Problem 65 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Answer: The polar form of the product of the complex numbers (1+i) and (1+√3i) is (2√2 * e^(i(7π/12))).

Step 1: Convert complex numbers to polar form

To convert a complex number in rectangular form a + bi to polar form, we must find its modulus (r) and its argument (θ), where r is the distance from the origin to the point representing a complex number in the complex plane, and θ is the angle between the positive x-axis and the line connecting the origin and the point representing the complex number. For the complex numbers (1+i) and (1+√3i), first find the modulus (r) for each: r = √(a^2 + b^2) For (1+i): r = √(1^2 + 1^2) = √2 For (1+√3i): r = √(1^2 + (3)^2) = 2 Next, find the argument (θ) for each: θ = arctan(b/a) For (1+i): θ = arctan(1/1) = π/4 For (1+√3i): θ = arctan(√3/1) = π/3 Thus, the polar form of (1+i) is (√2 * e^(iπ/4)) and the polar form of (1+√3i) is (2 * e^(iπ/3)).

Step 2: Multiply the complex numbers in polar form

To multiply two complex numbers in polar form, we need to multiply their moduli and add their arguments: (r1 * e^(iθ1)) * (r2 * e^(iθ2)) = (r1r2) * e^(i(θ1+θ2)) For the given exercise, we have: (√2 * e^(iπ/4)) * (2 * e^(iπ/3)) = (√2 * 2) * e^(i(π/4 + π/3))

Step 3: Simplify the result

Now, simplify the result obtained in the previous step: (2√2) * e^(i(7π/12)) The final answer is (2√2 * e^(i(7π/12))). Here, the polar form of the product of complex numbers (1+i) and (1+√3i) is obtained.

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part with \(i\) being the imaginary unit. The imaginary unit \(i\) has the property that \(i^2 = -1\).

• Real Part: The real part \(a\) is the component of the complex number along the real axis in the complex plane.
• Imaginary Part: The imaginary part \(b\) is the component along the imaginary axis.

Each complex number is essentially a vector in this two-dimensional system, and operations like addition or multiplication can be visualized as transformations in the complex plane. Understanding complex numbers is crucial as they reveal solutions to equations that do not have real number solutions, such as \(x^2 + 1 = 0\).

Modulus and Argument

The modulus and argument of a complex number are used in the conversion from rectangular to polar form.
The modulus of a complex number \(a + bi\) is its distance from the origin in the complex plane, calculated as \( \sqrt{a^2 + b^2} \). It represents the size or magnitude of the vector formed by the complex number.
The argument is the angle that this vector forms with the positive real axis. It is calculated using the arctan function: \( \theta = \arctan\left( \frac{b}{a} \right) \).

• The modulus tells you how far the number is from the origin, providing a measure of its size.
• The argument tells you the direction of the number in the complex plane.

Together, these values allow you to express complex numbers in polar form, which can simplify multiplication and division of complex numbers.

Rectangular to Polar Conversion

Converting complex numbers from rectangular form \(a + bi\) to polar form requires using both the modulus and the argument. In polar form, a complex number is expressed as \( r \cdot e^{i\theta} \), where \(r\) is the modulus and \(\theta\) is the argument.
The process is as follows:

• Calculate the modulus \(r\) using \( \sqrt{a^2 + b^2} \).
• Find the argument \(\theta\) using \( \arctan\left( \frac{b}{a} \right) \).
• Express the complex number in polar form as \( r \cdot e^{i\theta} \).

This method is particularly useful for operations like multiplication and division, as it converts these operations into simpler arithmetic involving the moduli and arguments:

• Multiplication: Multiply the moduli and add the arguments.
• Division: Divide the moduli and subtract the arguments.

Converting to polar form simplifies complex arithmetic by transforming these numbers into a form that highlights their magnitude and direction.