Problem 40 - Precalculus with Limits

Question

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(8 \left(\cos\ \dfrac{\pi}{2} + i\ \sin\ \dfrac{\pi}{2} \right)\)

Step-by-Step Solution

Answer

The standard form of the complex number is \( 8i \) and graphically, it's represented by the point \( (0, 8) \) in the complex plane.

Step 1: Convert Polar Form into Standard Form

To convert polar form into standard form, we express it in terms of \( r \left(\cos\ \theta + i\ \sin\ \theta \right) \) where \( r \) is the magnitude and \( \theta \) is the argument. Here \( r = 8 \) and \( \theta = \pi/2 \). Thus, we have \( 8 \left(\cos\ \pi/2 + i\ \sin\ \pi/2 \right) = 8 \cdot (0 + i \cdot 1) = 8i \)

Step 2: Graphical Representation of the Complex Number

The complex number \( 8i \) corresponds to the point \( (0, 8) \) in the complex plane. It lies 8 units above the origin. Simply draw a point in an x-y coordinate system at the point (0,8) to represent this complex number graphically.

Key Concepts

Polar Form

In mathematics, a complex number can be expressed in polar form, which offers a different perspective from its traditional representation. Polar form is given by the expression \( r(\cos \theta + i\sin \theta) \), where \( r \) is the magnitude (or modulus) of the complex number, and \( \theta \) is the argument (or angle).

• The magnitude \( r \) represents the distance of the complex number from the origin in the complex plane.
• The argument \( \theta \) indicates the angle formed with the positive x-axis.

Using polar form is particularly helpful in simplifying the multiplication and division of complex numbers as well as raising them to powers. By converting from polar form, calculations become more manageable as operations on the magnitude and angle are more intuitive.
For the exercise provided, the complex number is given in polar form as \( 8 \left( \cos \dfrac{\pi}{2} + i\sin \dfrac{\pi}{2} \right) \). Here, the magnitude \( r = 8 \) and the angle \( \theta = \dfrac{\pi}{2} \). This setup positions the complex number as being purely imaginary when converted into its standard form.

Standard Form

Complex numbers in their standard form are represented as \( a + bi \), where \( a \) and \( b \) are real numbers. Here,

• \( a \) is the real part of the complex number.
• \( b \) is the coefficient of the imaginary part with \( i \) denoting the imaginary unit.

The conversion from polar form to standard form is a straightforward process where calculations involve rewriting trigonometric expressions. For the given exercise, we convert \( 8 \left( \cos \dfrac{\pi}{2} + i\sin \dfrac{\pi}{2} \right) \) into standard form by evaluating \( \cos \dfrac{\pi}{2} \) and \( \sin \dfrac{\pi}{2} \). Knowing that \( \cos \dfrac{\pi}{2} = 0 \) and \( \sin \dfrac{\pi}{2} = 1 \), we find the standard form: \( 8i \).
This result shows that the complex number is entirely on the imaginary axis, with no real component, only moving along the y-axis.

Graphical Representation

To visualize complex numbers, we use the complex plane, similar to the Cartesian coordinate system. This plane has:

• The horizontal axis labeled as the real axis.
• The vertical axis labeled as the imaginary axis.

For the complex number \( 8i \), its graph rests at the point \((0, 8)\) on the complex plane. This means it lies directly above the origin, purely along the imaginary axis. Graphically representing complex numbers helps in understanding their position and relationships to other numbers more clearly.
In practice, you simply plot the point with the real component along the x-axis and the imaginary component along the y-axis. In this case, draw a point 8 units upward from the origin along the imaginary axis. This form of representation facilitates understanding transformations like rotations and dilations that complex numbers undergo.

Complex Plane

The complex plane is a two-dimensional plane used to visually depict complex numbers. It's structured with:

• The real part of the complex number plotted along the horizontal axis.
• The imaginary part plotted along the vertical axis.

The plane allows us to represent each complex number as a unique point. The way complex numbers map onto this plane is crucial for understanding complex operations visually.
In our exercise, the complex number \( 8i \) is located at \((0, 8)\). This positioning helps us not only graphically represent the number but also gain insights into its properties. By plotting complex numbers like vectors, we can easily perform operations such as addition or find distances using geometric interpretations. The complex plane thus serves as a powerful tool in both elementary and advanced studies of complex numbers.