Problem 36 - Precalculus with Limits

Question

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(\sqrt{8}(\cos\ 225^{\circ} + i\ \sin\ 225^{\circ})\)

Step-by-Step Solution

Answer

The standard form of the complex number \( \sqrt{8}(\cos\ 225^{\circ} + i\ \sin\ 225^{\circ}) \) is approximately \( -2 - 2i \). In the complex plane, this point is represented in the third quadrant, since both the real and imaginary parts of the complex number are negative.

Step 1: Calculate the real part of the complex number

Use the conversion formula \( a = r\cos\ θ \) to calculate the real part, \( a \) of the complex number. Here, \( r = \sqrt{8} \) and \( θ = 225^{\circ} \). So, calculate \( a = \sqrt{8}\cos(225^{\circ}) \) using a calculator.

Step 2: Calculate the imaginary part of the complex number

Use the conversion formula \( b = r\sin\ θ \) to calculate the imaginary part, \( b \), of the complex number. Here, \( r = \sqrt{8} \) and \( θ = 225^{\circ} \). So, calculate \( b = \sqrt{8}\sin(225^{\circ}) \) using a calculator.

Step 3: Write the standard form of the complex number

Now, write the standard form of the complex number as \( a + bi \). Your \( a \) and \( b \) are the real and imaginary parts you calculated in steps 1 and 2.

Step 4: Represent the complex number graphically

Draw a complex plane with a horizontal (real) and vertical (imaginary) axis. Mark the point (\( a, b \)) where \( a \) and \( b \) denote the real and imaginary parts of the complex number respectively. The origin (0,0) of the complex plane corresponds to the complex number 0.

Key Concepts

Standard Form

Complex numbers can be represented in various forms, and the most common one is called the **standard form**. This form is expressed as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The letter \( i \) is used to denote the imaginary unit, with the property that \( i^2 = -1 \).

Would you like to know why the standard form is so practical? Here are some reasons:

• The standard form is very straightforward and easy to use when performing operations like addition and subtraction of complex numbers.
• It simplifies understanding by presenting a direct way to see what part of the number is real and what part is imaginary.
• The visual representation of complex numbers in the standard form on the complex plane is clear and helps in recognizing relationships and operations between them.

In the example from the exercise, after transforming the polar form into standard form, you have \( a \) and \( b \) calculated, allowing you to write the number as \( a + bi \) and clearly see its components.

Polar to Rectangular Conversion

Complex numbers can also be expressed in polar form as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude, and \( \theta \) is the angle in degrees or radians. However, calculations often require converting these into rectangular form, which is the standard form \( a + bi \).

Here's how to easily convert polar coordinates to rectangular coordinates:

• To find the real part \( a \), use the formula \( a = r \cos \theta \). This calculation determines the horizontal component on the complex plane.
• To find the imaginary part \( b \), use the formula \( b = r \sin \theta \). This gives you the vertical component.
• Combine \( a \) and \( b \) to write the complex number in standard form: \( a + bi \).

These steps allow you to take the angle and magnitude provided in the polar form and break them down into the familiar rectangular coordinates, making it easier to work with in further operations like addition or graphing.

Complex Plane

The complex plane is a valuable visual tool for representing complex numbers. This plane is similar to a regular coordinate graph but with a horizontal axis representing the real part and a vertical axis for the imaginary component of complex numbers.

Why use a complex plane?

• It provides a visual interpretation of complex numbers, making it easier to understand how they relate to each other.
• The plane allows the graphical representation of operations with complex numbers, like addition or multiplication.
• Understanding the location and movement of a point in this plane helps in comprehending concepts like modulus and argument of a complex number.

In the exercise, plotting the point given by \( (a, b) \) on the complex plane helps illustrate the number's magnitude and direction. This visual aspect is very helpful for learning, as it connects algebraic expressions with geometric interpretations.