Problem 24 - Contemporary Precalculus

Question

In Exercises \(17-24,\) sketch the graph of the equation in the complex plane (z denotes a complex number of the form a \(+b i\) ). \(\operatorname{Im}(z)=-5 / 2\) [The imaginary part of \(z=a+b i\) is defined to be the number \(b \text { ( not bi) and is denoted } \operatorname{Im}(z) .]\)

Step-by-Step Solution

Answer

Answer: The graph is a horizontal line located at b = -5 / 2 in the complex plane, representing all complex numbers with an imaginary part equal to -5 / 2.

Step 1: Identify the equation of the graph

Given that the imaginary part of \(z = a + bi\) is given to be \(\operatorname{Im}(z) = -\frac{5}{2}\). We'll be sketching a graph of all the complex numbers that have this imaginary part.

Step 2: Setup the graph

We'll sketch the graph in the complex plane, where the horizontal axis represents the real part of the complex number (a) and the vertical axis represents the imaginary part of the complex number (b).

Step 3: Plot the line

Since the imaginary part of z is fixed at \(-\frac{5}{2}\), the graph will be a horizontal line located at \(b=-\frac{5}{2}\). For any real part a, z will have the form \(z = a - \frac{5}{2}i\). Plot this line on the complex plane.

Step 4: Analyze the graph

The horizontal line represents all complex numbers for which the imaginary part is \(-\frac{5}{2}\). The real part of each complex number on the line can take any value, so the line extends infinitely in both directions.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane that provides a visual representation of complex numbers. Unlike the traditional Cartesian plane you might know from algebra, the complex plane is specially designed to accommodate the unique nature of complex numbers, which consist of both a real and an imaginary part. Think of the complex plane as a map where every point is a unique complex number.

In the complex plane, the horizontal axis (often labeled as the 'x-axis') is associated with the real part of a complex number, and the vertical axis (typically referred to as the 'y-axis') is linked with the imaginary part. To plot a complex number such as \( a + bi \), where \( a \) is the real component and \( b \) is the imaginary coefficient, one would move \( a \) units along the horizontal axis and \( b \) units along the vertical axis, marking this spot on the plane. This graphical representation makes manipulating and understanding complex numbers a lot more intuitive.

Imaginary Part of Complex Numbers

In the realm of complex numbers, every number can be expressed in the form \( z = a + bi \) where \( a \) represents the real part, and \( bi \) represents the imaginary part. The term \( 'bi' \) signifies a real number \( b \) multiplied by the imaginary unit \( i \), which is defined as \( \sqrt{-1} \).

It's important to understand that the imaginary part refers only to the coefficient \( b \) before the imaginary unit, not the entire product \( bi \). This distinction is crucial when graphing complex numbers or performing operations with them. For example, if we have \( \operatorname{Im}(z) = -\frac{5}{2} \), the \( -\frac{5}{2} \) is the value of the imaginary part alone. It tells us how far to move up or down from the origin (\( 0, 0 \) point) along the vertical axis of the complex plane to locate all points with this imaginary component.

Complex Number Graph

Plotting a graph of a complex number, or a set of complex numbers, involves understanding both its real and imaginary parts. When you have an equation like \( \operatorname{Im}(z) = -\frac{5}{2} \), it's specifying a set of complex numbers with a constant imaginary part. On the complex plane, these numbers form a horizontal line because regardless of the value of their real part \( a \), the imaginary part remains fixed at \( -\frac{5}{2} \).

To graph this, simply draw a straight horizontal line through the vertical point \( -\frac{5}{2} \) on the y-axis. This line contains all points (complex numbers) \( a - \frac{5}{2}i \) where \( a \) can be any real number. The line is a visual representation of infinitely many complex numbers all sharing the same imaginary part, illustrating a key aspect of the relationship between algebra and geometry when dealing with complex numbers.