Argument: Known also as the phase angle or simply the angle, the argument is the counterclockwise angle made with the positive x-axis (real axis) and is calculated by \(\theta = \tan^{-1}(b/a)\).
  

These components become especially helpful when moving between different representations of a complex number and are essential for performing operations such as finding products or quotients in polar form.

Rectangular to Polar Conversion

The conversion from rectangular form to polar form is a fundamental concept in handling complex numbers, allowing easier arithmetic and more insightful geometric interpretations.

To begin the conversion:

• First, calculate the magnitude of the complex number. This is done using the formula \(r = \sqrt{a^2 + b^2}\). The magnitude is a real number representing the "length" of the complex vector from the origin in the complex plane.
  
• Next, determine the argument. The argument is \(\theta = \tan^{-1}(b/a)\), an angle in radians or degrees, indicating the direction of the complex number's vector relative to the positive x-axis.
  

Once you have \(r\) and \(\theta\), the polar form is expressed as \(r(\cos \theta + i \sin \theta)\). This form is advantageous when dealing with powers and roots of complex numbers, where multiplication and division translate into simple addition and subtraction of angles, respectively.

Mastering this conversion opens the door to more complex operations and visualizations of complex numbers.