Problem 67 - Algebra and Trigonometry

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

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

Step 1: Convert z1 to Polar Form

The complex number \(z_1 = -20\) is on the negative real axis. Therefore, its modulus is \(|z_1| = 20\) and the argument is \(\theta = \pi\) radians (or 180 degrees). Thus, \(z_1\) in polar form is \(20\text{cis}(\pi)\).

Step 2: Convert z2 to Polar Form

To convert \(z_2 = \sqrt{3} + i\) to polar form, first calculate the modulus: \(|z_2| = \sqrt{(\sqrt{3})^2 + 1^2} = 2\). Then, find the argument: \(\theta = \tan^{-1}(1/\sqrt{3}) = \pi/6\) radians (or 30 degrees). Therefore, \(z_2\) in polar form is \(2\text{cis}(\pi/6)\).

Step 3: Compute the Product z1z2

The product of two complex numbers in polar form is the product of their moduli and the sum of their arguments. So, \(|z_1 z_2| = 20 \times 2 = 40\) and \(\text{arg}(z_1 z_2) = \pi + \pi/6 = 7\pi/6\). Thus, \(z_1 z_2 = 40\text{cis}(7\pi/6)\).

Step 4: Compute the Quotient z1/z2

The quotient of two complex numbers in polar form is the quotient of their moduli and the difference of their arguments. So, \(|z_1 / z_2| = 20 / 2 = 10\) and \(\text{arg}(z_1 / z_2) = \pi - \pi/6 = 5\pi/6\). Thus, \(z_1 / z_2 = 10\text{cis}(5\pi/6)\).

Step 5: Compute the Quotient 1/z1

The reciprocal of a complex number in polar form has modulus \(1/|z_1|\) and argument \(-\theta\). Therefore, \(|1/z_1| = 1/20\) and \(\text{arg}(1/z_1) = -\pi\). Hence, \(1/z_1 = \frac{1}{20}\text{cis}(-\pi)\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are generally represented in the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. The imaginary unit \(i\) is defined as \(\sqrt{-1}\). Complex numbers expand our traditional number system. They are especially useful in engineering, physics, and applied mathematics where they help model and solve equations that do not have solutions in the real numbers alone.
Examples of complex numbers include:

• Purely real numbers like 3, which can be written as \(3 + 0i\).
• Purely imaginary numbers like \(2i\) or \(-i\).
• Mixed complex numbers like \(5 + 2i\) or \(-1 - i\).

Understanding complex numbers is the first step toward exploring more advanced concepts such as the polar form, which simplifies their multiplication and division.

Product and Quotient of Complex Numbers

The product and quotient of complex numbers are calculations that become much simpler when using polar form instead of the traditional rectangular form (\(a + bi\)). In polar form, a complex number is represented as \(r\text{cis}(\theta)\) or \(re^{i\theta}\), where \(r\) is the modulus and \(\theta\) is the argument.
When multiplying two complex numbers, such as \(z_1 = r_1\text{cis}(\theta_1)\) and \(z_2 = r_2\text{cis}(\theta_2)\), you simply multiply their moduli and add their arguments:

• Modulus: \(|z_1 z_2| = r_1 \times r_2\)
• Argument: \(\text{arg}(z_1 z_2) = \theta_1 + \theta_2\)

Similarly, dividing two complex numbers involves dividing their moduli and subtracting their arguments:

• Modulus: \(|z_1 / z_2| = r_1 / r_2\)
• Argument: \(\text{arg}(z_1 / z_2) = \theta_1 - \theta_2\)

This compact method avoids the more cumbersome expansion of the product or quotient into standard form, making calculations faster and more intuitive.

Modulus and Argument

The modulus and argument are key components of a complex number in polar form. The modulus, \(|z|\), of a complex number \(z = a + bi\) is its distance from the origin in the complex plane, calculated as \(|z| = \sqrt{a^2 + b^2}\). It gives us the magnitude or absolute value of the complex number.
The argument, \(\theta\), is the angle formed with the positive real axis in the complex plane. The argument can be determined using trigonometry, especially the arctangent function: \(\theta = \tan^{-1}(b/a)\). Depending on the quadrant where the number lies, adjustments may be necessary to find the correct angle.
Together, the modulus and argument allow us to express complex numbers in the succinct and powerful polar form:

• Example: For \(z = -20\), the modulus is 20, and the argument is \(\pi\).
• Example: For \(z = \sqrt{3} + i\), the modulus is 2, and the argument is \(\pi/6\).

Knowing these forms helps simplify various operations on complex numbers.

Reciprocal of Complex Number

Finding the reciprocal of a complex number means determining a number that when multiplied with the original complex number results in 1. In polar form, the reciprocal becomes quite straightforward to find.
For a complex number \(z = r\text{cis}(\theta)\), its reciprocal \(1/z\) has:

• Modulus : \(1/r\)
• Argument : \(-\theta\)

This is because multiplying \(z\) by its reciprocal should cancel out both the modulus and argument, resulting in 1. For example, with \(z_1 = 20\text{cis}(\pi)\), the reciprocal will be \(\frac{1}{20}\text{cis}(-\pi)\).
This technique sets up the operation in a very streamlined way, bypassing the need to multiply and divide individual real and imaginary parts, which can quickly become cumbersome.