Problem 43 - Algebra and Trigonometry

Question

In Exercises \(37-44,\) find the product of the complex numbers. Leave answers in polar form. $$ \begin{aligned} &z_{1}=1+i\\\ &z_{2}=-1+i \end{aligned} $$

Step-by-Step Solution

Answer

The product of the complex numbers is \(2(\cos(\pi) + i \sin(\pi))\)

Step 1: Convert to Polar Form

Convert the complex numbers to polar form. Complex number \(z_1 = 1 + i\) has modulus \(\sqrt{(1)^2 + (1)^2} = \sqrt{2}\) and argument \(\arctan(\frac{1}{1}) = \frac{\pi}{4}\) . Similarly, for \(z_2 = -1 + i\), the modulus is \(\sqrt{(-1)^2 + (1)^2} = \sqrt{2}\) and the argument is \(\arctan(\frac{1}{-1}) = \frac{3\pi}{4}\) . So, \(z_1 = \sqrt{2}(\cos(\frac{\pi}{4}) + i \sin(\frac{\pi}{4}))\) and \(z_2 = \sqrt{2}( \cos(\frac{3\pi}{4}) + i \sin(\frac{3\pi}{4}))\)

Step 2: Multiply the complex numbers

Now, to multiply the two complex numbers in polar form, simply multiply the moduli and add the arguments together. The modulus of the product is \( r = \sqrt{2} \times \sqrt{2} = 2 \) and the argument of the product is \( \theta = \frac{\pi}{4} + \frac{3\pi}{4} = \pi \)

Step 3: Write product in polar form

The final result is thus presented in polar form, using the formula \( r(\cos(\theta) + i \sin(\theta)) = 2(\cos(\pi) + i \sin(\pi)) \)

Key Concepts

Polar Form

When working with complex numbers, polar form can make calculations simpler. Instead of representing a complex number in rectangular form as \( a + bi \), where \( a \) and \( b \) are real numbers, polar form uses a different perspective. It features two components: the modulus \( r \) (or magnitude) and the argument \( \theta \) (or angle). The polar form is expressed as \( r(\cos(\theta) + i\sin(\theta)) \).
To convert a complex number from rectangular to polar form, you need its modulus and argument. Polar form is particularly useful when multiplying or dividing complex numbers, as it simplifies the process by focusing on these components rather than the real and imaginary parts.

Modulus

The modulus of a complex number, such as \( z = a + bi \), measures its "size" or "distance" from the origin in the complex plane. It is calculated as \( \sqrt{a^2 + b^2} \). This formula arises from the Pythagorean theorem, treating \( a \) and \( b \) as the legs of a right triangle.
Knowing the modulus is crucial when converting complex numbers into their polar form. In multiplication, the modulus of the product is simply the product of the moduli of the multiplicands. This property makes computations easier compared to working with the original rectangular form.

Argument

The argument of a complex number, denoted as \( \theta \), is the angle formed between the positive real axis and the line representing the complex number. It is measured in radians. The value of the argument can be found using the arctan function: \( \theta = \arctan\left(\frac{b}{a}\right) \).
Understanding the argument is essential when addressing multiplication in polar form. When two complex numbers are multiplied, you simply add their arguments. This ease of addition replaces the need for more complicated arithmetic when multiplying numbers in their rectangular form.

Arctan Function

The arctan function, often called the inverse tangent function, is key when determining the argument of a complex number. It helps find the angle \( \theta \) in the expression \( \theta = \arctan\left(\frac{b}{a}\right) \), where \( a \) and \( b \) are the real and imaginary components, respectively.
This function calculates the angle whose tangent is the given ratio. Therefore, it's invaluable for converting complex numbers from rectangular to polar form. Keep in mind that the output of the arctan function needs interpretation based on the signs of \( a \) and \( b \) to determine the correct quadrant of the angle on the complex plane.