Problem 71 - Contemporary Precalculus

Question

In Exercises \(65-72,\) convert to polar form and then multiply or divide. Express your answer in polar form. $$i(i+1)(-\sqrt{3}+i)$$

Step-by-Step Solution

Answer

#Answer# In polar form, the expression is \(2\sqrt{2}\operatorname{cis}\frac{5\pi}{4}\).

Step 1: Convert each complex number factor to polar form

We will convert each factor in the expression to polar form: - For \(i\), we have \(r = 1\) and \(\theta = \frac{\pi}{2}\). So, in polar form, \(i = 1 \operatorname{cis}\frac{\pi}{2}\). - For \(1 + i\), we have \(r = \sqrt{1^2+1^2} = \sqrt{2}\) and \(\theta = \arctan\frac{1}{1} = \frac{\pi}{4}\). So, in polar form, \(1 + i = \sqrt{2}\operatorname{cis}\frac{\pi}{4}\). - For \(- \sqrt{3} + i\), we have \(r = \sqrt{3^2+1^2} = 2\) and \(\theta = \arctan\frac{1}{-\sqrt{3}} = \frac{5\pi}{6}\). So, in polar form, \(-\sqrt{3}+i = 2\operatorname{cis}\frac{5\pi}{6}\).

Step 2: Multiply the factors in polar form

Now, it's time to multiply the factors: $$i(i + 1)( -\sqrt{3} + i) = (1\operatorname{cis}\frac{\pi}{2})(\sqrt{2}\operatorname{cis}\frac{\pi}{4})(2\operatorname{cis}\frac{5\pi}{6})$$ In polar form, we multiply the moduli (the \(r\) values) and sum the angles (the \(\theta\) values): \(r = 1\times\sqrt{2}\times 2 = 2\sqrt{2}\) \(\theta = \frac{\pi}{2}+\frac{\pi}{4}+\frac{5\pi}{6} = \frac{15\pi}{12} = \frac{5\pi}{4}\) So, our expression becomes: \(2\sqrt{2}\operatorname{cis}\frac{5\pi}{4}\) We have successfully expressed the expression in polar form.

Key Concepts

Polar Form Multiplication

Polar form multiplication is a process used in complex number arithmetic that vastly simplifies the multiplication of complex numbers. When complex numbers are expressed in their polar forms, the multiplication of these numbers becomes an operation that involves multiplying their magnitudes (also known as moduli) and adding their angles (also known as arguments).

The mathematical representation of a complex number in polar form is usually given as \( r \operatorname{cis} \theta \), where \( r \) is the magnitude of the complex number and \( \theta \) is the angle made with the positive real axis.

To multiply two complex numbers in polar form, \( r_1 \operatorname{cis} \theta_1 \) and \( r_2 \operatorname{cis} \theta_2 \), you calculate the new magnitude by multiplying \( r_1 \) and \( r_2 \) and the new angle by adding \( \theta_1 \) and \( \theta_2 \). The product is then \( r_1r_2 \operatorname{cis}(\theta_1 + \theta_2) \).

This method is incredibly efficient compared to the standard form multiplication which involves combining like terms and can get quite cumbersome especially for multiple factors.

Converting Complex Numbers to Polar Form

To perform operations like multiplication quickly, it is often helpful to convert complex numbers to polar form. This representation captures the essence of a complex number in terms of its magnitude and direction relative to the origin when plotted on a complex plane.

The conversion involves finding two key components: the magnitude \( r \), given by the square root of the sum of the squares of the real and imaginary parts, \( r = \sqrt{x^2 + y^2} \); and the angle \( \theta \), which is the arctan of the ratio of the imaginary part to the real part, \( \theta = \arctan(\frac{y}{x}) \). If the complex number has a nonpositive real part, an additional \( \pi \) might need to be added to the angle to place it in the correct quadrant.

For example, converting the complex number \(1+i\) to polar form involves calculating its magnitude \( r = \sqrt{1^2 + 1^2} = \sqrt{2} \) and the angle \( \theta = \arctan\left(\frac{1}{1}\right) = \frac{\pi}{4} \), leading to the polar form \( \sqrt{2} \operatorname{cis} \frac{\pi}{4} \).

Understanding how to properly convert complex numbers to polar form is crucial as it sets the stage for applying operations such as multiplication, division, raising powers, and even finding roots more effortlessly.

Complex Number Arithmetic

Complex number arithmetic involves operations such as addition, subtraction, multiplication, and division, using a combination of real and imaginary numbers. While addition and subtraction follow the intuitive approach—combining like terms with real parts added to real parts, and imaginary to imaginary—multiplication and division can benefit greatly from using polar form, particularly for higher order operations.

For instance, when multiplying complex numbers in standard form, you apply the distributive property (FOIL) which can get intricate with multiple terms. Polar form condenses the process, making the multiplication and division of complex numbers significantly less labor-intensive. After converting to polar form, these operations translate into a much simpler manipulation of magnitudes and addition or subtraction of angles.

Learning to operate with complex numbers in different forms is a valuable mathematical skill. It allows for flexibility in approach and can often provide computational advantages, as well as enhance understanding of the geometric interpretation of these operations in the complex plane.