Problem 62 - 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\(/ z_{1}\) . $$ z_{1}=\sqrt{2}-\sqrt{2} i, \quad z_{2}=1-i $$

Step-by-Step Solution

Answer

\(z_1 = 2(\cos(-\pi/4) + i \sin(-\pi/4)), z_2 = \sqrt{2}(\cos(-\pi/4) + i \sin(-\pi/4)) \); \(z_1z_2 = -2\sqrt{2}i\); \(z_1/z_2 = \sqrt{2}\); \(1/z_1 = \frac{\sqrt{2}}{4} + i \frac{\sqrt{2}}{4}\).

Step 1: Write Complex Numbers in Polar Form

First, we convert the given complex numbers \( z_1 = \sqrt{2} - \sqrt{2}i \) and \( z_2 = 1 - i \) into polar form. The polar form of a complex number \( z = a + bi \) is \( r (\cos \theta + i \sin \theta) \), where \( r = \sqrt{a^2 + b^2} \) is the modulus and \( \theta = \text{tan}^{-1}(b/a) \) is the argument. For \( z_1 \): - Modulus: \( r_1 = \sqrt{(\sqrt{2})^2 + (-\sqrt{2})^2} = \sqrt{4} = 2 \) - Argument: \( \theta_1 = \text{tan}^{-1}\left(\frac{-\sqrt{2}}{\sqrt{2}}\right) = \text{tan}^{-1}(-1) = -\frac{\pi}{4} \) Thus, \( z_1 = 2(\cos(-\pi/4) + i \sin(-\pi/4)) \). For \( z_2 \): - Modulus: \( r_2 = \sqrt{1^2 + (-1)^2} = \sqrt{2} \) - Argument: \( \theta_2 = \text{tan}^{-1}\left(\frac{-1}{1}\right) = \text{tan}^{-1}(-1) = -\frac{\pi}{4} \) Thus, \( z_2 = \sqrt{2}(\cos(-\pi/4) + i \sin(-\pi/4)) \).

Step 2: Calculate the Product \( z_1 z_2 \)

To find the product \( z_1 z_2 \), multiply their moduli and add their arguments:- Modulus: \( r = r_1 \times r_2 = 2 \times \sqrt{2} = 2\sqrt{2} \) - Argument: \( \theta = \theta_1 + \theta_2 = -\frac{\pi}{4} + (-\frac{\pi}{4}) = -\frac{\pi}{2} \) Therefore, the product is \( z_1 z_2 = 2\sqrt{2}(\cos(-\pi/2) + i \sin(-\pi/2)) \). Simplifying, \( z_1 z_2 = 2\sqrt{2}(0 - i) = -2\sqrt{2}i \).

Step 3: Calculate the Quotient \( z_1 / z_2 \)

For the quotient \( z_1 / z_2 \), divide their moduli and subtract their arguments:- Modulus: \( \frac{r_1}{r_2} = \frac{2}{\sqrt{2}} = \sqrt{2} \) - Argument: \( \theta = \theta_1 - \theta_2 = -\frac{\pi}{4} - (-\frac{\pi}{4}) = 0 \) Therefore, \( z_1 / z_2 = \sqrt{2}(\cos(0) + i \sin(0)) = \sqrt{2}(1 + 0i) = \sqrt{2} \).

Step 4: Calculate the Quotient \( 1 / z_1 \)

To find \( 1 / z_1 \), take the reciprocal of its modulus and change the sign of its argument:- Modulus: \( \frac{1}{r_1} = \frac{1}{2} \) - Argument: \( -\theta_1 = \pi/4 \) Thus, \( 1 / z_1 = \frac{1}{2}(\cos(\pi/4) + i \sin(\pi/4)) \).Simplifying, \( 1 / z_1 = \frac{1}{2}\left(\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{4} + i \frac{\sqrt{2}}{4} \).

