Problem 13 - A Graphical Approach to Precalculus with Limits

Question

Graph each complex number as a vector in the complex plane. Do not use a calculator. $$-8$$

Step-by-Step Solution

Answer

The complex number \(-8\) is represented by a vector from the origin to \((-8, 0)\) on the real axis.

Step 1: Identify the Real and Imaginary Parts

A complex number can be expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The given complex number is \(-8\), which can be rewritten as \(-8 + 0i\). Here, \(a = -8\) and \(b = 0\).

Step 2: Determine the Position in the Complex Plane

In the complex plane, the horizontal axis (real axis) represents the real part \(a\), and the vertical axis (imaginary axis) represents the imaginary part \(b\). For the complex number \(-8 + 0i\), it has a real part of \(-8\) and an imaginary part of \(0\).

Step 3: Plot the Complex Number

Locate the point corresponding to \(-8\) on the real axis, since the imaginary component is zero. Therefore, the point \((-8, 0)\) should be plotted on the real axis as a vector originating from the origin \((0, 0)\) and ending at \((-8, 0)\).

Step 4: Draw the Vector

Draw a line, or vector, that starts from the origin at \((0, 0)\) and extends horizontally to the left, reaching the point \((-8, 0)\). This vector represents the complex number \(-8\) in the complex plane.

Key Concepts

Complex Plane

The complex plane is a way to visually represent complex numbers, much like how you might plot points on a graph. It uses a two-dimensional plane where the horizontal axis represents the real part of a complex number and the vertical axis represents the imaginary part.
Each complex number can thus be thought of as a point or vector in this plane. For instance, with the complex number \(-8\), it is located entirely on the real axis, specifically at the point \((-8, 0)\). This means it doesn't move up or down since its imaginary part is zero. As such, understanding the complex plane requires familiarity with the Cartesian coordinate system, since this is essentially a complex plane with one dimension added for imaginary numbers.
The key takeaway here is the orderly representation of complex numbers, facilitating their interpretation and manipulation in mathematical operations.

Real and Imaginary Parts

Every complex number comprises two parts: the real part and the imaginary part. These parts can often be denoted as \(a + bi\), where \(a\) is the real component and \(b\) the imaginary one, coefficient to the imaginary unit \(i\).
For easier visualization:

• The real part is graphed on the horizontal axis.
• The imaginary part is graphed on the vertical axis.

Take the number \(-8\). It is all real, as it can be reformulated as \(-8 + 0i\).
Hence, its plotting occurs purely on the horizontal axis, emphasizing its lack of an imaginary portion. Grasping how to separate and identify these components is vital, as it sets the foundation for further exploration into complex number manipulations and applications.

Graphing Vectors

Graphing vectors in the complex plane involves drawing a directed line from the origin \((0, 0)\) to the point denoting the complex number, using its real and imaginary parts.
When dealing with complex numbers such as \(-8\), the process simplifies because the entire number lies on the real axis.

• To graph \(-8\), you begin at the origin and draw a horizontal line to the left, reaching \((-8, 0)\).
• The direction and length of this line embody both the magnitude and direction of the number within the plane.

This visualization becomes indispensable when handling more intricate complex numbers and operations, such as addition and multiplication. It transforms abstract mathematical concepts into tangible visual representations, simplifying comprehension and solution of problems involving complex numbers.