Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, plot the complex numbers on the complex plane. \(-3-4 i\)

Step-by-Step Solution

Answer

Plot the point at (-3, -4) on the complex plane.

Step 1: Define the Complex Number

Recognize that the complex number given is \(-3-4i\). In this complex number, the real part is \(-3\) and the imaginary part is \(-4\).

Step 2: Identify the Axes on the Complex Plane

The complex plane is a two-dimensional plane where the horizontal axis (x-axis) represents the real part of the number and the vertical axis (y-axis) represents the imaginary part of the number.

Step 3: Locate the Real Part

Plot a point at \(-3\) on the x-axis (real axis). This is moving 3 units to the left from the origin \((0,0)\) on the complex plane.

Step 4: Locate the Imaginary Part

From \(-3\) on the x-axis, move vertically to \(-4\) on the y-axis (imaginary axis). This involves moving 4 units down from the x-axis.

Step 5: Plot the Complex Number

The intersection point of \(-3\) on the horizontal axis and \(-4\) on the vertical axis is the location of the complex number \(-3-4i\) on the plot of the complex plane.

Key Concepts

Plotting Complex Numbers

Plotting complex numbers on the complex plane is similar to locating points on a traditional coordinate plane. However, in the complex plane, each complex number is represented as a point with a real part and an imaginary part.

• The horizontal axis of the complex plane is called the real axis. It corresponds to the real part of the complex number.
• The vertical axis is referred to as the imaginary axis and it corresponds to the imaginary part of the complex number.

To plot a complex number, follow these steps: 1. **Identify the real and imaginary parts** of the complex number. For example, in \(-3-4i\), the real part is \(-3\) and the imaginary part is \(-4\).
2. **Locate the point** corresponding to the real part on the horizontal axis. For \(-3\), move 3 units to the left from the origin (0,0).
3. **From this point**, move vertically to locate the imaginary part on the vertical axis. For \(-4\), move 4 units downward.
4. **Plot the intersection point**. This is where the complex number is located in the complex plane.
By following these steps, you can easily visualize complex numbers and their positions in the complex plane.

Real and Imaginary Parts

In mathematics, every complex number has two fundamental components: its real part and imaginary part. Understanding these components is critical when working with complex numbers.

• The real part is a real number found along the x-axis of the complex plane.
• The imaginary part is a real number multiplied by the imaginary unit \(i\), positioned along the y-axis.

For instance, in the complex number \(-3-4i\):
- The real part is \(-3\), indicating that it lies on the left side of the real axis.
- The imaginary part is \(-4\), represented on the negative side of the imaginary axis.

The imaginary unit \(i\) is essential here. It is defined as \(i = \sqrt{-1}\), making any number multiplied by \(i\) imaginary. The combination of both parts makes complex numbers unique by extending their representation into a two-dimensional plane.

Coordinate System in Mathematics

A coordinate system is a system used to uniquely determine the position of a point or other geometric element on a manifold such as the Euclidean space. Complex numbers use a specific type of coordinate system known as the complex plane.
The complex plane takes two number components (real and imaginary) and treats them as coordinates:

• The horizontal coordinate corresponds to the real part of a complex number.
• The vertical coordinate corresponds to the imaginary part.

This makes the complex plane an essential tool in both geometry and algebra because it facilitates the visualization and calculation of operations with complex numbers.
To envision this, consider a standard Cartesian coordinate system where each point (x, y) represents a position on the plane. Similarly, each complex number specified by its real and imaginary components maps to a unique point in the complex plane. For example, the complex number \(-3-4i\) is plotted as a coordinate point \((-3, -4)\).
By consistently applying this coordinate system, mathematicians and students alike can handle complex numbers with greater ease and precision.