Key Concepts

Complex Number Multiplication

Multiplying complex numbers in polar form is straightforward and elegant. When you have two complex numbers expressed in polar form, say, \( z_1 = r_1 (\cos \theta_1 + i \sin \theta_1) \) and \( z_2 = r_2 (\cos \theta_2 + i \sin \theta_2) \), multiplying them involves two simple steps:

• Multiply their moduli: \( r = r_1 \times r_2 \).
• Add their arguments: \( \theta = \theta_1 + \theta_2 \).

This means that the product \( z_1 z_2 \) is represented in polar form as \( r (\cos \theta + i \sin \theta) \).For example, if \( z_1 \) and \( z_2 \) are given as in the exercise, with moduli \( r_1 = 2 \) and \( r_2 = \sqrt{2} \), and arguments \( \theta_1 = -\frac{\pi}{4} \) and \( \theta_2 = -\frac{\pi}{4} \), the product \( z_1 z_2 \) has a modulus of \( 2\sqrt{2} \) and an argument of \( -\frac{\pi}{2} \). This simplifies to \( z_1 z_2 = -2\sqrt{2}i \) in rectangular form.

Complex Number Division

Dividing complex numbers in polar form is as intuitive as multiplying them. For two complex numbers \( z_1 \) and \( z_2 \), where \( z_1 = r_1 (\cos \theta_1 + i \sin \theta_1) \) and \( z_2 = r_2 (\cos \theta_2 + i \sin \theta_2) \):

• Divide their moduli: \( \frac{r_1}{r_2} \).
• Subtract their arguments: \( \theta = \theta_1 - \theta_2 \).

The quotient \( \frac{z_1}{z_2} \) can then be expressed in polar form as \( \frac{r_1}{r_2}(\cos \theta + i \sin \theta) \).Following the exercise, with \( r_1 = 2 \), \( r_2 = \sqrt{2} \), \( \theta_1 = -\frac{\pi}{4} \), and \( \theta_2 = -\frac{\pi}{4} \), the division \( \frac{z_1}{z_2} \) results in a modulus of \( \sqrt{2} \) and an argument of \( 0 \). This simplifies to \( \sqrt{2} \) in the rectangular coordinate form, representing a purely real number.

Modulus of a Complex Number

The modulus of a complex number, also known as its absolute value, measures the "size" or "distance" of the complex number from the origin in the complex plane. For a complex number \( z = a + bi \), the modulus \( r \) is given by the formula:\[r = \sqrt{a^2 + b^2}\]This formula ensures you're calculating the hypotenuse of the right triangle formed by the real and imaginary parts of the complex number. Essentially, the modulus is the length of the vector that represents the complex number in the complex plane.In our exercise example, to find the modulus of \( z_1 = \sqrt{2} - \sqrt{2}i \), you would calculate \( r_1 = \sqrt{2^2 + (-\sqrt{2})^2} = \sqrt{4} = 2 \). For \( z_2 = 1 - i \), the modulus is \( r_2 = \sqrt{1^2 + (-1)^2} = \sqrt{2} \). The modulus is crucial for converting complex numbers between different forms and for operations like multiplication and division.

Argument of a Complex Number

The argument of a complex number is a critical concept when dealing with their polar representation. It is the angle measured from the positive real axis to the line segment that represents the complex number in the complex plane. For a complex number \( z = a + bi \), the argument \( \theta \) can be found using:\[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]This calculation considers the relationship between the imaginary and the real part of the complex number to determine the angle of rotation. Remember that the argument may need adjustments depending on the quadrant in which the complex number resides.From the textbook exercise, for \( z_1 \) with real and imaginary parts \( \sqrt{2} \) and \(-\sqrt{2} \), the argument is \( \theta_1 = \tan^{-1}(-1) = -\frac{\pi}{4} \). Similarly, for \( z_2 = 1 - i \), since both are negative, \( \theta_2 = \tan^{-1}(-1) = -\frac{\pi}{4} \). Knowing the argument helps in comprehensively working with complex numbers in polar form.