Problem 12 - College Algebra with Corequisite Support

Question

For the following exercises, plot the complex numbers on the complex plane. \(-2+3 i\)

Step-by-Step Solution

Answer

Plot the point (-2, 3) on the complex plane.

Step 1: Identify the Components of the Complex Number

The complex number is given as \(-2 + 3i\). This number can be broken down into its real and imaginary components. The real part is \(-2\), and the imaginary part is \(3\).

Step 2: Set Up the Complex Plane

Before plotting, understand that the complex plane is similar to a coordinate system. The horizontal axis represents the real part of the complex number, while the vertical axis represents the imaginary part.

Step 3: Plot the Real Part

Start at the origin (0,0) on the complex plane. Since the real part is \(-2\), move 2 units to the left along the horizontal axis.

Step 4: Plot the Imaginary Part

From the position \(-2\) on the horizontal axis, move 3 units up because the imaginary part \(3i\) is positive. Now you should be at the point \((-2, 3)\).

Step 5: Mark the Point on the Complex Plane

The point \((-2, 3)\) represents the complex number \(-2 + 3i\). Mark this point on the complex plane to complete the plot.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used to graphically represent complex numbers. It is a fundamental tool in mathematics that helps visually interpret the relationship between real and imaginary numbers. In this plane:

• The horizontal axis is called the real axis and is used to represent the real component of complex numbers.
• The vertical axis is known as the imaginary axis and is used to represent the imaginary component.

The complex plane resembles a coordinate system, similar to the Cartesian plane, where each point represents a unique complex number. When plotting a complex number such as \(-2 + 3i\), you treat it as an ordered pair: the real part \(-2\) and the imaginary part \(3\). You simply plot this as \((-2, 3)\) on the complex plane by moving 2 units to the left (real) and 3 units up (imaginary) from the origin.This graphical method makes it easier to identify how complex numbers interact with one another, as well as facilitating operations like addition, subtraction, and finding magnitudes. It effectively allows complex numbers to be treated with similar methods as vectors.

Imaginary Numbers

Imaginary numbers expand the concept of numbers beyond the real number line. They are numbers that give a negative result when squared, a behavior unlike any real number. The most basic imaginary unit is denoted as \(i\), where \(i^2 = -1\).When you multiply \(i\) with any real number, you get an imaginary number. For example, in the complex number \(-2 + 3i\), \(3i\) is the imaginary component, \(3\) being a scalar multiple of \(i\). Imaginary numbers, while initially theoretical, find significant applications in various fields such as engineering, control systems, and physics due to their properties and the solutions they provide to equations that lack real solutions.By understanding imaginary numbers, you can leverage their properties when mapping values onto the complex plane or performing operations involving complex numbers. They are integral to fully grasping the complex number system and engaging with advanced mathematical concepts.

Real and Imaginary Components

Understanding the real and imaginary components of a complex number is crucial to using and interpreting complex numbers effectively. Originally presented as \(a + bi\), a complex number consists of:

• The real component \(a\), which lies on the horizontal axis of the complex plane.
• The imaginary component \(b\), paired with the imaginary unit \(i\), and plotted along the vertical axis.

In the example, \(-2 + 3i\), \(-2\) is the real component and \(3i\) is the imaginary component. To plot this on the complex plane, the real component \(-2\) indicates a leftward direction on the horizontal axis, while the imaginary component \(3\) indicates an upward direction on the imaginary axis.Recognizing these components simplifies using complex numbers in computational tasks. It empowers you to visualize complex operations and interpret the geometric implications of complex numbers. Utilizing real and imaginary components allows performing algebraic and geometric transformations which are key to solving complex equations and understanding their solutions.