Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Graph each complex number as a vector in the complex plane. Do not use a calculator. $$-4 i$$

Step-by-Step Solution

Answer

The complex number \(-4i\) is a vector on the y-axis with coordinates (0, -4).

Step 1: Understand the Complex Number

The given complex number is \(-4i\).This number is purely imaginary and can be written as \(0 - 4i\). It has no real part.

Step 2: Identify Coordinates

In the complex plane, the real part corresponds to the x-axis (horizontal), and the imaginary part corresponds to the y-axis (vertical).For \(-4i\), the real part is 0, and the imaginary part is -4.Thus, its coordinates are \((0, -4)\).

Step 3: Plot the Vector in the Complex Plane

On the complex plane, locate the point (0, -4). This point lies directly on the y-axis at -4, indicating a downward position.Draw a directed line from the origin (0,0) to the point (0,-4). This line represents the vector of the complex number \(-4i\).

Step 4: Label the Vector

Label the vector with the complex number \(-4i\) to clearly identify which complex number this vector represents in the complex plane.

Key Concepts

Imaginary Numbers

Imaginary numbers are a fascinating concept in mathematics. Unlike real numbers, which you can find on the number line, imaginary numbers exist beyond it. An imaginary number is created when you take the square root of a negative number. The most basic imaginary unit is represented by the letter \(i\). This unit is defined as \(i = \sqrt{-1}\).

When you multiply \(i\) by itself, you get a negative real number:

• \(i^2 = -1\)

Beyond \(i\), any multiple of it is also considered an imaginary number, such as \(2i\), \(-3i\), or \(-4i\). In essence, these numbers extend the real number system into the imaginary realm. When an imaginary number is combined with a real number, it forms a complex number. Imaginary numbers are crucial for understanding more profound areas like electronic engineering and signal processing.

Complex Plane

The complex plane gives us a visual way to understand complex numbers, including imaginary numbers like \(-4i\). This plane uses two axes to represent complex numbers:

• The horizontal axis is called the real axis.
• The vertical axis is known as the imaginary axis.

Each point on this plane corresponds to a complex number where the real part is plotted on the x-axis and the imaginary part on the y-axis. In our example, with o real part and an imaginary part of \(-4i\), we locate the point (0, -4) on the imaginary axis. The complex plane not only simplifies the visualization of complex numbers but also opens up the potential for performing operations like addition or multiplication. Understanding the complex plane is crucial for fields like quantum physics, control theory, and complex dynamics.

Vectors

Vectors might initially seem like they belong only in the realm of physics, but they play an important role with complex numbers too. In the context of the complex plane, a complex number can be represented as a vector. This vector starts from the origin (0,0) and extends to the point described by the complex number’s coordinates.
In our example, the complex number \(-4i\) translates to a vector pointing directly downward from the origin to the point (0, -4).

• The real component (0) shows no movement horizontally.
• The imaginary component (-4) indicates movement 4 units downward.

The length of this vector can also be calculated using the formula: \(|a + bi| = \sqrt{a^2 + b^2}\). With \(a = 0\) and \(b = -4\), the vector's length evaluates to 4. Vectors make complex numbers tangible and applicable in diverse areas, such as in computer graphics, navigation, and robotics.