Problem 7 - Precalculus

Question

Graph each complex number in the complex plane. $$-3 i$$

Step-by-Step Solution

Answer

Plot \((0, -3)\) on the complex plane.

Step 1: Understand the Complex Number Format

A complex number is written as \( a + bi \) where \( a \) is the real part and \( b \) is the imaginary part. In the complex plane, the horizontal axis (x-axis) represents the real part and the vertical axis (y-axis) represents the imaginary part.

Step 2: Identify the Real and Imaginary Parts

For the complex number \( -3i \), identify \( a = 0 \) (since there is no real part) and \( b = -3 \). This means you have \( 0 + (-3)i \).

Step 3: Plot the Point on the Complex Plane

In the complex plane, locate the point at \( x = 0 \) (real part) and \( y = -3 \) (imaginary part). This corresponds to the point \((0, -3)\), which you will plot on the plane.

Key Concepts

Understanding the Complex Plane

The complex plane is a two-dimensional plane used to visualize complex numbers. Imagine it as a graph where each point represents a complex number, much like how points on a Cartesian plane represent ordered pairs. Here's how it works:

• The horizontal axis, or x-axis, represents the real part of the complex number.
• The vertical axis, or y-axis, stands for the imaginary part of the complex number.

This representation allows us to plot any complex number on the plane simply by identifying its real and imaginary components. Each point on this plane corresponds to a unique complex number, making it a highly effective way to visualize operations involving complex numbers.

Imaginary and Real Parts of a Complex Number

A complex number is expressed in the form \(a + bi\), where:

• \(a\) is the real part of the complex number.
• \(bi\) is the imaginary part. Here, \(b\) is a real number, and \(i\) is the imaginary unit, defined as \(i^2 = -1\).

The real part \(a\) and the imaginary part \(b\) can be any real number, which allows complex numbers to extend the number system beyond real numbers. By decomposing complex numbers into these parts, we gain a better understanding of their structure and properties. For example, in the complex number \(-3i\), the real part is \(0\) and the imaginary part is \(-3\), essentially positioning the number entirely on the imaginary axis.

Plotting Complex Numbers

Plotting complex numbers involves locating them on the complex plane based on their real and imaginary parts. Let's walk through the process using the example of the complex number \(-3i\):

• Identify the real part \(a\), which is \(0\) here.
• Identify the imaginary part \(b\), which is \(-3\) in this case.
• To plot \(-3i\), place a point at the coordinates \((0, -3)\) on the complex plane. This means you move zero units along the horizontal axis and three units downward along the vertical axis to reach the point.

By using the complex plane, you easily plot and analyze complex numbers graphically, which enhances understanding of their magnitude and direction. The visualization of complex numbers as points or vectors provides an intuitive grasp of their behavior in mathematical operations.