Problem 67 - Precalculus Mathematics for Calculus

Question

Write \(z_{1}\) and \(z_{2}\) in polar form, and then find the product \(z_{1} z_{2}\) and the quotients \(z_{1} / z_{2}\) and \(1 / z_{1}\). $$z_{1}=-20, \quad z_{2}=\sqrt{3}+i$$

Step-by-Step Solution

Answer

\( z_1 = 20\text{cis}(\pi) \), \( z_2 = 2\text{cis}(\frac{\pi}{6}) \); \( z_1 z_2 = 40\text{cis}(\frac{7\pi}{6}) \); \( \frac{z_1}{z_2} = 10\text{cis}(\frac{5\pi}{6}) \); \( \frac{1}{z_1} = -\frac{1}{20} \).

Step 1: Expressing z1 in Polar Form

The complex number \( z_1 = -20 \) has only a real part and no imaginary part, so we can write it as \( -20 + 0i \). In polar form, this number has a magnitude (modulus) \( |z_1| = 20 \) and an angle (argument) \( \theta = \pi \) (since it's on the negative real axis). Thus, \( z_1 = 20 \text{cis}(\pi) \).

Step 2: Expressing z2 in Polar Form

The complex number \( z_2 = \sqrt{3} + i \) can be expressed in polar form by calculating its magnitude and argument. The magnitude is \( |z_2| = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{4} = 2 \). The argument is \( \theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \). Thus, \( z_2 = 2\text{cis}\left(\frac{\pi}{6}\right) \).

Step 3: Calculating the Product z1z2

The product of two complex numbers in polar form \( z_1 = r_1\text{cis}(\theta_1) \) and \( z_2 = r_2\text{cis}(\theta_2) \) is given by \( z_1z_2 = r_1r_2\text{cis}(\theta_1 + \theta_2) \). Here, \( r_1 = 20 \) and \( r_2 = 2 \), so their product is \( 40 \). Their angles sum to \( \pi + \frac{\pi}{6} = \frac{7\pi}{6} \). Therefore, \( z_1 z_2 = 40\text{cis}\left(\frac{7\pi}{6}\right) \).

Step 4: Calculating the Quotient z1/z2

The quotient of two complex numbers in polar form \( z_1 = r_1\text{cis}(\theta_1) \) and \( z_2 = r_2\text{cis}(\theta_2) \) is given by \( \frac{z_1}{z_2} = \frac{r_1}{r_2}\text{cis}(\theta_1 - \theta_2) \). With \( r_1 = 20 \) and \( r_2 = 2 \), the magnitude is \( 10 \). The angle difference is \( \pi - \frac{\pi}{6} = \frac{5\pi}{6} \). Thus, \( \frac{z_1}{z_2} = 10\text{cis}\left(\frac{5\pi}{6}\right) \).

Step 5: Calculating the Quotient 1/z1

The reciprocal of a complex number in polar form \( z = r\text{cis}(\theta) \) is given by \( \frac{1}{z} = \frac{1}{r}\text{cis}(-\theta) \). For \( z_1 = 20\text{cis}(\pi) \), the reciprocal is \( \frac{1}{20} \text{cis}(-\pi) \). This corresponds to the complex number \( -\frac{1}{20} \).

Key Concepts

Polar Coordinates

Complex numbers can be represented in different forms, one of which includes the polar coordinate form. In this representation, a complex number is expressed in terms of its distance from the origin (called the modulus) and the angle it makes with the positive real axis (known as the argument). For example, any complex number can be written as \(z = r\text{cis}(\theta)\), where \(r\) is the modulus and \( \theta \) is the argument.

• The modulus, \(r\), indicates how far the number is from the origin in the complex plane.
• The argument, \( \theta \), is the angle formed between the positive real axis and the line joining the origin with the point representing the complex number.

These components make it easier to perform operations such as multiplication and division. Indeed, transforming a complex number to polar form helps simplify computation, especially when dealing with large numbers or fractions.

Modulus and Argument

The modulus and argument form the backbone of the polar representation of complex numbers. Let's break these down further:- **Modulus**: The modulus of a complex number \(z = x + yi\) is given by \(|z| = \sqrt{x^2 + y^2}\). This value represents the "size" or "length" of the vector from the origin to the point \((x, y)\) on the complex plane.- **Argument**: The argument, or angle, tells us where exactly the line representing the complex number is oriented with respect to the real axis. It is denoted as \( \theta = \tan^{-1}(\frac{y}{x})\). This angle is usually measured in radians.
Using these, you can easily convert rectangular form to polar form, which is particularly handy when dealing with the product or quotient of complex numbers.

Multiplication and Division of Complex Numbers

When multiplying or dividing complex numbers, the polar form provides a simple and efficient method. In polar form, the operations can be simplified using the properties of modulus and argument.

• **Multiplication**: To multiply two complex numbers, \(z_1 = r_1\text{cis}(\theta_1)\) and \(z_2 = r_2\text{cis}(\theta_2)\), you multiply their moduli and add their arguments: \(z_1z_2 = r_1r_2\text{cis}(\theta_1 + \theta_2)\). This takes advantage of the identity for addition of angles.
• **Division**: To divide the numbers, divide their moduli and subtract the arguments: \(\frac{z_1}{z_2} = \frac{r_1}{r_2}\text{cis}(\theta_1 - \theta_2)\). This provides a quick way of determining the result of division without directly dealing with real and imaginary parts.

This method reduces the complexity of calculations and is especially important when performing operations on more complex equations or expressions involving complex numbers.