Problem 67 - Contemporary Precalculus

Question

In Exercises \(65-72,\) convert to polar form and then multiply or divide. Express your answer in polar form. $$\frac{1+i}{1-i}$$

Step-by-Step Solution

Answer

The polar form of the division of the given complex numbers is \(\cos{\frac{\pi}{2}} + i\sin{\frac{\pi}{2}}\).

Step 1: Convert to polar form

To convert the complex numbers to polar form, we need to find their magnitude and angle (argument). The polar form of a complex number is given by \(r(\cos{\theta} + i\sin{\theta})\), where \(r\) is the magnitude and \(\theta\) is the angle (argument). For the complex number \(1+i\), we have: Magnitude: \(r_1 = \sqrt{1^2 + 1^2} = \sqrt{2}\) Angle: \(\theta_1 = \tan^{-1} \frac{1}{1} = \frac{\pi}{4}\) Thus, the polar form of \(1+i\) is \(\sqrt{2}(\cos{\frac{\pi}{4}} + i\sin{\frac{\pi}{4}})\). For the complex number \(1-i\), we have: Magnitude: \(r_2 = \sqrt{1^2 + (-1)^2} = \sqrt{2}\) Angle: \(\theta_2 = \tan^{-1} \frac{-1}{1} = -\frac{\pi}{4}\) Thus, the polar form of \(1-i\) is \(\sqrt{2}(\cos{-\frac{\pi}{4}} + i\sin{-\frac{\pi}{4}})\).

Step 2: Divide the complex numbers in polar form

To divide two complex numbers in polar form, we will use the following rule: $$\frac{r_1(\cos{\theta_1} + i\sin{\theta_1})}{r_2(\cos{\theta_2} + i\sin{\theta_2})} = \frac{r_1}{r_2}(\cos{(\theta_1-\theta_2)} + i\sin{(\theta_1-\theta_2)})$$ Applying this rule to our complex numbers, we get: $$\frac{\sqrt{2}(\cos{\frac{\pi}{4}} + i\sin{\frac{\pi}{4}})}{\sqrt{2}(\cos{-\frac{\pi}{4}} + i\sin{-\frac{\pi}{4}})} = \frac{\sqrt{2}}{\sqrt{2}}(\cos{(\frac{\pi}{4} - (-\frac{\pi}{4}))} + i\sin{(\frac{\pi}{4} - (-\frac{\pi}{4}))})$$

Step 3: Simplify the expression

Now, we will simplify the expression to find the result in polar form. $$\frac{\sqrt{2}}{\sqrt{2}}(\cos{(\frac{\pi}{4} - (-\frac{\pi}{4}))} + i\sin{(\frac{\pi}{4} - (-\frac{\pi}{4}))}) = 1(\cos{\frac{\pi}{2}} + i\sin{\frac{\pi}{2}})$$ The result is: \(\cos{\frac{\pi}{2}} + i\sin{\frac{\pi}{2}}\) Therefore, the polar form of the division of the given complex numbers is \(\cos{\frac{\pi}{2}} + i\sin{\frac{\pi}{2}}\).

Key Concepts

Polar Form

Complex numbers can be expressed in a more useful manner known as polar form, especially when dealing with multiplication and division. In polar form, a complex number is represented as \[ r(\cos{\theta} + i\sin{\theta}) \] where:

• \( r \) is the magnitude (or modulus) of the complex number, representing its distance from the origin in the complex plane.
• \( \theta \) is the argument (or angle), indicating the direction of the number from the positive real axis.

This form is very handy for performing operations like multiplication and division because it simplifies the calculations by focusing on the magnitudes and the angles rather than their real and imaginary parts. This conversion requires you to find both the magnitude and the angle, which we'll explore next.
Understanding this representation is crucial as it greatly simplifies complex arithmetic, helping to visualize operations in terms of geometric transformations.

Magnitude and Argument

The two key components of the polar form are the magnitude and the argument of a complex number.
The magnitude, \( r \), of a complex number \( a + bi \) is found using the formula:\[ r = \sqrt{a^2 + b^2} \]This formula gives you the "length" from the origin to the point \((a, b)\) in the complex plane.
The argument, \( \theta \), is the angle made with the positive real axis, determined by:\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]For example, for the complex number \(1+i\), the magnitude is \( \sqrt{2} \) and the angle is \( \frac{\pi}{4} \). For \(1-i\), the magnitude is also \( \sqrt{2} \), but the angle is \( -\frac{\pi}{4} \).
Having both the magnitude and argument allows you to fully convert a complex number from its rectangular form to polar form, making operations like division much simpler.

Division of Complex Numbers

Dividing complex numbers is made straightforward with polar form. When you divide two complex numbers in polar form:

• First, divide their magnitudes.
• Second, subtract their arguments.

The formula for division is:\[\frac{r_1(\cos{\theta_1} + i\sin{\theta_1})}{r_2(\cos{\theta_2} + i\sin{\theta_2})} = \frac{r_1}{r_2} \left(\cos{(\theta_1 - \theta_2)} + i\sin{(\theta_1 - \theta_2)}\right)\]In our exercise, the division of \( \frac{1+i}{1-i} \) simplifies to the division of their magnitudes, \( \frac{\sqrt{2}}{\sqrt{2}} = 1 \), and the subtraction of their arguments, \( \frac{\pi}{4} - (-\frac{\pi}{4}) = \frac{\pi}{2} \). Thus, the result is simply:\[ \cos{\frac{\pi}{2}} + i\sin{\frac{\pi}{2}} \]This method showcases how polar forms not only make the process clean but also emphasize the rotational nature of complex number multiplication and division.