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Chapter 8: Problem 616

For the following exercises, plot the complex number in the complex plane. $$-1+3 i$$

Short Answer

Expert verified
Plot the point at (-1, 3) on the complex plane.

Step by step solution

01

Understand the Complex Number

The complex number given is \(-1 + 3i\).We can divide it into real and imaginary parts: the real part is \(-1\), and the imaginary part is \(3\).
02

Identify Coordinates

On the complex plane, the x-axis represents the real part, and the y-axis represents the imaginary part. So, the coordinates for \(-1 + 3i\) are \((-1, 3)\).
03

Plot the Point on the Complex Plane

To plot this number, move \(-1\) unit along the x-axis (to the left because it's negative) and move \(3\) units up along the y-axis. Mark this point on the complex plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Plane
The complex plane is a way of visualizing complex numbers like \(-1 + 3i\). Similar to the Cartesian coordinate system, it uses two perpendicular axes.
The horizontal axis, called the real axis, represents the real component of the complex number. Meanwhile, the vertical axis, known as the imaginary axis, represents the imaginary component. By using this plane, we can easily plot complex numbers and perform operations like addition or subtraction visually. This two-dimensional space allows us to treat complex numbers as points or vectors, offering a geometrical interpretation to complex analyses.
Real and Imaginary Parts
To work with complex numbers effectively, it's crucial to understand their structure. A complex number is typically written as \(a + bi\), where
  • \(a\) is the real part.
  • \(b\) is the imaginary part (multiplied by the imaginary unit \(i\)).
In our example of \(-1 + 3i\),
  • \(-1\) is the real part.
  • \(3\) is the imaginary part.
Each component uniquely defines where the complex number will fall on the complex plane. Recognizing these parts is the first step in plotting any complex number. This decomposition allows us to analyze and operate on complex numbers more comprehensively.
Coordinate System
The coordinate system for complex numbers functions similarly to the traditional one used for plotting points on a graph. The real part of a complex number corresponds to the x-coordinate, while the imaginary part corresponds to the y-coordinate.
This transformation allows us to describe our complex number \(-1 + 3i\) positioned at the coordinate \((-1, 3)\).
  • The x-coordinate, \(-1\), shows the position along the horizontal real axis.
  • The y-coordinate, \(3\), shows the position along the vertical imaginary axis.
Understanding this coordinate system helps us interpret complex numbers as geometric entities, facilitating operations like multiplication and addition through visual tools.
Plotting Points
Plotting points for complex numbers involves translating the real and imaginary parts into a visual representation on the complex plane. To plot \(-1 + 3i\), we follow these steps:
  • Start at the origin \((0, 0)\) of the complex plane.
  • Move \(-1\) unit left since the real part is negative.
  • Then move \(3\) units up to account for the positive imaginary part.
  • Mark this location, which corresponds to the point \((-1, 3)\).
Plotting is more than just a mechanical step; it provides a visual confirmation of where the complex number resides and how it relates to other numbers on the plane. This graphical approach is invaluable in analyzing and understanding complex number operations.

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