Problem 11 - College Algebra

Question

For the following exercises, plot the complex numbers on the complex plane. $$ 1-2 i $$

Step-by-Step Solution

Answer

Plot the point (1, -2) on the complex plane.

Step 1: Understand the Complex Number

A complex number is in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. In this case, the complex number is \(1 - 2i\), so \(a = 1\) and \(b = -2\).

Step 2: Identify the Coordinate on the Complex Plane

On the complex plane, the real part \(a\) is plotted on the horizontal axis, and the imaginary part \(b\) is plotted on the vertical axis. So for \(1 - 2i\), plot the point \((1, -2)\) where 1 is on the horizontal axis and -2 is on the vertical axis.

Step 3: Draw the Plot

Create a complex plane with a horizontal (real axis) and a vertical (imaginary axis). Mark the point \((1, -2)\) on the plane. You should see the point being 1 unit to the right of the origin and 2 units down.

Key Concepts

Understanding the Complex Plane

The complex plane is a visual representation that helps in understanding complex numbers. It comprises two axes, similar to the standard coordinate plane. However, in the complex plane, the horizontal axis represents the real part of a number, while the vertical axis denotes the imaginary part.
This setup allows us to "see" complex numbers as points or vectors. Each complex number corresponds to one unique point on this plane. If you've ever graphed points on a Cartesian coordinate system, you've done something similar. But instead of having just \(x\) and \(y\) axes, we have real and imaginary axes for the complex plane.
Another helpful way to think about the complex plane is to see it like a map. Just as every location on a map can be identified by coordinates, every complex number can be pinpointed on the complex plane. This makes the complex plane crucial in both mathematical understanding and applications.

Real and Imaginary Parts

A complex number is expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.
Much like how we view points in mathematics as \((x, y)\) coordinates, complex numbers have real and imaginary parts which can be represented as \(a\) and \(b\).

• The real part \(a\) tells us how far along the horizontal axis the number is.
• The imaginary part \(b\) shows the distance along the vertical axis.

For example, in the complex number \(1 - 2i\), \(1\) is the real part, meaning it is one unit on the horizontal axis.
Conversely, \(-2\) is the imaginary part, so it's two units downward on the vertical axis. This noticeable difference between the complexities allows for intricate but precise mathematical operations.

Plotting Complex Numbers

Plotting complex numbers on the complex plane is straightforward.
Once you grasp that a complex number \(a + bi\) corresponds to a point \((a, b)\), you can begin plotting.
Here's a simple guide:

• Identify the real part, \(a\), and place it on the horizontal axis.
• Then, locate the imaginary part, \(b\), on the vertical axis.
• Mark the intersection of both these values as a point on the plane.

For the complex number \(1 - 2i\), start by marking \(1\) along the horizontal axis. Next, move downward by \(2\) units on the vertical axis to reflect \(-2\).
Finally, place a dot at this location, making the complex number's position clear.
This simple act of plotting helps visualize complex numbers, ensuring better comprehension and ease in handling more advanced operations.