Problem 22 - Algebra and Trigonometry

Question

In Exercises \(11-26,\) plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$ 3 \sqrt{2}-3 i \sqrt{2} $$

Step-by-Step Solution

Answer

The Polar form of the complex number \(3 \sqrt{2}-3 i \sqrt{2}\) is \(6 \left(\cos(-π/4) + i \sin(-π/4)\right)\).

Step 1: Identify the Real and Imaginary Parts

From the complex number, \(3 \sqrt{2}-3 i \sqrt{2}\), one can deduce that the real part, \(a\), is \(3 \sqrt{2}\), and the imaginary part, \(b\), is \(-3 \sqrt{2}\).

Step 2: Calculate the modulus \(r\)

Having identified \(a\) and \(b\), the modulus, \(r\), can be found using the formula \(r=\sqrt{a^{2}+b^{2}}\). Substituting \(a=3 \sqrt{2}\) and \(b=-3 \sqrt{2}\) into this formula gives \(r=3\sqrt{2}\sqrt{2}=6\).

Step 3: Calculate the argument \(θ\)

Next, the argument, \(θ\), is determined using the formula \(θ = \arctan(\frac{b}{a})\). Since \(b = -3 \sqrt{2}\) and \(a=3 \sqrt{2}\), the calculation is \(θ=\arctan(\frac{-3\sqrt{2}}{3\sqrt{2}})=\arctan(-1)=-\frac{π}{4}\).

Step 4: Write in Polar Form

The complex number in polar form is given by \(r(\cos θ + i \sin θ)\). Substituting the obtained values of \(r\) and \(θ\), the polar form is \(6(\cos(-π/4) + i \sin(-π/4))\).

Step 5: Plot the Complex Number

Plot the complex number \(3 \sqrt{2}-3 i \sqrt{2}\) on the complex plane. The real and imaginary parts correspond to the x and y coordinates, respectively. This plotting helps to visualize the complex number geometrically.

Key Concepts

Polar Form

The polar form of a complex number provides a unique way to express complex numbers. Unlike the Cartesian form, which utilizes real and imaginary parts (like in the format around the x and y axes), the polar form focuses on the modulus and argument.
The polar form expresses a complex number as \( r(\cos \theta + i \sin \theta) \) or more compactly, \( re^{i\theta} \). Here, \( r \) is the modulus, and \( \theta \) is the argument of the complex number. This approach highlights the magnitude and direction of the number on the complex plane:

• It allows for straightforward multiplication and division operations.
• It highlights rotational and scaling aspects of complex numbers.

The transformation to polar form is crucial for understanding the deeper connections between algebraic and geometric interpretations of complex numbers.

Modulus of Complex Number

The modulus of a complex number is a measure of its size or magnitude. For a complex number given in the format \( a + bi \), the modulus is calculated using the formula: \[ r = \sqrt{a^2 + b^2} \]This is equivalent to the length of the vector from the origin to the point \( (a, b) \) in the complex plane. Some important aspects of the modulus include:

• It is always a non-negative real number.
• It represents the distance of the complex number from the origin.

For example, for the complex number \( 3\sqrt{2} - 3i\sqrt{2} \), the modulus is calculated as \( r = 6 \). The modulus serves as a key component in converting to polar form, demonstrating how far the number is from the origin on the complex plane.

Argument of Complex Number

The argument of a complex number, often denoted as \( \theta \), describes the angle formed with the positive x-axis when the complex number is represented on the complex plane. The argument provides insight into the direction in which the complex number lies from the origin.
To find the argument, we use:

• \( \theta = \arctan\left(\frac{b}{a}\right) \)
• The value of \( \theta \) considers the quadrant in which the complex number resides.

For the complex number \( 3\sqrt{2} - 3i\sqrt{2} \), the argument \( \theta \) is calculated as \( -\frac{\pi}{4} \). The negative value indicates a direction below the x-axis. Understanding the argument is critical for applications involving rotations and transformations in the polar coordinate system.

Complex Plane

The complex plane is a two-dimensional space helpful for visualizing complex numbers. It resembles the Cartesian coordinate plane, with the horizontal axis representing real parts and the vertical axis representing imaginary parts of complex numbers.
Important characteristics of the complex plane include:

• The x-axis (real axis) consists of all real numbers.
• The y-axis (imaginary axis) consists of all imaginary numbers.
• Any point on this plane represents a complex number \( a + bi \).

Plotting a complex number helps to visualize operations, such as addition, subtraction, as well as the transformation needed to convert to polar form. For instance, the point corresponding to \( 3\sqrt{2} - 3i\sqrt{2} \) on the complex plane provides a clear geometric representation, showing its relation to the origin and how it translates to polar coordinates.